%I #41 Sep 08 2022 08:44:41

%S 1,81,625,2401,6561,14641,28561,50625,83521,130321,194481,279841,

%T 390625,531441,707281,923521,1185921,1500625,1874161,2313441,2825761,

%U 3418801,4100625,4879681,5764801,6765201,7890481,9150625,10556001,12117361,13845841,15752961,17850625

%N a(n) = (2*n+1)^4.

%C a(n) is the number of ordered pairs of lattice points (vectors in R^2 with integer coordinates) that are in or on a square centered at the origin with side length 2*n. - _Geoffrey Critzer_, Apr 20 2013

%H Vincenzo Librandi, <a href="/A016756/b016756.txt">Table of n, a(n) for n = 0..10000</a>

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

%F From _Wolfdieter Lang_, Mar 12 2017: (Start)

%F G.f.: (1+76*x+230*x^2+76*x^3+x^4)/(1-x)^5; see row n=5 of A060187.

%F E.g.f.: (1 + 80*x + 232*x^2 + 128*x^3 + 16*x^4)*exp(x); see row n=4 of A154537. (End)

%F Sum_{n>=0} 1/a(n) = Pi^4/96 (A300707). - _Amiram Eldar_, Oct 10 2020

%F From _Amiram Eldar_, Jan 28 2021: (Start)

%F Product_{n>=0} (1 + 1/a(n)) = (cos(Pi/sqrt(2)) + cosh(Pi/sqrt(2)))/2.

%F Product_{n>=1} (1 - 1/a(n)) = Pi*cosh(Pi/2)/8. (End)

%e a(1) = 81 because there are 9 lattice points in or on the 2 x 2 square centered at the origin, so there are 9*9 =81 ordered pairs. - _Geoffrey Critzer_, Apr 20 2013

%t Table[(2n+1)^4,{n,0,25}] (* _Geoffrey Critzer_, Apr 20 2013 *)

%t LinearRecurrence[{5,-10,10,-5,1},{1,81,625,2401,6561},30] (* _Harvey P. Dale_, Mar 24 2020 *)

%o (Magma) [(2*n+1)^4: n in [0..40]]; // _Vincenzo Librandi_, Sep 07 2011

%o (PARI) vector(40, n, n--; (2*n+1)^4) \\ _G. C. Greubel_, Sep 15 2018

%Y Cf. A016755, A060187, A154537, A300707.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_