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%I #39 Sep 08 2022 08:45:00
%S 1,1,4,5,11,14,24,30,45,55,76,91,119,140,176,204,249,285,340,385,451,
%T 506,584,650,741,819,924,1015,1135,1240,1376,1496,1649,1785,1956,2109,
%U 2299,2470,2680,2870,3101,3311,3564,3795,4071,4324,4624,4900,5225,5525
%N Number of nonnegative integer 2 X 2 matrices with sum of elements equal to n, under row and column permutations.
%C An interleaved sequence of pyramidal and polygonal numbers: a(2n)= A006527(n+1), a(2n+1)=A000330(n+1) - _Paul Barry_, Mar 17 2003
%C a(n) is also the number of solutions to the equation XOR(x1, x2, ..., xn) = 0 such that each xi is a 2-bit binary number and xi >= xj for i >= j. For example, a(2) = 4 since (x1, x2) = { (00, 00), (01, 01), (10, 10), (11, 11) }. - _Ramasamy Chandramouli_, Jan 17 2009
%C These are also the "spreading numbers" alpha_4(n). See Babcock et al. for precise definition.
%H G. C. Greubel, <a href="/A053307/b053307.txt">Table of n, a(n) for n = 0..5000</a>
%H B. Babcock and A. van Tuyl, <a href="http://arxiv.org/abs/1109.5847">Revisiting the spreading and covering numbers</a>, arXiv preprint arXiv:1109.5847 [math.AC], 2011-2013.
%H John Machacek, <a href="https://arxiv.org/abs/2010.11112">Unique maximum independent sets in graphs on monomials of a fixed degree</a>, arXiv:2010.11112 [math.CO], 2020.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-4,1,2,-1).
%F G.f.: (x^2-x+1)/((1-x^2)^2*(1-x)^2).
%F a(n) = (n+2)*(2*n^2 + 8*n + 15 + 9*(-1)^n)/48. - _Vaclav Kotesovec_, Mar 16 2014
%t Table[(n+2)*(2*n^2 + 8*n + 15 + 9*(-1)^n)/48,{n,0,20}] (* _Vaclav Kotesovec_, Mar 16 2014 *)
%o (PARI) for(n=0,30, print1((n+2)*(2*n^2 + 8*n + 15 + 9*(-1)^n)/48, ", ")) \\ _G. C. Greubel_, May 31 2018
%o (Magma) [(n+2)*(2*n^2 + 8*n + 15 + 9*(-1)^n)/48: n in [0..30]]; // _G. C. Greubel_, May 31 2018
%Y Row 2 of A318795.
%Y Row 4 of A202175.
%Y Cf. A081283, A081284.
%K easy,nonn
%O 0,3
%A _Vladeta Jovovic_, Mar 05 2000
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