A081283 -id:A081283 - cp's OEIS frontend

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

Vladeta Jovovic, Mar 05 2000

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.

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).

Row 2 of A318795.
Row 4 of A202175.
Cf. A081283, A081284.

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

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

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

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

A081284 An interleaved sequence of pyramidal and polygonal numbers.

Original entry on oeis.org

0, 1, 1, 6, 7, 22, 27, 62, 77, 147, 182, 308, 378, 588, 714, 1044, 1254, 1749, 2079, 2794, 3289, 4290, 5005, 6370, 7371, 9191, 10556, 12936, 14756, 17816, 20196, 24072, 27132, 31977, 35853, 41838, 46683, 53998, 59983, 68838, 76153, 86779, 95634

Offset: 0

Paul Barry, Mar 17 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,3,-8,-2,12,-2,-8,3,2,-1).

Cf. A081283, A081282.

CoefficientList[Series[x (1 + x^3) / ((1 - x) (1 - x^2)^5), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 07 2013 *)

G.f.: x*(1+x^3)/((1-x)*(1-x^2)^5).
a(2*n) = A005585(n); a(2*n+1) = A081282(n+1).
G.f.: (x*(x^2-x+1))/((x+1)^4*(x-1)^6). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 28 2009
a(n) = (2*n+1-(-1)^n)*(2*n+5-(-1)^n)*(2*n+9-(-1)^n)*(2*n^2+15*n+51+3*(n-5)*(-1)^n)/30720. - Luce ETIENNE, Mar 13 2015

cp's OEIS Frontend

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

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