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A235367 Sum of positive even numbers up to n^2.

%I A235367 #34 Sep 08 2022 08:46:06
%S A235367 0,6,20,72,156,342,600,1056,1640,2550,3660,5256,7140,9702,12656,16512,
%T A235367 20880,26406,32580,40200,48620,58806,69960,83232,97656,114582,132860,
%U A235367 154056,176820,202950,230880,262656,296480,334662,375156,420552,468540,522006,578360,640800,706440,778806
%N A235367 Sum of positive even numbers up to n^2.
%C A235367 Consider a square array of side n in which we write the integers from 1 to n in any order. This sequence gives the sum of the even numbers in the array.
%H A235367 Vincenzo Librandi, <a href="/A235367/b235367.txt">Table of n, a(n) for n = 1..1000</a>
%H A235367 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-6,0,6,-2,-2,1).
%F A235367 a(n) = (n^4 + 2n^2)/4 if n is even, a(n) = (n^4 - 1)/4 if n is odd.
%F A235367 a(n) = ((n^2 - (n^2 mod 2))/4)(n^2 + 2 - (n^2 mod 2)). - _Alonso del Arte_, Jan 16 2014
%F A235367 a(n) = A110660(n^2). - _Michel Marcus_, Jan 18 2014
%F A235367 G.f.: -2*x^2*(3*x^4+4*x^3+10*x^2+4*x+3) / ((x-1)^5*(x+1)^3). - _Colin Barker_, Jan 18 2014
%e A235367 a(1) = 0 because there are no even numbers between 1 and itself.
%e A235367 a(2) = 6 because between 1 and 2^2 there are the even numbers 2 and 4, which add up to 6.
%e A235367 a(3) = 20 because between 1 and 3^2 there are the even numbers 2, 4, 6 and 8, which add up to 20.
%t A235367 Table[((n^2 - Mod[n^2, 2])/4)(n^2 + 2 - Mod[n^2, 2]), {n, 40}] (* _Alonso del Arte_, Jan 16 2014 *)
%o A235367 (PARI) a(n) = sum(i=1, n, i^2*(!(i % 2))); \\ _Michel Marcus_, Jan 18 2014
%o A235367 (Magma) [&+[i: i in [0..n^2 by 2]]: n in [1..50]]; // _Bruno Berselli_, Oct 26 2018
%Y A235367 Cf. A002620, A083374.
%K A235367 nonn,easy
%O A235367 1,2
%A A235367 _Réjean Labrie_, Jan 07 2014
%E A235367 Corrected by _Vincenzo Librandi_, Jan 18 2014