A266443 Decimal representation of the n-th iteration of the "Rule 25" elementary cellular automaton starting with a single ON (black) cell.

1, 1, 10, 67, 116, 1671, 232, 32015, 464, 522783, 928, 8385599, 1856, 134211711, 3712, 2147471615, 7424, 34359714303, 14848, 549755765759, 29696, 8796092925951, 59392, 140737488162815, 118784, 2251799813300223, 237568, 36028797018193919, 475136

Offset

0,3

Links

Robert Price, Table of n, a(n) for n = 0..1000

Eric Weisstein's World of Mathematics, Elementary Cellular Automaton

Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Index entries for sequences related to cellular automata

Index to Elementary Cellular Automata

Index entries for linear recurrences with constant coefficients, signature (0,19,0,-50,0,32).

Formula

From Colin Barker, Dec 30 2015 and Apr 16 2019: (Start)

a(n) = 19*a(n-2)-50*a(n-4)+32*a(n-6) for n>9.

G.f.: (1+x-9*x^2+48*x^3-24*x^4+448*x^5-1504*x^6+3584*x^7+1536*x^8-4096*x^9) / ((1-x)*(1+x)*(1-4*x)*(1+4*x)*(1-2*x^2)). (End)

a(n) = (1 - (-1)^n)*(2*4^n - 1)/2 + ((1 + (-1)^n)*(94 + 29*2^(1/2))/4 - 47)*2^((n+1)/2) for n>3. Therefore: for even n>2, a(n) = 29*2^(n/2); for odd n>3, a(n) = 2^(2*n+1) - 47*2^((n+1)/2) - 1. [Bruno Berselli, Dec 30 2015]

Mathematica

rule=25; rows=20; ca=CellularAutomaton[rule, {{1}, 0}, rows-1, {All, All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]], {rows-k+1, rows+k-1}], {k, 1, rows}]; (* Truncated list of each row *) Table[FromDigits[catri[[k]], 2], {k, 1, rows}] (* Decimal Representation of Rows *)

Prog

(Python) print([1, 1, 10, 67] + [2*4**n-47*2**((n+1)//2)-1 if n%2 else 29*2**(n//2) for n in range(4, 30)]) # Karl V. Keller, Jr., Jul 05 2021

Crossrefs

Cf. A266441.

Sequence in context: A377917 A231886 A278090 * A108275 A086443 A172066

Adjacent sequences: A266440 A266441 A266442 * A266444 A266445 A266446

Keyword

nonn,easy

Author

Robert Price, Dec 29 2015

Status

approved