A117872 -id:A117872 - cp's OEIS frontend

A126768 Equal-parity block sizes in sequence A117872.

1, 1, 1, 2, 2, 4, 2, 1, 1, 3, 3, 2, 2, 1, 1, 3, 2, 3, 1, 2, 1, 1, 2, 2, 2, 1, 2, 2, 2, 3, 3, 1, 1, 1, 1, 4, 1, 2, 1, 4, 2, 3, 1, 2, 2, 1, 1, 1, 3, 3, 1, 3, 2, 1, 2, 4, 2, 2, 4, 2, 4, 1, 2, 1, 1, 3, 1, 2, 2, 1, 2, 1, 1, 2, 3, 2, 1, 4, 3, 1, 3, 1, 5, 6, 3, 1, 4, 5, 5, 1, 5, 3, 1, 1, 6, 1, 1, 4, 1, 2, 2, 2, 3, 1, 1, 1, 1, 1, 7, 1, 1, 2, 1, 1, 1, 3, 1, 2, 2, 2, 3, 1, 1, 1, 1

Offset: 1

Greg Huber, Feb 16 2007

nonn

From a suggestion due to D. R. Hofstadter.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

import Data.List (group)
a126768 n = a126768_list !! n
a126768_list = map length $ group a117872_list
-- Reinhard Zumkeller, Aug 15 2013

Corrected terms from Greg Huber, Aug 21 2013

A007501 a(0) = 2; for n >= 0, a(n+1) = a(n)*(a(n)+1)/2.

Original entry on oeis.org

2, 3, 6, 21, 231, 26796, 359026206, 64449908476890321, 2076895351339769460477611370186681, 2156747150208372213435450937462082366919951682912789656986079991221

Offset: 0

N. J. A. Sloane, Robert G. Wilson v

Number of nonisomorphic complete binary trees with leaves colored using two colors. - Brendan McKay, Feb 01 2001

With a(0) = 2, a(n+1) is the number of possible distinct sums between any number of elements in {1,...,a(n)}. - Derek Orr, Dec 13 2014

			Example for depth 2 (the nonisomorphic possibilities are AAAA, AAAB, AABB, ABAB, ABBB, BBBB):
         o
        / \
       /   \
      o     o
     / \   / \
    /   \ /   \
    A   B B   B

W. H. Cutler, Subdividing a Box into Completely Incongruent Boxes, J. Rec. Math., 12 (1979), 104-111.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 0..12
G. L. Honaker, Jr., 41041 (another Prime Pages' Curiosity)
J. C. Kieffer, Hierarchical Type Classes and Their Entropy Functions, in 2011 First International Conference on Data Compression, Communications and Processing, pp. 246-254; Digital Object Identifier: 10.1109/CCP.2011.36.
J. V. Post, Math Pages [wayback copy]
J.S. Seneschal, Iteration of Complete Graphs
Stephan Wagner, Enumeration of highly balanced trees

Cf. A000217, A006893.
Cf. A117872 (parity), A275342 (2-adic valuation).
Cf. A129440.
Cf. A013589 (start=4), A050542 (start=5), A050548 (start=7), A050536 (start=8), A050909 (start=9).
Cf.  A108225, A145272.

a007501 n = a007501_list !! n
a007501_list = iterate a000217 2  -- Reinhard Zumkeller, Aug 15 2013

f[n_Integer] := n(n + 1)/2; NestList[f, 2, 10]

PARI
```
a(n)=if(n<1,2,a(n-1)*(1+a(n-1))/2)
```

a(n) = A006893(n+1) + 1.
a(n+1) = A000217(a(n)). - Reinhard Zumkeller, Aug 15 2013
a(n) ~ 2 * c^(2^n), where c = 1.34576817070125852633753712522207761954658547520962441996... . - Vaclav Kotesovec, Dec 17 2014
a(n) = A145272(n) + a(n-1). - J.S. Seneschal, Jul 17 2025

A076436 Square board sizes for which the lights-out problem has a unique solution (counting solutions differing only by rotation and reflection as distinct).

Original entry on oeis.org

1, 2, 3, 6, 7, 8, 10, 12, 13, 15, 18, 20, 21, 22, 25, 26, 27, 28, 31, 36, 37, 38, 40, 42, 43, 45, 46, 48, 51, 52, 55, 56, 57, 58, 60, 63, 66, 68, 70, 72, 73, 75, 76, 78, 80, 81, 82, 85, 86, 87, 88, 90, 91, 93, 96, 97, 100, 102, 103, 105, 106, 108, 110, 111, 112, 115, 116, 117, 120

Offset: 1

Eric W. Weisstein, Oct 11 2002

nonn

These are also the boards where any starting configuration can be turned off. - Robert Cowen (robert.cowen(AT)gmail.com), Jan 06 2007. [Comment corrected by Sune Kristian Jakobsen (sunejakobsen(AT)hotmail.com), Feb 04 2008]

N. J. A. Sloane and Thomas Buchholz, Table of n, a(n) for n = 1..1000 [terms were extended by N. J. A. Sloane (based on A117870), May 14 2006; terms 70 through 1000 were computed by Thomas Buchholz, May 16 2014]
K. Sutner, Linear cellular automata and the Garden-of-Eden, Math. Intelligencer, 11 (No. 2, 1989), 49-53.
Eric Weisstein's World of Mathematics, Lights Out Puzzle

Cf. A075462, A076437, A117872. Complement of A117870.

Positive integer n is in this sequence iff A159257(n)=0. [Max Alekseyev, Sep 25 2009]

More terms from N. J. A. Sloane (based on A117870), May 14 2006, and Thomas Buchholz, May 16 2014

A117870 Square board sizes for which the lights out problem does not have a unique solution (counting solutions differing only by rotation and reflection as distinct).

Original entry on oeis.org

4, 5, 9, 11, 14, 16, 17, 19, 23, 24, 29, 30, 32, 33, 34, 35, 39, 41, 44, 47, 49, 50, 53, 54, 59, 61, 62, 64, 65, 67, 69, 71, 74, 77, 79, 83, 84, 89, 92, 94, 95, 98, 99, 101, 104, 107, 109, 113, 114, 118, 119, 123, 124, 125, 126, 128, 129, 131, 134, 135, 137, 139, 143

Offset: 1

N. J. A. Sloane, May 14 2006

nonn

Numbers k such that a k X k parity pattern exists (see A118141). - Don Knuth, May 11 2006

Max Alekseyev and Thomas Buchholz, Table of n, a(n) for n = 1..1000 [terms were extended by Max Alekseyev, Sep 17 2009; terms 64 through 1000 were computed by Thomas Buchholz, May 16 2014]
Jaap's puzzle page, The Mathematics of Lights Out
K. Sutner, Linear cellular automata and the Garden-of-Eden, Math. Intelligencer, 11 (No. 2, 1989), 49-53.
Eric Weisstein's World of Mathematics, Lights-Out Puzzle
Wikipedia, Lights Out (game)

Cf. A075462, A076437, A117872. Complement of A076436.

a(n) = A093614(n) - 1.

Contains positive integers k such that A159257(k) > 0. - Max Alekseyev, Sep 17 2009

More terms from Max Alekseyev, Sep 17 2009, and Thomas Buchholz, May 16 2014

A275342 2-adic valuation of iterated triangular numbers, starting with 2.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 1, 0, 0, 0, 0, 2, 1, 0, 1, 0, 0, 0, 3, 2, 1, 0, 0, 2, 1, 0, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 1, 0, 0, 2, 1, 0, 2, 1, 0, 0, 2, 1, 0, 0, 0, 3, 2, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 1, 0, 0, 0, 1, 0, 0, 2, 1, 0, 1, 0, 3, 2, 1, 0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 2, 1, 0, 0, 0, 0, 2, 1, 0, 0, 4, 3, 2, 1, 0, 0, 4, 3, 2, 1, 0, 2, 1, 0, 1, 0, 0, 0, 1

Offset: 1

Jeffrey Shallit, Jul 23 2016

nonn

			t(1) = 2, t(2) = 3, t(3) = 6, t(4) = 21, so a(1) = 1, a(2) = 0, a(3) = 1, a(4) = 0.

Cf. A007501, A007814, A117872.

Writing t(1) = 2, t(n+1) = t(n)(t(n)+1)/2, the sequence is nu_2 (t(n)), where nu_2 (x) is the exponent of the highest power of 2 dividing x.

a(n) = A007814(A007501(n)). - Michel Marcus, Jul 23 2016

A145273 Parity sequence of A006893.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1

Offset: 1

Greg Huber, Oct 06 2008

Complement of the parity sequence of iterated triangular numbers (A117872).

Cf. A145272. Complement of A117872

cp's OEIS Frontend

A126768 Equal-parity block sizes in sequence A117872.

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A007501 a(0) = 2; for n >= 0, a(n+1) = a(n)*(a(n)+1)/2.

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A076436 Square board sizes for which the lights-out problem has a unique solution (counting solutions differing only by rotation and reflection as distinct).

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A117870 Square board sizes for which the lights out problem does not have a unique solution (counting solutions differing only by rotation and reflection as distinct).

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A275342 2-adic valuation of iterated triangular numbers, starting with 2.

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A145273 Parity sequence of A006893.

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