A248917 a(n) = 2^n * n^2 + 1.

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

Offset: 0

Author

Paul Curtz, Oct 22 2014

Keywords

• nonn
• easy

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)).

Examples

			a(3) = 9 * 8 + 1 = 73.
a(4) = 16 * 16 + 1 = 257.
a(5) = 25 * 32 + 1 = 801.
		

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).

Crossrefs

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

Programs

Magma

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

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 *)

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<Charles R Greathouse IV, Oct 22 2014

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