A158138 Number of nondecreasing integer sequences of length 4 with sum zero and sum of absolute values 2n.

1, 4, 6, 11, 13, 22, 24, 35, 39, 52, 56, 73, 77, 96, 102, 123, 129, 154, 160, 187, 195, 224, 232, 265, 273, 308, 318, 355, 365, 406, 416, 459, 471, 516, 528, 577, 589, 640, 654, 707, 721, 778, 792, 851, 867, 928, 944, 1009, 1025, 1092, 1110, 1179, 1197, 1270, 1288

Offset: 1

Author

R. H. Hardin, Mar 13 2009

Keywords

• nonn

Comments

a(n) = A000041(n)^2 for n<=2

a(n) = A000041(n)^2 - cumulative A000712(2*n-1-length), 0 <= 2*n-1-length <= floor(n/2) [empirical].

Examples

			For n = 6, we count the possible concatenations of the 4 pairs in the list (-6,0),(-5,-1),(-4,-2),(-3,-3) with their negative reversed correspondants (starting with (-6,0,0,6)), giving (6/2 + 1)^2 = 16 quadruples, plus the 3 quadruples (-6,1,1,4), (-6,1,2,3), (-6,2,2,2) and their 3 negative reversed correspondants, giving a total of 22 possibilities. - _Georg Fischer_, Apr 20 2022
		

Links

• R. H. Hardin, Table of n, a(n) for n=1..999

Crossrefs

Cf. A069905, A158139-A158184 (for length 5..50).

Programs

AWK

# empirical
function a(n) { s=1; for(i=1; i
				

Formula

a(n) = (floor(n/2) + 1)^2 + 2*A069905(n). - Georg Fischer, Apr 20 2022