A185129 Smaller member of a pair of triangular numbers whose sum and difference are triangular.

15, 105, 378, 780, 2145, 1485, 5460, 7875, 29403, 21945, 70125, 105570, 61425, 37950, 255255, 306153, 61425, 667590, 749700, 522753, 1016025, 353220, 176715, 1471470, 1445850, 1747515, 246753, 794430, 749700, 514605, 3499335, 2953665, 5073705, 635628, 8382465

Offset

1,1

References

Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966, p. 197, nr. 8.

Links

Table of n, a(n) for n=1..35.

Example

a(2) = 105, since the pair of triangular numbers 171 = 18*(18+1)/2 and 105 = 14*(14+1)/2 produce the sum 276 = 23*(23+1)/2 and the difference 66 = 11*(11+1)/2 which are both triangular numbers.

Mathematica

kmax=2000; TriangularQ[n_]:=IntegerQ[(Sqrt[1+8n]-1)/2]; A000217[n_]:=n(n+1)/2; a={}; For[k=1, k<=kmax, k++, For[h=1, A000217[h]<A000217[k], h++, If[TriangularQ[A000217[k] - A000217[h]] && TriangularQ[A000217[k]+A000217[h]], AppendTo[a, A000217[h]]]]]; a (* Stefano Spezia, Sep 02 2024 *)

Prog

(PARI) lista(n) = {v = vector(nn, n, n*(n+1)/2); for (n=2, nn, for (k=1, n-1, if (ispolygonal(v[n]+v[k], 3) && ispolygonal(v[n]-v[k], 3), print1(v[k], ", ")); ); ); } \\ Michel Marcus, Jan 08 2015

Crossrefs

Cf. A000217, A185128, A185223, A185233, A185243, A185253, A185257, A185258.

Sequence in context: A160892 A061550 A174385 * A090454 A240440 A256287

Adjacent sequences: A185126 A185127 A185128 * A185130 A185131 A185132

Keyword

nonn

Author

Martin Renner, Jan 20 2012

Status

approved