A082660 Number of ways n can be expressed as the sum of a square and a triangular number.

1, 1, 1, 1, 1, 1, 2, 0, 0, 3, 1, 1, 0, 1, 2, 1, 1, 0, 3, 0, 1, 2, 0, 1, 1, 2, 0, 2, 1, 1, 2, 1, 0, 0, 1, 1, 4, 0, 1, 2, 0, 1, 0, 1, 2, 3, 0, 0, 1, 1, 1, 2, 1, 1, 2, 1, 1, 0, 2, 0, 2, 0, 0, 3, 1, 1, 2, 0, 0, 4, 1, 1, 0, 1, 1, 0, 1, 1, 2, 1, 1, 3, 0, 1, 2, 0, 2, 0, 0, 0, 4, 2, 0, 2, 1, 1, 0, 0, 0, 2, 1, 2, 2, 1, 1

Offset: 1

Author

Jason Earls, May 17 2003

Keywords

• nonn

Comments

It is assumed here that 0 is a square but not a triangular number. - Amiram Eldar, Dec 08 2019

The greedy inverse (positions of the first occurrence of n) is 1, 7, 10, 37, 136, 235, 1225, 631, 2116, 4789, 11026, 3997, 148240, 19045, 20827, 25876, ... - R. J. Mathar, Apr 28 2020

Examples

			a(631) = 8 because:
1. 631 = 6 + 625
2. 631 = 55 + 576
3. 631 = 190 + 441
4. 631 = 231 + 400
5. 631 = 406 + 225
6. 631 = 435 + 196
7. 631 = 595 + 36
8. 631 = 630 + 1
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000217, A000290, A082657, A082658, A082659.

Programs

Maple

A082660 := proc(n)
    local a,tidx,t;
    a := 0 ;
    for tidx from 1 do
        t := A000217(tidx) ;
        if t > n then
            break;
        end if;
        if issqr(n-t) then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Apr 28 2020

Mathematica

a[n_] := Length @ Solve[x^2 + y (y + 1)/2 == n && x >= 0 && y > 0, {x, y}, Integers]; Array[a, 100] (* Amiram Eldar, Dec 08 2019 *)