%I #26 Mar 29 2024 02:22:49
%S 0,1,10,25,68,117,222,325,520,697,1010,1281,1740,2125,2758,3277,4112,
%T 4785,5850,6697,8020,9061,10670,11925,13848,15337,17602,19345,21980,
%U 23997,27030,29341,32800,35425,39338,42297,46692,50005,54910,58597,64040,68121,74130
%N a(n) = n*(n^2 + 1) if n is even, otherwise (n - 1/2)*(n^2 + 1).
%C Sum of the numbers along the diagonals of an n X n square array whose elements are the numbers from 1..n^2, listed in increasing order by rows (see example). - _Wesley Ivan Hurt_, May 15 2021
%D T. A. Gulliver, Sequences from Arrays of Integers, Int. Math. Journal, Vol. 1, No. 4, pp. 323-332, 2002.
%H G. C. Greubel, <a href="/A071289/b071289.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-3,-3,3,1,-1).
%F From _R. J. Mathar_, Oct 19 2010: (Start)
%F a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7).
%F G.f.: x*(1 + 9*x + 12*x^2 + 16*x^3 + 7*x^4 + 3*x^5)/((1+x)^3*(1-x)^4). (End)
%F a(n) = (n^2+1)*(4*n-1+(-1)^n)/4. - _Wesley Ivan Hurt_, May 15 2021
%F E.g.f.: (1/4)*( (1 - x + x^2)*exp(-x) + (-1 + 7*x + 11*x^2 + 4*x^3)*exp(x) ). - _G. C. Greubel_, Mar 29 2024
%e From _Wesley Ivan Hurt_, May 15 2021: (Start)
%e [1 2 3 4 5]
%e [1 2 3 4] [6 7 8 9 10]
%e [1 2 3] [5 6 7 8] [11 12 13 14 15]
%e [1 2] [4 5 6] [9 10 11 12] [16 17 18 19 20]
%e [1] [3 4] [7 8 9] [13 14 15 16] [21 22 23 24 25]
%e ------------------------------------------------------------------------
%e n 1 2 3 4 5
%e ------------------------------------------------------------------------
%e a(n) 1 10 25 68 117
%e ------------------------------------------------------------------------
%e (End)
%t Array[If[EvenQ[#],#(#^2+1),(#-1/2)(#^2+1)]&,50,0] (* _Harvey P. Dale_, Sep 20 2013 *)
%o (Magma) [(n^2+1)*(4*n-1+(-1)^n)/4: n in [0..50]]; // _G. C. Greubel_, Mar 29 2024
%o (SageMath) [(n^2+1)*(2*n-(n%2))//2 for n in range(51)] # _G. C. Greubel_, Mar 29 2024
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_, Jun 12 2002