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A096951 Sum of odd powers of 2 and of 3 divided by 5. 7
1, 7, 55, 463, 4039, 35839, 320503, 2876335, 25854247, 232557151, 2092490071, 18830313487, 169464432775, 1525146340543, 13726182847159, 123535108753519, 1111813831298023, 10006315891747615, 90056808665990167 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 22 18:50 EDT 2019. Contains 328319 sequences. (Running on oeis4.)