A076465 Sum of squares of numbers that can be written as t*n + u*(n+1) for nonnegative integers t,u in exactly n ways.

1, 571, 12938, 115270, 630755, 2543401, 8307796, 23249388, 57792165, 130790935, 274285726, 540036146, 1008233863, 1798831685, 3085968040, 5116005976, 8229746121, 12889413363, 19711057330, 29503047070, 43311380651, 62472570721, 88674907388, 124028940100

Offset: 1

Author

Floor van Lamoen, Oct 13 2002

Keywords

• easy
• nonn

References

• Fred. Schuh, Vragen betreffende een onbepaalde vergelijking, Nieuw Tijdschrift voor Wiskunde, 52 (1964-1965) 193-198.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Crossrefs

Cf. A076389, A076459-A076464.

Programs

Magma

[n*(n+1)*(6*n^6+12*n^5-5*n^4-16*n^3+5*n+1)/6: n in [1..50]]; // Vincenzo Librandi, Dec 30 2013

Maple

seq(1/6*n*(n+1)*(6*n^6+12*n^5-5*n^4-16*n^3+5*n+1),n=1..25);

Mathematica

CoefficientList[Series[(1 + 562 x + 7835 x^2 + 19300 x^3 + 11255 x^4 + 1354 x^5 + 13 x^6)/(1 - x)^9, {x, 0, 50}], x] (* Vincenzo Librandi, Dec 30 2013 *)
LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,571,12938,115270,630755,2543401,8307796,23249388,57792165},30] (* Harvey P. Dale, Sep 05 2015 *)

Formula

a(n) = n*(n+1)*(6*n^6+12*n^5-5*n^4-16*n^3+5*n+1)/6.

G.f.: x*(1+562*x+7835*x^2+19300*x^3+11255*x^4+1354*x^5+13*x^6)/(1-x)^9.

a(1)=1, a(2)=571, a(3)=12938, a(4)=115270, a(5)=630755, a(6)=2543401, a(7)=8307796, a(8)=23249388, a(9)=57792165, a(n)=9*a(n-1)- 36*a(n-2)+ 84*a(n-3)-126*a(n-4)+126*a(n-5)-84*a(n-6)+36*a(n-7)-9*a(n-8)+a(n-9). - Harvey P. Dale, Sep 05 2015

Extensions

More terms from Vincenzo Librandi, Dec 30 2013