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

1,7

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

Links

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

Example

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

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 *)

Crossrefs

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

Sequence in context: A162934 A303908 A351592 * A134673 A131636 A188023

Adjacent sequences: A082657 A082658 A082659 * A082661 A082662 A082663

Keyword

nonn

Author

Jason Earls, May 17 2003

Status

approved