login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

a(n) = (6^n - 2^n)^2 / 16.
3

%I #26 Oct 04 2024 20:51:38

%S 1,64,2704,102400,3748096,135675904,4893282304,176265625600,

%T 6346852335616,228502162898944,8226263614357504,296147719133593600,

%U 10661344637077159936,383808727914259677184,13817118056668205154304,497416296261117961830400,17906987220053014721069056

%N a(n) = (6^n - 2^n)^2 / 16.

%H G. C. Greubel, <a href="/A144843/b144843.txt">Table of n, a(n) for n = 1..630</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (52,-624,1728).

%F From _R. J. Mathar_, Sep 24 2008: (Start)

%F a(n) = 81*36^(n-2) + 4^(n-2) - 18*12^(n-2).

%F G.f.: x*(1+12*x)/((1-4*x)*(1-12*x)*(1-36*x)). (End)

%F a(n) = A016129(n-1)^2. - _Philippe Deléham_, Nov 26 2008

%F a(n) = 4^(n-2) * (3^n - 1)^2. - _Harvey P. Dale_, Apr 15 2020

%F E.g.f.: (1/16)*exp(4*x)*(1 - 2*exp(8*x) + exp(32*x)). - _G. C. Greubel_, Oct 03 2024

%t Table[(6^n-2^n)^2/16, {n,20}] (* _Harvey P. Dale_, Apr 15 2020 *)

%o (Magma) [4^(n-2)*(3^n-1)^2: n in [1..30]]; // _G. C. Greubel_, Oct 03 2024

%o (SageMath) [4^(n-2)*(3^n-1)^2 for n in range(1,31)] # _G. C. Greubel_, Oct 03 2024

%Y Cf. A016129.

%K nonn,easy

%O 1,2

%A Al Hakanson (hawkuu(AT)gmail.com), Sep 22 2008

%E More terms from _R. J. Mathar_, Sep 24 2008