A270444 Expansion of 2*(1+2*x) / (1-8*x+4*x^2).

2, 20, 152, 1136, 8480, 63296, 472448, 3526400, 26321408, 196465664, 1466439680, 10945654784, 81699479552, 609813217280, 4551707820032, 33974409691136, 253588446248960, 1892809931227136, 14128125664821248, 105453765593661440

Offset: 1

Author

Altug Alkan, Mar 17 2016

Keywords

• nonn

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) = 20 because (1 + sqrt(3))^3 + (1 - sqrt(3))^3 = 20.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (8, -4).

Crossrefs

Cf. A077444, A080040, A080041, A107903.

Programs

Mathematica

CoefficientList[Series[2(1+2x)/(1-8x+4x^2),{x,0,30}],x] (* or *) LinearRecurrence[{8,-4},{2,20},30] (* Harvey P. Dale, Jun 09 2020 *)

PARI

Vec(2*(1+2*x)/(1-8*x+4*x^2) + O(x^100))

Formula

G.f.: 2*(1+2*x)/(1-8*x+4*x^2).

a(n) = (1+sqrt(3))^(2*n-1) + (1-sqrt(3))^(2*n-1).

a(n) = 2 * A107903(n-1).