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

0,3

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).

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

Mathematica

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

Prog

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

Crossrefs

Row 2 of A318795.

Row 4 of A202175.

Cf. A081283, A081284.

Sequence in context: A050018 A347513 A125577 * A076065 A176115 A066898

Adjacent sequences: A053304 A053305 A053306 * A053308 A053309 A053310

Keyword

easy,nonn

Author

Vladeta Jovovic, Mar 05 2000

Status

approved