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A266443
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Decimal representation of the n-th iteration of the "Rule 25" elementary cellular automaton starting with a single ON (black) cell.
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3
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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
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OFFSET
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0,3
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LINKS
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FORMULA
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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]
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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