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

0,2

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. (7-36x)/(1-13x+36x^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).

Formula

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

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

a(n+1) = 4*a(n) + 3^(2*n+1), a(0) = 1. - Reinhard Zumkeller, Mar 07 2008

Mathematica

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

Prog

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

Crossrefs

Cf. A074614 for sum of even powers of 2 and of 3. A007689 for sum of powers of 2 and powers of 3.

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

Sequence in context: A097189 A049028 A224274 * A113714 A246459 A152262

Adjacent sequences: A096948 A096949 A096950 * A096952 A096953 A096954

Keyword

nonn,easy

Author

Wolfdieter Lang, Jul 16 2004

Status

approved