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A266590 Decimal representation of the n-th iteration of the "Rule 37" elementary cellular automaton starting with a single ON (black) cell. 1
1, 2, 14, 65, 56, 1799, 224, 31775, 896, 520319, 3584, 8372735, 14336, 134154239, 57344, 2147229695, 229376, 34358722559, 917504, 549751750655, 3670016, 8796076769279, 14680064, 140737423343615, 58720256, 2251799553638399, 234881024, 36028795978776575 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

REFERENCES

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

LINKS

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

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 Elementary Cellular Automata

Index entries for linear recurrences with constant coefficients, signature (0,21,0,-84,0,64).

FORMULA

From Colin Barker, Jan 02 2016: (Start)

a(n) = 21*a(n-2)-84*a(n-4)+64*a(n-6) for n>7.

G.f.: (1+2*x-7*x^2+23*x^3-154*x^4+602*x^5+160*x^6-672*x^7) / ((1-x)*(1+x)*(1-2*x)*(1+2*x)*(1-4*x)*(1+4*x)).

(End)

MATHEMATICA

rule=37; 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 *)

Join[{1, 2}, LinearRecurrence[{0, 21, 0, -84, 0, 64}, {14, 65, 56, 1799, 224, 31775}, 26]] (* Vincenzo Librandi, Jan 02 2016 *)

PROG

(PARI) Vec((1+2*x-7*x^2+23*x^3-154*x^4+602*x^5+160*x^6-672*x^7) / ((1-x)*(1+x)*(1-2*x)*(1+2*x)*(1-4*x)*(1+4*x)) + O(x^40)) \\ Colin Barker, Jan 02 2016

(MAGMA) I:=[1, 2, 14, 65, 56, 1799, 224, 31775]; [n le 8 select I[n] else 21*Self(n-2)-84*Self(n-4)+64*Self(n-6): n in [1..30]]; // Vincenzo Librandi, Jan 02 2016

CROSSREFS

Cf. A266588.

Sequence in context: A167555 A222445 A181394 * A196977 A254197 A222715

Adjacent sequences:  A266587 A266588 A266589 * A266591 A266592 A266593

KEYWORD

nonn,easy

AUTHOR

Robert Price, Jan 01 2016

STATUS

approved

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Last modified October 21 13:05 EDT 2018. Contains 316422 sequences. (Running on oeis4.)