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

1,1

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

Links

Table of n, a(n) for n=1..52.

Example

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

Prog

(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

Crossrefs

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

Sequence in context: A059390 A179243 A350686 * A162918 A105018 A154488

Adjacent sequences: A064822 A064823 A064824 * A064826 A064827 A064828

Keyword

nonn

Author

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

Extensions

Offset changed to 1 by Michel Marcus, Jan 14 2014

Status

approved