A020514 a(n) = 1^n + 2^n + 4^n + 8^n + 16^n.

5, 31, 341, 4681, 69905, 1082401, 17043521, 270549121, 4311810305, 68853957121, 1100586419201, 17600780175361, 281543712968705, 4504149450301441, 72061992352890881, 1152956690052710401, 18447025552981295105, 295150156996346511361, 4722384497336874434561

Offset: 0

Author

Simon Plouffe and N. J. A. Sloane

Keywords

• nonn
• easy

Comments

5th cyclotomic polynomial evaluated at 2^n.

Links

• T. D. Noe, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (31,-310,1240,-1984,1024).

Crossrefs

Cf. A082771, A006097, A022166.

Programs

Maple

with(numtheory,cyclotomic):seq(cyclotomic(5,2^i),i=0..24);

Mathematica

With[{c=2^Range[0,4]},Table[Total[c^n],{n,0,20}]] (* Harvey P. Dale, May 27 2012 *)

PARI

a(n)=1^n+2^n+4^n+8^n+16^n \\ Charles R Greathouse IV, Oct 07 2015

Sage

[sigma(16,n)for n in range(0,16)] # Zerinvary Lajos, Jun 04 2009

Formula

G.f.: 1/(1-x)+1/(1-2*x)+1/(1-4*x)+1/(1-8*x)+1/(1-16*x). - Philippe Deléham, Apr 06 2013

E.g.f.: exp(x) + exp(2*x) + exp(4*x) + exp(8*x) + exp(16*x). - Philippe Deléham, Apr 06 2013

a(n) = 31*a(n-1) - 310*a(n-2) + 1240*a(n-3) - 1984*a(n-4) + 1024*a(n-5) with a(0) = 5, a(1) = 31, a(2) = 341, a(3) = 4681, a(4) = 69905. - Philippe Deléham, Apr 06 2013

a(n) = (2^(5*n) - 1)/( 2^n - 1). Exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 31*x + 651*x^2 + 11811*x^3 + ... is the o.g.f. for the 4th subdiagonal of triangle A022166, essentially A006097. - Peter Bala, Apr 07 2015