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

1,2

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].

Links

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

Formula

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

Example

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

Prog

(AWK) # empirical
function a(n) { s=1; for(i=1; i<n; i++) { if(i%2==0)s+=2*int((i+5)/6); else s+=(i+2)+2*int((i+1)/6); } return s; }

Crossrefs

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

Sequence in context: A287565 A163417 A247336 * A096833 A153357 A310591

Adjacent sequences: A158135 A158136 A158137 * A158139 A158140 A158141

Keyword

nonn,changed

Author

R. H. Hardin, Mar 13 2009

Status

approved