A101692 A modular binomial sum transform of 2^n.

1, 1, 5, 1, 5, 17, 85, 1, 5, 17, 85, 257, 1285, 4369, 21845, 1, 5, 17, 85, 257, 1285, 4369, 21845, 65537, 327685, 1114129, 5570645, 16843009, 84215045, 286331153, 1431655765, 1, 5, 17, 85, 257, 1285, 4369, 21845, 65537, 327685, 1114129, 5570645, 16843009, 84215045

Offset: 0

Author

Paul Barry, Dec 11 2004

Keywords

• nonn
• easy

Comments

a(2^n) is 1, 5, 5, 5, 5, ...

a(2^n+1) is 5, 1, 17, 17, 17, ...

a(2*(2^n+1)) is 5, 85, 85, 85, ...

a(2^n)*a(2^n+1) is 5, 5, 85, 85, 85, ...

Also, decimal representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 4", based on the 5-celled von Neumann neighborhood. Initialized with a single black (ON) cell at stage zero. - Robert Price, Nov 03 2016

References

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

Links

• Stefano Spezia, Table of n, a(n) for n = 0..1500
• Robert Price, Diagrams of first 20 stages of the cellular automaton
• 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
• Index entries for sequences related to cellular automata
• Index to 2D 5-Neighbor Cellular Automata
• Index to Elementary Cellular Automata

Crossrefs

Cf. A001045, A048896.

Cf. A277916, A277917, A277918.

Programs

Mathematica

a[n_]:= Sum[Mod[Binomial[2*n+2, k],2]*2^k,{k,0,n}]; Array[a,43,0] (* Stefano Spezia, Aug 04 2025 *)

Formula

a(n) = Sum_{k=0..n} (binomial(2*n+2, k) mod 2)*2^k.

a(n) = Sum_{k=0..n} A128937(n, k)*2^(n-k). - Philippe Deléham, Oct 09 2007