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
Keywords
Examples
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.
References
- Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966, p. 197, no. 8.
Links
- 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.
Programs
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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 *)
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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
Extensions
Edited by N. J. A. Sloane, Dec 28 2024
Comments