This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A185129 #32 Dec 29 2024 12:48:53
%S A185129 15,105,378,780,2145,1485,5460,7875,29403,21945,70125,105570,61425,
%T A185129 37950,255255,306153,61425,667590,749700,522753,1016025,353220,176715,
%U A185129 1471470,1445850,1747515,246753,794430,749700,514605,3499335,2953665,5073705,635628,8382465
%N A185129 a(n) = smaller member of n-th pair of distinct, positive, triangular numbers whose sum and difference are also triangular numbers.
%C A185129 See A185128 for further information.
%D A185129 Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966, p. 197, no. 8.
%H A185129 N. J. A. Sloane, <a href="/A185128/a185128.jpg">Annotated scan of Beiler's Table 81</a>, based on page 197 of Beiler's "Recreations in the Theory of Numbers: The Queen of Mathematics Entertains", New York, Dover, First ed., 1964.
%e A185129 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.
%t A185129 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 *)
%o A185129 (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
%Y A185129 Cf. A000217, A185128, A185223, A185233, A185243, A185253, A185257, A185258.
%K A185129 nonn
%O A185129 1,1
%A A185129 _Martin Renner_, Jan 20 2012
%E A185129 Edited by _N. J. A. Sloane_, Dec 28 2024