A061262 Smallest number representable as the sum of 3 triangular numbers in exactly n ways.

0, 3, 12, 21, 52, 57, 91, 121, 136, 211, 192, 226, 409, 331, 367, 406, 511, 507, 886, 637, 772, 721, 871, 952, 1102, 1066, 1227, 1192, 1641, 1621, 1396, 1381, 1501, 1732, 1792, 1927, 1942, 2401, 2611, 2551, 2422, 2557, 2887, 2821, 3136, 3271, 3607, 3376

Offset: 1

Author

Ed Pegg Jr, Apr 24 2001

Keywords

• easy
• nice
• nonn

Comments

Fermat claimed, Euler tried, Gauss proved (July 10, 1796) that every number can be represented as a sum of three triangular numbers. I'm considering 0 as a triangular number here. If at first you do not succeed, tri + tri + tri again.

Conjecture: for n large enough, 1 < a(n)/n^2 < 2. - Benoit Cloitre, May 10 2003

Conjecture: No term a(n) with n > 2 is congruent to 0 or 3 modulo 5. - Zhi-Wei Sun, Feb 25 2015

Examples

			57 is the smallest number that can be represented by exactly 6 different triangular triple sums: {6, 6, 5}, {7, 7, 1}, {8, 5, 3}, {8, 6, 0}, {9, 3, 3}, {10, 1, 1}.
		

Links

• T. D. Noe, Table of n, a(n) for n=1..1000

Crossrefs

Cf. A000217, A053614, A060773, A002636.

Cf. A124978.

Programs

Mathematica

a = Table[ n(n + 1)/2, {n, 0, 85} ]; b = {0}; c = Table[0, {3655} ]; Do[ b = Append[b, a[[i] ] + a[[j]] + a[[k]]], {k, 1, 85}, {j, 1, k}, {i, 1, j} ]; b = Delete[b, 1]; b = Sort[b]; l = Length[b]; Do[ If[b[[n]] < 3655, c[[b[[n]] + 1]]++ ], {n, 1, l} ]; Do[ k = 1; While[ c[[k]] != n, k++ ]; Print[k - 1], {n, 1, 48} ]