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A248917
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a(n) = 2^n * n^2 + 1.
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3
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1, 3, 17, 73, 257, 801, 2305, 6273, 16385, 41473, 102401, 247809, 589825, 1384449, 3211265, 7372801, 16777217, 37879809, 84934657, 189267969, 419430401, 924844033, 2030043137, 4437573633, 9663676417, 20971520001, 45365592065, 97844723713, 210453397505, 451508436993
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OFFSET
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0,2
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COMMENTS
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Binomial transform of A118239 (Engel expansion of cosh(1)).
Table of successive differences of a(n):
1, 3, 17, 73, 257, 801, 2305,...
2, 14, 56, 184, 544, 1504,...
12, 42, 128, 360, 960,...
30, 86, 232, 600,...
56, 146, 368,...
90, 222,...
132,...
etc.
Via b(n) = 0, 0, 0 followed by A055580(n), i.e., 0, 0, 0, 1, 7, 31, 111, ... (the main sequence for the recurrence), a link can be found between a(n) and A002064(n): c(n) = b(n+1) - 2*b(n) = 0, 0, 1, 5, 17, 49, 129, ... (the main sequence for the signature (5, -8, 4)).
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LINKS
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FORMULA
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a(n) = 4*a(n-1) - 4*a(n-2) + 2^(n+1) + 1.
G.f.: -(12*x^3-14*x^2+4*x-1) / ((x-1)*(2*x-1)^3). - Colin Barker, Oct 22 2014
E.g.f.: exp(x) + 2*x*(1 + 2*x)*exp(2*x). - G. C. Greubel, Oct 28 2016
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EXAMPLE
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a(3) = 9 * 8 + 1 = 73.
a(4) = 16 * 16 + 1 = 257.
a(5) = 25 * 32 + 1 = 801.
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MATHEMATICA
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LinearRecurrence[{7, -18, 20, -8}, {1, 3, 17, 73}, 25] (* G. C. Greubel, Oct 28 2016 *)
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PROG
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(PARI) Vec(-(12*x^3-14*x^2+4*x-1)/((x-1)*(2*x-1)^3) + O(x^100)) \\ Colin Barker, Oct 22 2014
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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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