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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<<n + 1 \\ Charles R Greathouse IV, Oct 22 2014

(MAGMA) [2^n*n^2+1: n in [0..30]]; // Vincenzo Librandi, Oct 29 2016

CROSSREFS

Cf. A000225, A002064 (Cullen numbers), A006784, A007758, A055580, A118239, A168298.

Sequence in context: A317452 A128948 A049181 * A282400 A179596 A191587

Adjacent sequences:  A248914 A248915 A248916 * A248918 A248919 A248920

KEYWORD

nonn,easy

AUTHOR

Paul Curtz, Oct 22 2014

STATUS

approved

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Last modified June 20 08:06 EDT 2019. Contains 324229 sequences. (Running on oeis4.)