A270445 Expansion of 2*x*(1+4*x) / (1-12*x+16*x^2).

2, 32, 352, 3712, 38912, 407552, 4268032, 44695552, 468058112, 4901568512, 51329892352, 537533612032, 5629125066752, 58948963008512, 617321555034112, 6464675252273152, 67698958146732032, 708952693724413952, 7424248994345254912

Offset: 1

Author

Altug Alkan, Mar 17 2016

Keywords

• nonn
• easy

Comments

If p is an odd prime, a((p+1)/2) == 2 mod p. In other words, a((p+1)/2) - 2^p is divisible by p where p is an odd prime.

Examples

			a(2) = 32 because (1 + sqrt(5))^3 + (1 - sqrt(5))^3 = 32.
		

Links

• Colin Barker, Table of n, a(n) for n = 1..950
• Index entries for linear recurrences with constant coefficients, signature (12,-16).

Crossrefs

Cf. A077444, A087131.

Programs

PARI

Vec(2*x*(1+4*x)/(1-12*x+16*x^2) + O(x^50)) \\ Colin Barker, Mar 17 2016

Formula

a(n) = 12*a(n-1) - 16*a(n-2) for n>2. G.f.: 2*x*(1+4*x) / (1-12*x+16*x^2). - Colin Barker, Mar 17 2016

a(n) = (1+sqrt(5))^(2*n-1) + (1-sqrt(5))^(2*n-1).