A226389 Triangular numbers representable as triangular(x)*triangular(y)+1.

1, 10, 91, 136, 946, 1711, 1891, 5671, 8911, 10585, 11026, 11935, 13861, 19306, 21736, 26335, 32131, 36856, 44551, 49141, 65341, 107416, 138601, 239086, 305371, 351541, 366796, 459361, 849556, 873181, 933661, 1100386, 1413721, 1516411, 1904176, 2297296, 2467531, 3837835

Offset: 1

Author

Alex Ratushnyak, Jun 06 2013

Keywords

• nonn

Comments

Cases with x=y are included.

The associated indices in A000217 are 1, 4, 13, 16, 43, 58, 61, 106, 133, 145, 148, 154,...

Examples

			10 = 3 * 3 + 1,
91 = 6 * 15 + 1,
3837835 = 378 * 10153 + 1.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000217, A188630, A225390.

Programs

Maple

A000217inv:=proc(n) local t1; t1:=floor(sqrt(2*n)); if n = t1*(t1+1)/2 then return t1 ; else return -1; end if; end proc:
A000217 := proc(n)
    n*(n+1)/2 ;
end proc:
isA226389 := proc(n)
    local Tx, Ty;
    if n = 1 then
        return true;
    elif A000217inv(n) >= 0 then
        for x from 0 do
            Tx := A000217(x) ;
            if Tx+1 > n then
                return false;
            end if;
            for y from 0 to x do
                Ty := A000217(y) ;
                if Tx*Ty+1 >n then
                    break;
                elif Tx*Ty+1 = n then
                    return true;
                end if;
            end do:
        end do:
    else
        false;
    end if;
end proc:
for n from 0 do
    Tn := A000217(n) ;
    if isA226389(Tn) then
        printf("%d,\n",Tn) ;
    end if;
end do: # R. J. Mathar, Jun 06 2013