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A166956
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a(n) = 2^n +(-1)^n - 2.
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6
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0, -1, 3, 5, 15, 29, 63, 125, 255, 509, 1023, 2045, 4095, 8189, 16383, 32765, 65535, 131069, 262143, 524285, 1048575, 2097149, 4194303, 8388605, 16777215, 33554429, 67108863, 134217725, 268435455, 536870909, 1073741823, 2147483645, 4294967295, 8589934589
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OFFSET
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0,3
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COMMENTS
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The inverse binomial transform yields 0,-1,5,-7,17,-31,..., a sign alternating variant of A014551.
In a table of a(n) and higher-order differences in successive rows, the main diagonal contains 0, 4, 8, 16, ... (zero followed by A020707).
Similar to the decimal representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 899", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero, which begins with 1,3,5,15,29,63,125. - Robert Price, Aug 08 2017
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REFERENCES
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S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
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LINKS
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FORMULA
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a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3).
G.f.: x*(5*x -1)/((1-x)*(1-2*x)*(1+x)).
E.g.f.: exp(2*x) - 2*exp(x) + exp(-x). - G. C. Greubel, May 29 2016
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MATHEMATICA
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LinearRecurrence[{2, 1, -2}, {0, -1, 3}, 20] (* G. C. Greubel, May 29 2016 *)
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PROG
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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