A064825 Numbers which are the sums of three positive triangular numbers in exactly two different ways.

12, 17, 19, 21, 22, 23, 26, 27, 28, 31, 32, 33, 34, 35, 39, 41, 42, 43, 44, 46, 47, 51, 54, 56, 62, 64, 65, 68, 74, 80, 88, 89, 90, 92, 95, 106, 113, 123, 128, 137, 141, 146, 153, 164, 179, 194, 200, 218, 245, 281, 326, 335

Offset: 1

Author

Jonathan Ayres (jonathan.ayres(AT)btinternet.com), Oct 23 2001

Keywords

• nonn

Comments

Checked up to 1000000 but haven't found any other values.

A063993(a(n)) = 2. - Reinhard Zumkeller, Jul 20 2012

Without the qualifier "positive" in the Name, sequence A071530 would result. - Jon E. Schoenfield, Jan 01 2020

No further terms <= 3*10^7. - Michael S. Branicky, Dec 17 2021

Examples

			19 = 10 + 6 + 3 = 15 + 3 + 1.
		

Crossrefs

Cf. A063993, A064816, A002097, A071530, A111638.

Programs

Magma

[k:k in [1..350]|#RestrictedPartitions(k, 3, {m*(m+1) div 2:m in [1..200]}) eq 2 ]; // Marius A. Burtea, Jan 01 2020

Python

from collections import Counter
from itertools import count, takewhile, combinations_with_replacement as mc
def aupto(N):
    tris = takewhile(lambda x: x <= N, (i*(i+1)//2 for i in count(1)))
    sum3 = filter(lambda x: x <= N, (sum(c) for c in mc(tris, 3)))
    sum3counts = Counter(sum3)
    return sorted(k for k in sum3counts if sum3counts[k] == 2)
print(aupto(1000)) # Michael S. Branicky, Dec 17 2021

Extensions

Offset changed to 1 by Michel Marcus, Jan 14 2014