cp's OEIS Frontend

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

%I A096951 #29 Aug 04 2025 01:10:23
%S A096951 1,7,55,463,4039,35839,320503,2876335,25854247,232557151,2092490071,
%T A096951 18830313487,169464432775,1525146340543,13726182847159,
%U A096951 123535108753519,1111813831298023,10006315891747615,90056808665990167,810511140554958031,7294599715238808391
%N A096951 Sum of odd powers of 2 and of 3 divided by 5.
%C A096951 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)).
%C A096951 Divisibility of 2^(2*n+1) + 3^(2*n+1) by 5 is proved by induction.
%C A096951 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
%H A096951 Vincenzo Librandi, <a href="/A096951/b096951.txt">Table of n, a(n) for n = 0..200</a>
%H A096951 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (13,-36).
%F A096951 a(n) = (2^(2*n+1) + 3^(2*n+1))/5.
%F A096951 G.f.: (1-6*x)/((1-4*x)*(1-9*x)).
%F A096951 From _Reinhard Zumkeller_, Mar 07 2008: (Start)
%F A096951 a(n+1) = 4*a(n) + 3^(2*n+1), a(0) = 1.
%F A096951 a(n) = A138233(n)/5. (End)
%F A096951 From _Elmo R. Oliveira_, Aug 02 2025: (Start)
%F A096951 E.g.f.: exp(4*x)*(2 + 3*exp(5*x))/5.
%F A096951 a(n) = 13*a(n-1) - 36*a(n-2).
%F A096951 a(n) = A015441(2*n+1). (End)
%t A096951 LinearRecurrence[{13, -36},{1, 7},19] (* _Ray Chandler_, Jul 14 2017 *)
%o A096951 (Magma) [(2^(2*n+1) + 3^(2*n+1))/5: n in [0..30]]; // _Vincenzo Librandi_, May 31 2011
%Y A096951 Cf. A074614 (sum of even powers of 2 and of 3), A007689 (sum of powers of 2 and powers of 3).
%Y A096951 Cf. A000108, A015441, A096952, A138233.
%K A096951 nonn,easy
%O A096951 0,2
%A A096951 _Wolfdieter Lang_, Jul 16 2004