Data.Rivers.Ecology

This module is an attempt to construct many and varied examples of Rivers and Streams. At the moment, the concept of what Rivers are (or are not) is not entirely clear in the mind of any particular person on Earth, myself foremost amoung the befuddled.

Primarily because I lay claim to inventing the idea, this is a worrisome situation. Nevertheless, whatever these things are, regular old Data.Stream streams (and kin) are subsets (or sub-classes, or.. sub-something) of these things, and hence they must qualify to be mapped in this ecosystem (by definition).

As of now, these example originate primarily from three excellent papers on Streams and their properties. More precisey, they originate from my haphazard and occasionally mindless transcription of what I saw from these documents. Therefore, the authors of the following papers deserve much of the credit, but bear none of the responsibility of the contents of this module.

CONTRACTIVE

• Graham Hutton and Mauro Jaskelioff, "Representing contractive functions on streams",
• Submitted to the Journal of Functional Programming (October 2011).
• Link: http://www.cs.nott.ac.uk/~gmh/bib.html#contractive
• PDF: http://www.cs.nott.ac.uk/~gmh/contractive.pdf

PEARL-UFP

• Ralph Hinze, "Functional pearl: streams and unique fixed points".
• Proceedings of the 13th ACM SIGPLAN international conference on Functional Programming (ICFP '08) (22-24 September 2008). pp. 189-200. (c) ACM
• Link: http://www.cs.ox.ac.uk/ralf.hinze/publications/index.html#B9
• PDF: http://www.cs.ox.ac.uk/ralf.hinze/publications/ICFP08.pdf

PROVING-UFP

• Ralf Hinze and Daniel W. H. James, "Proving The Unique Fixed-Point Principle Correct"
• Proceeding of the 16th ACM SIGPLAN international conference on Functional programming (ICFP '11) (September 2011). pp. 359-371. (c) ACM
• Link: http://www.cs.ox.ac.uk/people/daniel.james/unique.html
• PDF: http://www.cs.ox.ac.uk/people/daniel.james/unique/unique-conf.pdf

This module should clearly document the behavior of all functions. In fact, the code in this module is generally not intended to be imported and used directly. Instead the purpose of this module is to:

1. show *many* examples of Streams, especially non-trivial ones (ie, more complicated than fibionacci)
2. provide visual (and eventually, pictoral) *proof* of equality (insofar that this is possible)
3. ... more things
4. ... should go here
5. ... becuase there's a point to all of this, right?

Note: To accomidate the documentation, the text-width of this document is 129 characters.

As a witness to the correctness of the examples, I include the result of running doctest:

 $ doctest Data/Rivers/Ecology.hs
 Cases: 74  Tried: 74  Errors: 0  Failures: 0

Silly Streams

Constant Zeroes

sZero :: S Integer

>>> stake 30 $ sZero
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]

gZero :: G Integer Integer

>>> stake 30 $ fwdFix gZero
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]

rZero :: G Integer Integer

>>> stake 30 $ revFix rZero
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]

cZero :: S Integer

>>> stake 30 $ cZero
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]

cZeroH :: Num a => ([a], Int) -> a

cZeroT :: Num t => ([a], t) -> a -> ([a], t)

Constant Ones

sOne :: S Integer

Believe it or not, this is in OEIS:

>>> take 30 $ fromOEIS "A000012"
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]

>>> stake 30 $ sOne
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]

sOnes :: S Integer

>>> stake 30 $ sOnes
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]

gOne :: G Integer Integer

gOne' :: G Integer Integer

>>> stake 30 $ fwdFix gOne
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]

rOne :: G Integer Integer

>>> stake 30 $ revFix rOne
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]

cOne :: S Integer

>>> stake 30 $ cOne
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]

cOneH :: t -> t

cOneT :: t -> t -> t

The Natural Numbers

sNat :: S Integer

>>> take 30 $ fromOEIS "A000027"
[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]

>>> stake 30 $ sNat
[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29]

sNatIt :: S Integer

>>> stake 30 $ siterate (+1) 0
[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29]

natnat :: S Integer

>>> stake 30 $ natnat
[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29]

gNat :: G Integer Integer

gNat' :: [a] -> Int

>>> stake 30 $ fwdFix gNat
[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29]

rNat :: G Integer Integer

>>> stake 30 $ revFix rNat
[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29]

cNat :: S Integer

>>> stake 30 $ cNat
[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29]

cNatH :: t -> t

cNatT :: Num a => a -> t -> a

Natural Numbers from Binary

bin :: S Integer

>>> stake 30 $ bin
[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29]

Simple Streams

Power Streams

Powers of (-2)

revPowersOfN2 :: S Integer

>>> take 15 $ fromOEIS "A122803"
[1,2,4,8,16,32,64,128,256,512,1024,2048,4096,8192,16384]

>>> stake 15 $ revPowersOfN2
[1,-2,4,-8,16,-32,64,-128,256,-512,1024,-2048,4096,-8192,16384]

a122803 :: G Integer Integer

Powers of (2) - Bool

pot :: S Bool

>>> stake 15 $ pot
[True,True,False,True,False,False,False,True,False,False,False,False,False,False,False]

Powers of (3)

revPowersOf3 :: S Integer

>>> take 15 $ fromOEIS "A000244"
[1,3,9,27,81,243,729,2187,6561,19683,59049,177147,531441,1594323,4782969]

>>> stake 15 $ revPowersOf3
[1,3,9,27,81,243,729,2187,6561,19683,59049,177147,531441,1594323,4782969]

pothree :: G Integer Integer

Squares

a000290 :: S Integer

>>> take 30 $ fromOEIS "A000290"
[0,1,4,9,16,25,36,49,64,81,100,121,144,169,196,225,256,289,324,361,400,441,484,529,576,625,676,729,784,841]

>>> stake 30 $ bsum $ 2 * sNat + 1
[0,1,4,9,16,25,36,49,64,81,100,121,144,169,196,225,256,289,324,361,400,441,484,529,576,625,676,729,784,841]

Factorial Numbers

sFac :: S Integer

>>> take 10 $ fromOEIS "A000142"
[1,1,2,6,24,120,720,5040,40320,362880]

>>> stake 10 $ sFac
[1,1,2,6,24,120,720,5040,40320,362880]

The Family of Fibionacci (and Lucas!)

Forever Fibionacci

sFib :: S Integer

>>> take 29 $ fromOEIS "A000045"
[0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,10946,17711,28657,46368,75025,121393,196418,317811]

>>> stake 29 $ sFib
[0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,10946,17711,28657,46368,75025,121393,196418,317811]

sFib2 :: S Integer

>>> stake 29 $ sFib2
[0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,10946,17711,28657,46368,75025,121393,196418,317811]

revGfibs :: G Integer Integer

>>> stake 29 $ revFix revGfibs
[0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,10946,17711,28657,46368,75025,121393,196418,317811]

cFib :: S Integer

cFibH :: (a, b) -> a

cFibT :: Num a => (a, a) -> a -> (a, a)

>>> stake 29 $ cFib
[0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,10946,17711,28657,46368,75025,121393,196418,317811]

gFibs :: G Integer Integer

>>> stake 29 $ fwdFix gFibs
[0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,10946,17711,28657,46368,75025,121393,196418,317811]

Lonely Lucas

sLucas :: S Integer

>>> take 28 $ fromOEIS "A000032"
[2,1,3,4,7,11,18,29,47,76,123,199,322,521,843,1364,2207,3571,5778,9349,15127,24476,39603,64079,103682,167761,271443,439204]

>>> stake 28 $ sLucas
[2,1,3,4,7,11,18,29,47,76,123,199,322,521,843,1364,2207,3571,5778,9349,15127,24476,39603,64079,103682,167761,271443,439204]

rLucas :: G Integer Integer

>>> stake 28 $ revFix rLucas
[2,1,3,4,7,11,18,29,47,76,123,199,322,521,843,1364,2207,3571,5778,9349,15127,24476,39603,64079,103682,167761,271443,439204]

under (4*n + 1)

fib4np1 :: S Integer

The Fibionacci (4n + 1) and Lucas (4n + 1) numbers

drop0L is evidently a bisect twice function

>>> stake 10 $ drop0L sFib
[1,5,34,233,1597,10946,75025,514229,3524578,24157817]

>>> take 10 $ fromOEIS "A033889"
[1,5,34,233,1597,10946,75025,514229,3524578,24157817]

luc4np1 :: S Integer

>>> stake 10 $ drop0L sLucas
[1,11,76,521,3571,24476,167761,1149851,7881196,54018521]

>>> take 10 $ fromOEIS "A056914"
[1,11,76,521,3571,24476,167761,1149851,7881196,54018521]

under (2*n)

fib2n :: S Integer

The Fib (2*n) and Lucas (2*n) numbers

dromIp1L is evidently a bisect function

>>> stake 20 $ dropIp1L sFib
[0,1,3,8,21,55,144,377,987,2584,6765,17711,46368,121393,317811,832040,2178309,5702887,14930352,39088169]

>>> take 20 $ fromOEIS "A001906"
[0,1,3,8,21,55,144,377,987,2584,6765,17711,46368,121393,317811,832040,2178309,5702887,14930352,39088169]

luc2n :: S Integer

>>> stake 20 $ dropIp1L sLucas
[2,3,7,18,47,123,322,843,2207,5778,15127,39603,103682,271443,710647,1860498,4870847,12752043,33385282,87403803]

>>> take 20 $ fromOEIS "A005248"
[2,3,7,18,47,123,322,843,2207,5778,15127,39603,103682,271443,710647,1860498,4870847,12752043,33385282,87403803]

under (sum)

fibpfib :: S Integer

>>> stake 21 $ [1] <<| plus sFib sFib
[1,0,2,2,4,6,10,16,26,42,68,110,178,288,466,754,1220,1974,3194,5168,8362]

>>> take 21 $ fromOEIS "A006355"
[1,0,2,2,4,6,10,16,26,42,68,110,178,288,466,754,1220,1974,3194,5168,8362]

under (diff)

dxFib :: S Integer

>>> take 20 $ fromOEIS "A039834"
[1,1,0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597]

>>> stake 20 $ [1] <<| diff sFib
[1,1,0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597]

dxLucas :: S Integer

>>> take 20 $ fromOEIS "A061084"
[1,2,1,3,4,7,11,18,29,47,76,123,199,322,521,843,1364,2207,3571,5778]

>>> stake 20 $ diff sLucas
[-1,2,1,3,4,7,11,18,29,47,76,123,199,322,521,843,1364,2207,3571,5778]

Padovans and Perrins, and Pells

Padovan

padovanPP11010 :: S Integer

>>> take 30 $ fromOEIS "A000931"
[1,0,0,1,0,1,1,1,2,2,3,4,5,7,9,12,16,21,28,37,49,65,86,114,151,200,265,351,465,616]

>>> stake 30 $ padovanPP11010
[1,0,0,1,0,1,1,1,2,2,3,4,5,7,9,12,16,21,28,37,49,65,86,114,151,200,265,351,465,616]

padovan :: S Integer

>>> take 30 $ fromOEIS "A134816"
[1,1,1,2,2,3,4,5,7,9,12,16,21,28,37,49,65,86,114,151,200,265,351,465,616,816,1081,1432,1897,2513]

>>> stake 30 $ padovan
[1,1,1,2,2,3,4,5,7,9,12,16,21,28,37,49,65,86,114,151,200,265,351,465,616,816,1081,1432,1897,2513]

rpadovan :: G Integer Integer

>>> stake 30 $ revFix rpadovan
[1,1,1,2,2,3,4,5,7,9,12,16,21,28,37,49,65,86,114,151,200,265,351,465,616,816,1081,1432,1897,2513]

d2xpadovan :: S Integer

>>> take 31 $ fromOEIS "A133034"
[1,0,1,1,1,0,0,1,0,1,1,1,2,2,3,4,5,7,9,12,16,21,28,37,49,65,86,114,151,200,265]

>>> stake 30 $ diff $ diff $ revFix rpadovan
[0,1,-1,1,0,0,1,0,1,1,1,2,2,3,4,5,7,9,12,16,21,28,37,49,65,86,114,151,200,265]

Perrin

perrin :: S Integer

>>> take 30 $ fromOEIS "A001608"
[3,0,2,3,2,5,5,7,10,12,17,22,29,39,51,68,90,119,158,209,277,367,486,644,853,1130,1497,1983,2627,3480]

>>> stake 30 $ perrin
[3,0,2,3,2,5,5,7,10,12,17,22,29,39,51,68,90,119,158,209,277,367,486,644,853,1130,1497,1983,2627,3480]

rperrin :: G Integer Integer

>>> stake 30 $ revFix rperrin
[3,0,2,3,2,5,5,7,10,12,17,22,29,39,51,68,90,119,158,209,277,367,486,644,853,1130,1497,1983,2627,3480]

Pell

cPell :: S Integer

Generating Function:

z -------------------- z^2 + 2z - 1

>>> take 20 $ fromOEIS "A000129"
[0,1,2,5,12,29,70,169,408,985,2378,5741,13860,33461,80782,195025,470832,1136689,2744210,6625109]

cPellH :: (a, a) -> a

cPellT :: Num a => (a, a) -> a -> (a, a)

>>> stake 20 $ cPell
[0,1,2,5,12,29,70,169,408,985,2378,5741,13860,33461,80782,195025,470832,1136689,2744210,6625109]

rpell :: G Integer Integer

>>> stake 20 $ revFix rpell
[0,1,2,5,12,29,70,169,408,985,2378,5741,13860,33461,80782,195025,470832,1136689,2744210,6625109]

Who invited the J's?

Jacobsthal Sequence

jacob :: S Integer

>>> take 20 $ fromOEIS "A001045"
[0,1,1,3,5,11,21,43,85,171,341,683,1365,2731,5461,10923,21845,43691,87381,174763]

>>> stake 20 $ jacob
[0,1,1,3,5,11,21,43,85,171,341,683,1365,2731,5461,10923,21845,43691,87381,174763]

rjacob :: G Integer Integer

>>> stake 20 $ revFix rjacob
[0,1,1,3,5,11,21,43,85,171,341,683,1365,2731,5461,10923,21845,43691,87381,174763]

Jacobsthal-Lucas Sequence

jacobl :: S Integer

>>> take 20 $ fromOEIS "A014551"
[2,1,5,7,17,31,65,127,257,511,1025,2047,4097,8191,16385,32767,65537,131071,262145,524287]

>>> stake 20 $ jacobl
[2,1,5,7,17,31,65,127,257,511,1025,2047,4097,8191,16385,32767,65537,131071,262145,524287]

rjacobl :: G Integer Integer

>>> stake 20 $ revFix rjacobl
[2,1,5,7,17,31,65,127,257,511,1025,2047,4097,8191,16385,32767,65537,131071,262145,524287]

Josephus Numbers

jos :: S Integer

>>> take 30 $ drop 1 $ fromOEIS "A006257"
[1,1,3,1,3,5,7,1,3,5,7,9,11,13,15,1,3,5,7,9,11,13,15,17,19,21,23,25,27,29]

>>> stake 30 $ jos
[1,1,3,1,3,5,7,1,3,5,7,9,11,13,15,1,3,5,7,9,11,13,15,17,19,21,23,25,27,29]

josAlt :: S Integer

>>> stake 30 $ josAlt
[1,1,3,1,3,5,7,1,3,5,7,9,11,13,15,1,3,5,7,9,11,13,15,17,19,21,23,25,27,29]

Basically Binary (Base 2)

Most Significant Bit

msb :: S Integer

>>> take 30 $ drop 1 $ fromOEIS "A053644"
[1,2,2,4,4,4,4,8,8,8,8,8,8,8,8,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16]

>>> stake 30 $ msb
[1,2,2,4,4,4,4,8,8,8,8,8,8,8,8,16,16,16,16,16,16,16,16,16,16,16,16,16,16,16]

Thue-Morse Sequence

athue :: S Integer

>>> take 30 $ fromOEIS "A010060"
[0,1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0,1,1,0,1,0]

>>> stake 30 $ athue
[0,1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0,1,1,0,1,0]

thue :: S Integer

thue' :: S Integer

>>> stake 30 $ thue
[0,1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0,1,1,0,1,0]

Something with twos!

binlike1 :: S Integer

>>> stake 30 $ binlike1
[0,1,0,1,2,0,0,1,2,2,4,0,0,0,0,1,2,2,4,2,4,4,8,0,0,0,0,0,0,0]

Something with twos! (2)

binlike2 :: S Integer

>>> stake 30 $ binlike2
[0,1,1,0,1,2,0,2,1,0,2,2,0,4,2,0,1,4,0,2,2,0,2,4,0,4,4,0,2,8]

Who knows what this is?

lsb :: S Integer

>>> take 30 $ fromOEIS "A000035"
[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1]

>>> stake 30 $ bsum 0 |~| 1
[0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1]

Binary Sum Carry

sumcarry :: S Integer

>>> take 30 $ fromOEIS "A011371"
[0,0,1,1,3,3,4,4,7,7,8,8,10,10,11,11,15,15,16,16,18,18,19,19,22,22,23,23,25,25]

>>> stake 30 $ sumcarry
[0,0,1,1,3,3,4,4,7,7,8,8,10,10,11,11,15,15,16,16,18,18,19,19,22,22,23,23,25,25]

Fractals

Apollonian Fractal

D2: (-1,2,2,3)

apolloD2 :: Floating a => [a] -> a

>>> take 20 $ fromOEIS "A060790"
[1,2,2,3,15,38,110,323,927,2682,7754,22403,64751,187134,540822,1563011,4517183,13054898,37729362,109039875]

>>> stake 20 $ revFix apolloD2
[2.0,2.0,3.0,15.0,38.0,110.0,323.0,927.0,2682.0,7754.0,22403.0,64751.0,187134.0,540822.0,1563011.0,4517183.0,1.3054898e7,3.7729362e7,1.09039875e8,3.15131087e8]

apolloD2alt :: Floating a => [a] -> a

>>> stake 20 $ revFix apolloD2alt
[2.0,2.0,3.0,-1.0,2.0,2.0,3.0,-1.0,2.0,2.0,3.0,-1.0,2.0,2.0,3.0,-1.0,2.0,2.0,3.0,-1.0]

apd2 :: Floating a => [a] -> a

D3: (-0,0,1,1)

fr :: Num t => (t, t, t, t, t, t, t, t) -> (t, t, t, t, t, t, t, t)

apd3 :: Coalg (Integer, (Integer, Integer, Integer, Integer, Integer, Integer, Integer, Integer)) Integer Integer

proy :: (a, a, a, a, a, a, a, a) -> Integer -> a

streamApD3 :: S Integer

>>> stake 32 $ streamApD3
[0,0,1,1,1,2,2,3,4,8,9,9,15,32,32,33,56,120,121,121,209,450,450,451,780,1680,1681,1681,2911,6272,6272,6273]

streamApD3' :: S (Integer, Integer, Integer, Integer, Integer, Integer, Integer, Integer)

>>> stake 4 $ streamApD3'
[(0,0,1,1,1,2,2,3),(4,8,9,9,15,32,32,33),(56,120,121,121,209,450,450,451),(780,1680,1681,1681,2911,6272,6272,6273)]

Some Fractal Numbers

frac :: S Integer

>>> take 30 $ fromOEIS "A025480"
[0,0,1,0,2,1,3,0,4,2,5,1,6,3,7,0,8,4,9,2,10,5,11,1,12,6,13,3,14,7]

>>> stake 30 $ frac
[0,0,1,0,2,1,3,0,4,2,5,1,6,3,7,0,8,4,9,2,10,5,11,1,12,6,13,3,14,7]

Greatest Odd Divisor

god :: S Integer

>>> take 30 $ fromOEIS "A000265"
[1,1,3,1,5,3,7,1,9,5,11,3,13,7,15,1,17,9,19,5,21,11,23,3,25,13,27,7,29,15]

>>> stake 30 $ god
[1,1,3,1,5,3,7,1,9,5,11,3,13,7,15,1,17,9,19,5,21,11,23,3,25,13,27,7,29,15]

Dunno

blah :: S Integer

diverges!!

The Zoo

Moser-de-Bruijn sequence

Fermi Dirac Numbers

5-Smooth (Hamming) (Regular) Numbers

hamming :: S Integer

>>> take 30 $ fromOEIS "A051037"
[1,2,3,4,5,6,8,9,10,12,15,16,18,20,24,25,27,30,32,36,40,45,48,50,54,60,64,72,75,80]

>>> stake 30 $ hamming
[1,2,3,4,5,6,8,9,10,12,15,16,18,20,24,25,27,30,32,36,40,45,48,50,54,60,64,72,75,80]

montest :: S Integer

Dunno this one either?

a092323 :: S Integer

>>> take 30 $ fromOEIS "A092323"
[0,1,1,3,3,3,3,7,7,7,7,7,7,7,7,15,15,15,15,15,15,15,15,15,15,15,15,15,15,15]

>>> stake 30 $ (diff sNat - msb)
[0,-1,-1,-3,-3,-3,-3,-7,-7,-7,-7,-7,-7,-7,-7,-15,-15,-15,-15,-15,-15,-15,-15,-15,-15,-15,-15,-15,-15,-15]

Carry Bits

carry :: S Integer

>>> take 30 $ fromOEIS "A007814"
[0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1]

>>> stake 30 $ carry
[0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1]

altCarry :: S Integer

FIXME: Incorrect!

tree :: Integral a => a -> S a

>>> stake 30 $ tree 0
[0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4,0,1,0,2,0,1,0,3,0,1,0,2,0,1]

Non-Negative Integers. Repeated. Floor (n/2)

a004526 :: S Integer

>>> take 30 $ fromOEIS "A004526"
[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9,10,10,11,11,12,12,13,13,14,14]

>>> stake 30 $ a004526
[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9,10,10,11,11,12,12,13,13,14,14]

Square Pyramidal Numbers

a000330 :: S Integer

>>> take 16 $ 0 : fromOEIS "A000330"
[0,0,1,5,14,30,55,91,140,204,285,385,506,650,819,1015]

>>> stake 16 $ a000330
[0,0,1,5,14,30,55,91,140,204,285,385,506,650,819,1015]

The Perturbation Method

iterate k = 2^n-k

iterk2nk :: S Integer

>>> take 21 $ 1 : fromOEIS "A078008"
[1,1,0,2,2,6,10,22,42,86,170,342,682,1366,2730,5462,10922,21846,43690,87382,174762]

iterk2nkH :: (a, b) -> a

iterk2nkT :: Num a => (a, a) -> a -> (a, a)

>>> stake 21 $ [1,1] <<| iterk2nk
[1,1,0,2,2,6,10,22,42,86,170,342,682,1366,2730,5462,10922,21846,43690,87382,174762]

What the heck is this?

a090017 :: G Integer Integer

>>> take 15 $ fromOEIS "A090017"
[0,1,4,18,80,356,1584,7048,31360,139536,620864,2762528,12291840,54692416,243353344]

>>> stake 15 $ revFix a090017
[0,1,4,18,80,356,1584,7048,31360,139536,620864,2762528,12291840,54692416,243353344]

Horadam (0,1,4,2) Numbers

horadam0142 :: G Integer Integer

>>> take 15 $ fromOEIS "A085449"
[0,1,2,8,24,80,256,832,2688,8704,28160,91136,294912,954368,3088384]

>>> stake 15 $ revFix horadam0142
[0,1,2,8,24,80,256,832,2688,8704,28160,91136,294912,954368,3088384]

Some random thing I found.

a002605 :: G Integer Integer

>>> take 15 $ fromOEIS "A002605"
[0,1,2,6,16,44,120,328,896,2448,6688,18272,49920,136384,372608]

>>> stake 15 $ revFix a002605
[0,1,2,6,16,44,120,328,896,2448,6688,18272,49920,136384,372608]