A096951 Sum of odd powers of 2 and of 3 divided by 5.

1, 7, 55, 463, 4039, 35839, 320503, 2876335, 25854247, 232557151, 2092490071, 18830313487, 169464432775, 1525146340543, 13726182847159, 123535108753519, 1111813831298023, 10006315891747615, 90056808665990167, 810511140554958031, 7294599715238808391

Offset: 0

Author

Wolfdieter Lang, Jul 16 2004

Keywords

• nonn
• easy

Comments

Sequence appears in A096952 (upper bounds for Lagrange remainder in Taylor expansion of log((1+x)/(1-x)) for x=1/3, i.e., for log(2)).

Divisibility of 2^(2*n+1) + 3^(2*n+1) by 5 is proved by induction.

The sequence a(n+1), with g.f. (7-36*x)/(1-13*x+36*x^2) and formula (27*9^n + 8*4^n)/5, is the Hankel transform of C(n) + 6*C(n+1), where C(n) is A000108(n). - Paul Barry, Dec 06 2006

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (13,-36).

Crossrefs

Cf. A074614 (sum of even powers of 2 and of 3), A007689 (sum of powers of 2 and powers of 3).

Cf. A000108, A015441, A096952, A138233.

Programs

Magma

[(2^(2*n+1) + 3^(2*n+1))/5: n in [0..30]]; // Vincenzo Librandi, May 31 2011

Mathematica

LinearRecurrence[{13, -36},{1, 7},19] (* Ray Chandler, Jul 14 2017 *)

Formula

a(n) = (2^(2*n+1) + 3^(2*n+1))/5.

G.f.: (1-6*x)/((1-4*x)*(1-9*x)).

From Reinhard Zumkeller, Mar 07 2008: (Start)

a(n+1) = 4*a(n) + 3^(2*n+1), a(0) = 1.

a(n) = A138233(n)/5. (End)

From Elmo R. Oliveira, Aug 02 2025: (Start)

E.g.f.: exp(4*x)*(2 + 3*exp(5*x))/5.

a(n) = 13*a(n-1) - 36*a(n-2).

a(n) = A015441(2*n+1). (End)