A084213 Binomial transform of A081250.

1, 4, 18, 76, 312, 1264, 5088, 20416, 81792, 327424, 1310208, 5241856, 20969472, 83881984, 335536128, 1342160896, 5368676352, 21474770944, 85899214848, 343597121536, 1374389010432, 5497557090304, 21990230458368, 87960926027776

Offset: 0

Author

Paul Barry, May 19 2003

Keywords

• easy
• nonn

Comments

When 5*2^n - 1 is prime, that is, n is in A001770, then a(n+1) is in A136539. - Farideh Firoozbakht and M. F. Hasler, Nov 03 2012

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..400
• Index entries for linear recurrences with constant coefficients, signature (6,-8)

Programs

Magma

[5*4^n/4-2^n/2+0^n/4: n in [0..30]]; // Vincenzo Librandi, Jun 15 2011

Maple

seq(coeff(series((1-2*x+2*x^2)/((1-2*x)*(1-4*x)),x,n+1), x, n), n = 0 .. 25); # Muniru A Asiru, Oct 09 2018

Mathematica

Table[If[n==0, 1, 2^(n-2)*(5*2^n - 2)], {n,0,30}] (* G. C. Greubel, Oct 08 2018 *)
CoefficientList[Series[(1 - 2*x + 2*x^2)/((1-2*x)*(1-4*x)), {x, 0, 50}], x] (* or *)
CoefficientList[Series[(5*Exp[4*x] - 2*Exp[2*x] + 1)/4, {x, 0, 50}], x]*Table[k!, {k, 0, 50}] (* Stefano Spezia, Oct 11 2018 *)

PARI

vector(30, n, n--; (5*4^n - 2^(n+1) + 0^n)/4) \\ G. C. Greubel, Oct 08 2018

Formula

a(n) = (5*4^n - 2^(n+1) + 0^n)/4.

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

E.g.f.: (5*exp(4*x) - 2*exp(2*x) + 1)/4.

a(n+1) = 2^n*(5*2^n - 1) for all n >= 0. - M. F. Hasler, Nov 03 2012