

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
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OFFSET

0,2


COMMENTS

Sequence appears in A096952 (upper bounds for Lagrange remainder in Taylor expansion of log((1+x)/(1x)) 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. (736x)/(113x+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.: (16*x)/((14*x)*(19*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



