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 A248917 a(n) = 2^n * n^2 + 1. 3
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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)). LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (7,-18,20,-8). FORMULA a(n) = 4*a(n-1) - 4*a(n-2) + 2^(n+1) + 1. a(n) = A007758(n) + 1. a(n) = 7*a(n-1) - 18*a(n-2) + 20*a(n-3) - 8*a(n-4). - Jean-François Alcover, Oct 22 2014 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 EXAMPLE a(3) = 9 * 8 + 1 = 73. a(4) = 16 * 16 + 1 = 257. a(5) = 25 * 32 + 1 = 801. MATHEMATICA Table[n^2 * 2^n + 1, {n, 0, 31}] (* Alonso del Arte, Oct 22 2014 *) LinearRecurrence[{7, -18, 20, -8}, {1, 3, 17, 73}, 25] (* G. C. Greubel, Oct 28 2016 *) PROG (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 (PARI) a(n)=n^2<

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Last modified August 3 11:57 EDT 2020. Contains 336198 sequences. (Running on oeis4.)