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%I #46 Sep 08 2022 08:46:23
%S 0,0,4,14,32,64,108,174,256,368,500,670,864,1104,1372,1694,2048,2464,
%T 2916,3438,4000,4640,5324,6094,6912,7824,8788,9854,10976,12208,13500,
%U 14910,16384,17984,19652,21454,23328,25344,27436,29678,32000,34480,37044,39774
%N a(n) = (-2 - (-1)^n*(-2 + n) + n + 2*n^3)/4.
%C For n > 1, a(n) is the subdiagonal sum of the matrix M(n) whose permanent is A322277(n).
%C All the terms of this sequence are even numbers (A005843).
%H Stefano Spezia, <a href="/A323723/b323723.txt">Table of n, a(n) for n = 0..10000</a>
%H Christian Krause, <a href="https://github.com/ckrause/loda">LODA, an assembly language, a computational model and a tool for mining integer sequences</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-4,1,2,-1).
%F O.g.f.: 2*x^2*(2 + 3*x + x^3)/((1 - x)^4*(1 + x)^2).
%F E.g.f.: (1/4)*exp(-x)*(2 + x)*(1 + exp(2*x)*(-1 + 2*x + 2* x^2)).
%F a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) for n > 5.
%F a(n) = (6 + n + n^3 + 12*floor((n - 3)/2) + 4*floor((n - 3)/2)^2 - 2*(1 + n)*floor((n - 1)/2)/2.
%F a(n) = (-2 - A033999(n)*(-2 + n) + n + A033431(n))/4.
%F a(n) = n^3/2 for even n; a(n) = (n - 1)*(n^2 + n + 2)/2 otherwise. - _Bruno Berselli_, Feb 06 2019
%F a(n) = 2*A004526(n*A000982(n)). [Found by _Christian Krause_'s LODA miner] - _Stefano Spezia_, Dec 12 2021
%p a:=n->(-2 - (-1)^n*(-2 + n) + n + 2*n^3)/4: seq(a(n), n=0..50);
%t a[n_]:=(6 + n + n^3 + 12 Floor[1/2 (-3 + n)] + 4 Floor[1/2 (-3 + n)]^2 - 2 (1 + n) Floor[1/2 (-1 + n)])/2; Array[a,50,0]
%o (GAP) Flat(List([0..50], n -> (-2-(-1)^n*(-2+n)+n+2*n^3)/4));
%o (Magma) [(-2-(-1)^n*(-2+n)+n+2*n^3)/4: n in [0..50]];
%o (Maxima) makelist((-2-(-1)^n*(-2+n)+n+2*n^3)/4, n, 0, 50);
%o (PARI) a(n) = (-2-(-1)^n*(-2+n)+n+2*n^3)/4;
%o (Python) [(-2-(-1)**n*(-2+n)+n+2*n**3)/4 for n in range(50)]
%Y Cf. A000982, A004526, A005843, A033431, A033999, A317614, A322277, A323724, A325655.
%K nonn,easy
%O 0,3
%A _Stefano Spezia_, Jan 25 2019
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