A179441 Number of solutions to a+b+c < d+e with each of a,b,c,d,e in {1..n+1}.

1, 21, 121, 432, 1182, 2723, 5558, 10368, 18039, 29689, 46695, 70720, 103740, 148071, 206396, 281792, 377757, 498237, 647653, 830928, 1053514, 1321419, 1641234, 2020160, 2466035, 2987361, 3593331, 4293856, 5099592, 6021967, 7073208, 8266368, 9615353, 11134949, 12840849

Offset: 1

Author

Bobby Milazzo, Jul 14 2010

Keywords

• nonn
• easy

Examples

			a(1) = 1 since from {1,2} there is only one solution: {1,1,1} for a,b,c and {2,2} for d,e.
a(2) = 21 since from {1,2,3} there are 21 ways to select a,b,c,d,e such that a+b+c < d+e.
		

References

• Mathematics and Computer Education 1988 - 89 #261 Unsolved.

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A197083.

Programs

Mathematica

k=10;
Table[p=Expand[Sum[x^k,{k,1,n}]^2 Sum[1/x^k,{k,1,n}]^3];
Twowins=Drop[CoefficientList[p,x],1]//Total,{n,2,k}]

PARI

a(n)=(27*n^5 + 80*n^4 + 65*n^3 - 20*n^2 - 32*n)/120 \\ Andrew Howroyd, Apr 15 2021

Formula

a(n) = (1/120)*(27*n^5 + 80*n^4 + 65*n^3 - 20*n^2 - 32*n).

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n > 6.

G.f.: x*(1 + 15*x + 10*x^2 + x^3)/(1 - x)^6.

Extensions

Name edited and terms a(24) and beyond from Andrew Howroyd, Apr 15 2021