%I #49 Apr 01 2021 06:01:48

%S 0,4,64,324,1024,2500,5184,9604,16384,26244,40000,58564,82944,114244,

%T 153664,202500,262144,334084,419904,521284,640000,777924,937024,

%U 1119364,1327104,1562500,1827904,2125764,2458624,2829124,3240000,3694084,4194304,4743684,5345344

%N a(n) = 4*n^4.

%C Nonnegative integers a(n) such that (-a(n))^(1/4) is a Gaussian integer, since (n + n*i)^4 = -4*n^4

%C For n > 1, a(n) + k^4 is not prime for any k. - _Derek Orr_, May 31 2014

%C Suppose the vertices of a triangle are (T(n), T(n+j)), (T(n+2*j), T(n+3*j)) and (T(n+4*j), T(n+5*j)) where T(n) is the n-th triangular number. Then the area of this triangle will be a(j). - _Charlie Marion_, Mar 06 2021

%H G. C. Greubel, <a href="/A141046/b141046.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%F a(n) = 4*n^4.

%F a(n) = A008586(A000583(n)) = A000290(A005843(A000290(n))). - _Reinhard Zumkeller_, Jan 25 2012

%F G.f.: 4*x*(1 + x)*(1 + 10*x + x^2)/(1 - x)^5. - _Chai Wah Wu_, Jun 22 2016

%F From _G. C. Greubel_, Jun 22 2016: (Start)

%F a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).

%F E.g.f.: 4*x*(1 + 7*x + 6*x^2 + x^3)*exp(x). (End)

%F a(n) = A001105(n)^2. - _Bruce J. Nicholson_, Apr 03 2017

%F From _Amiram Eldar_, Jan 29 2021: (Start)

%F Sum_{n>=1} 1/a(n) = Pi^4/360.

%F Sum_{n>=1} (-1)^(n+1)/a(n) = 7*Pi^4/2880.

%F Product_{n>=1} (1 + 1/a(n)) = 2*cosh(Pi/2)^2/Pi^2.

%F Product_{n>=1} (1 - 1/a(n)) = 2*sin(Pi/sqrt(2))*sinh(Pi/sqrt(2))/Pi^2. (End)

%t Table[4 n^4, {n, 0, 20}]

%o (Haskell)

%o a141046 = (* 4) . (^ 4) -- _Reinhard Zumkeller_, Jan 25 2012

%o (PARI) a(n)=4*n^4 \\ _Charles R Greathouse IV_, Jan 26 2012

%Y Cf. A000290, A000583, A001105, A005843, A008586.

%K easy,nonn

%O 0,2

%A _Fredrik Johansson_, Jul 31 2008