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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, 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
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
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Jul 16 2004
STATUS
approved