A053307 Number of nonnegative integer 2 X 2 matrices with sum of elements equal to n, under row and column permutations.

1, 1, 4, 5, 11, 14, 24, 30, 45, 55, 76, 91, 119, 140, 176, 204, 249, 285, 340, 385, 451, 506, 584, 650, 741, 819, 924, 1015, 1135, 1240, 1376, 1496, 1649, 1785, 1956, 2109, 2299, 2470, 2680, 2870, 3101, 3311, 3564, 3795, 4071, 4324, 4624, 4900, 5225, 5525

Offset: 0

Author

Vladeta Jovovic, Mar 05 2000

Keywords

• easy
• nonn

Comments

An interleaved sequence of pyramidal and polygonal numbers: a(2n)= A006527(n+1), a(2n+1)=A000330(n+1) - Paul Barry, Mar 17 2003

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

These are also the "spreading numbers" alpha_4(n). See Babcock et al. for precise definition.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• B. Babcock and A. van Tuyl, Revisiting the spreading and covering numbers, arXiv preprint arXiv:1109.5847 [math.AC], 2011-2013.
• John Machacek, Unique maximum independent sets in graphs on monomials of a fixed degree, arXiv:2010.11112 [math.CO], 2020.
• Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Crossrefs

Row 2 of A318795.

Row 4 of A202175.

Cf. A081283, A081284.

Programs

Magma

[(n+2)*(2*n^2 + 8*n + 15 + 9*(-1)^n)/48: n in [0..30]]; // G. C. Greubel, May 31 2018

Mathematica

Table[(n+2)*(2*n^2 + 8*n + 15 + 9*(-1)^n)/48,{n,0,20}] (* Vaclav Kotesovec, Mar 16 2014 *)

PARI

for(n=0,30, print1((n+2)*(2*n^2 + 8*n + 15 + 9*(-1)^n)/48, ", ")) \\ G. C. Greubel, May 31 2018

Formula

G.f.: (x^2-x+1)/((1-x^2)^2*(1-x)^2).

a(n) = (n+2)*(2*n^2 + 8*n + 15 + 9*(-1)^n)/48. - Vaclav Kotesovec, Mar 16 2014