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

%I A082660 #14 Apr 28 2020 10:43:24
%S A082660 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,
%T A082660 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,
%U A082660 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
%N A082660 Number of ways n can be expressed as the sum of a square and a triangular number.
%C A082660 It is assumed here that 0 is a square but not a triangular number. - _Amiram Eldar_, Dec 08 2019
%C A082660 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
%H A082660 Amiram Eldar, <a href="/A082660/b082660.txt">Table of n, a(n) for n = 1..10000</a>
%e A082660 a(631) = 8 because:
%e A082660 1. 631 = 6 + 625
%e A082660 2. 631 = 55 + 576
%e A082660 3. 631 = 190 + 441
%e A082660 4. 631 = 231 + 400
%e A082660 5. 631 = 406 + 225
%e A082660 6. 631 = 435 + 196
%e A082660 7. 631 = 595 + 36
%e A082660 8. 631 = 630 + 1
%p A082660 A082660 := proc(n)
%p A082660     local a,tidx,t;
%p A082660     a := 0 ;
%p A082660     for tidx from 1 do
%p A082660         t := A000217(tidx) ;
%p A082660         if t > n then
%p A082660             break;
%p A082660         end if;
%p A082660         if issqr(n-t) then
%p A082660             a := a+1 ;
%p A082660         end if;
%p A082660     end do:
%p A082660     a ;
%p A082660 end proc: # _R. J. Mathar_, Apr 28 2020
%t A082660 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 *)
%Y A082660 Cf. A000217, A000290, A082657, A082658, A082659.
%K A082660 nonn
%O A082660 1,7
%A A082660 _Jason Earls_, May 17 2003