A185129 a(n) = smaller member of n-th pair of distinct, positive, triangular numbers whose sum and difference are also triangular numbers.

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

Comments

See A185128 for further information.

References

Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966, p. 197, no. 8.

Links

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

N. J. A. Sloane, Annotated scan of Beiler's Table 81, based on page 197 of Beiler's "Recreations in the Theory of Numbers: The Queen of Mathematics Entertains", New York, Dover, First ed., 1964.

Example

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

Extensions

Edited by N. J. A. Sloane, Dec 28 2024

Status

approved