A156232 -id:A156232 - cp's OEIS frontend

A032085 Number of reversible strings with n beads of 2 colors. If more than 1 bead, not palindromic.

2, 1, 2, 6, 12, 28, 56, 120, 240, 496, 992, 2016, 4032, 8128, 16256, 32640, 65280, 130816, 261632, 523776, 1047552, 2096128, 4192256, 8386560, 16773120, 33550336, 67100672, 134209536, 268419072, 536854528

Offset: 1

Christian G. Bower

a(n) is also the number of induced subgraphs with odd number of edges in the path graph P(n) if n>0. - Alessandro Cosentino (cosenal(AT)gmail.com), Feb 06 2009
A common recurrence of the bisections A020522 and A006516 means a(n+4) = 6*a(n+2) - 8*a(n), n>1. - Yosu Yurramendi, Aug 07 2008
Also, the decimal representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 566", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. - Robert Price, Jul 05 2017

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
M. Archibald, A. Blecher, A. Knopfmacher, M. E. Mays, Inversions and Parity in Compositions of Integers, J. Int. Seq., Vol. 23 (2020), Article 20.4.1.
C. G. Bower, Transforms (2)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1022
S. J. Cyvin et al., Theory of polypentagons, J. Chem. Inf. Comput. Sci., 33 (1993), 466-474.
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Wolfram Research, Wolfram Atlas of Simple Programs
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (2,2,-4).

Cf. A005418, A016116. Essentially the same as A122746.
Row sums of triangle A034877.
Cf. A289404, A289405, A052551.

[2] cat [2^(n-1)-2^Floor((n-1)/2) : n in [2..40]]; // Wesley Ivan Hurt, Jul 03 2020

Join[{2}, LinearRecurrence[{2, 2, -4}, {1, 2, 6}, 29]] (* Jean-François Alcover, Oct 11 2017 *)

a(n)=([0,1,0; 0,0,1; -4,2,2]^(n-1)*[2;1;2])[1,1] \\ Charles R Greathouse IV, Oct 21 2022

"BHK" (reversible, identity, unlabeled) transform of 2, 0, 0, 0, ...
a(n) = 2^(n-1)-2^floor((n-1)/2), n > 1. - Vladeta Jovovic, Nov 11 2001
G.f.: 2*x+x^2/((1-2*x)*(1-2*x^2)). - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 25 2004
a(n) = A005418(n+1)-A016116(n+2), n>1. - Yosu Yurramendi, Aug 07 2008
a(n+1) = A077957(n) + 2*a(n), n>1. a(n+2) = A000079(n+1) + 2*a(n), n>1. - Yosu Yurramendi, Aug 10 2008
First differences: a(n+1)-a(n) = A007179(n) = A156232(n+2)/4, n>1. - Paul Curtz, Nov 16 2009
a(n) = 2*(a(n-1) bitwiseOR a(n-2)), n>3. - Pierre Charland, Dec 12 2010
a(n) = 2*a(n-1) + 2*a(n-2) - 4*a(n-3). - Wesley Ivan Hurt, Jul 03 2020

A204696 G.f.: (32x^7/(1-2x) + 16x^5 + 24x^6)/(1-2*x^2).

Original entry on oeis.org

0, 0, 0, 0, 0, 16, 24, 64, 112, 256, 480, 1024, 1984, 4096, 8064, 16384, 32512, 65536, 130560, 262144, 523264, 1048576, 2095104, 4194304, 8384512, 16777216, 33546240, 67108864, 134201344, 268435456, 536838144, 1073741824, 2147418112, 4294967296, 8589803520, 17179869184, 34359476224, 68719476736, 137438429184, 274877906944

Offset: 0

N. J. A. Sloane, Jan 18 2012

nonn

Thomas W. Cusick, and Pantelimon Stanica, Fast evaluation, weights and nonlinearity of rotation-symmetric functions, Discrete Math. 258 (2002), no. 1-3, 289-301. See Eq. (14).

Essentially the same as A156232.

A350520 The number of degree-n^2 polynomials over Z/2Z that can be written as f(f(x)) where f is a polynomial.

Original entry on oeis.org

1, 1, 3, 8, 14, 32, 60, 128, 248, 512, 1008, 2048, 4064, 8192, 16320, 32768, 65408, 131072, 261888, 524288, 1048064, 2097152, 4193280, 8388608, 16775168, 33554432, 67104768, 134217728, 268427264

Offset: 0

Peter Kagey, Jan 03 2022

			For n = 2, there are a(2) = 3 degree 4 polynomials of the form f(f(x)):
  x^4         = f(f(x)) when f(x) = x^2 or f(x) = x^2 + 1,
  x^4 + x     = f(f(x)) when f(x) = x^2 + x, and
  x^4 + x + 1 = f(f(x)) when f(x) = x^2 + x + 1.

Cf. A033991, A156212, A156232.

Conjecture:
  a(2n) = A033991(2^(n-1)) = 4^n - 2^(n-1) for n >= 1;
  a(2n+1) = 2^(2n+1) for n >= 1.
Conjecture from Hugo Pfoertner, Jan 09 2022: Terms starting at 3 coincide with {A156232}/8.

a(0) prepended and a(11)-a(28) from Martin Ehrenstein, Jan 14 2022

cp's OEIS Frontend

A032085 Number of reversible strings with n beads of 2 colors. If more than 1 bead, not palindromic.

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A204696 G.f.: (32x^7/(1-2x) + 16x^5 + 24x^6)/(1-2*x^2).

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A350520 The number of degree-n^2 polynomials over Z/2Z that can be written as f(f(x)) where f is a polynomial.

Views

Author

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Examples

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Extensions

cp's OEIS Frontend

A032085 Number of reversible strings with n beads of 2 colors. If more than 1 bead, not palindromic.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A204696 G.f.: (32*x^7/(1-2*x) + 16*x^5 + 24*x^6)/(1-2*x^2).

Views

Author

Keywords

Links

Crossrefs

A350520 The number of degree-n^2 polynomials over Z/2Z that can be written as f(f(x)) where f is a polynomial.

Views

Author

Keywords

Examples

Crossrefs

Formula

Extensions

A204696 G.f.: (32x^7/(1-2x) + 16x^5 + 24x^6)/(1-2*x^2).