A112352 -id:A112352 - cp's OEIS frontend

A112353 Triangular numbers that are the sum of three distinct positive triangular numbers.

10, 28, 45, 55, 66, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275, 1326, 1378, 1431, 1485

Offset: 1

Rick L. Shepherd, Sep 05 2005

nonn

Subsequence of A112355: it doesn't require the three positive triangular numbers to be distinct.

			45 is a term because 45 = 3 + 6 + 36 and these four numbers are distinct triangular numbers (A000217(9) = A000217(2) + A000217(3) + A000217(8)).

Cf. A000217 (triangular numbers), A112352 (triangular numbers that are the sum of two distinct positive triangular numbers), A112355.

trnos=Accumulate[Range[200]];
Take[Union[Select[Total/@Subsets[trnos,{3}],MemberQ[trnos,#]&]],50]  (* Harvey P. Dale, Jan 15 2011 *)

A350367 Triangular numbers that are the sum of two distinct nonzero triangular numbers in more than one way.

Original entry on oeis.org

231, 276, 406, 666, 861, 1081, 1225, 1431, 1711, 1891, 2211, 2556, 3081, 3741, 3916, 4186, 4371, 4560, 4656, 5151, 5356, 5671, 5886, 6786, 7021, 7381, 7875, 8001, 8128, 8256, 8778, 9316, 10731, 11781, 12246, 12561, 12720, 13366, 13861, 14196, 14706, 15576

Offset: 1

Shyam Sunder Gupta, Dec 27 2021

nonn

			231 = 21 + 210 = 78 + 153.
276 = 45 + 231 = 66 + 210 = 105 + 171.

Intersection of A000217 and A262749.

Cf. A089982, A112352.

(P=PolygonalNumber)[3,Select[Range@176,Length@Select[Subsets[P[3,Range[s=#]],{2}],Total@#==P[3,s]&]>1&]] (* Giorgos Kalogeropoulos, Dec 31 2021 *)

from collections import Counter
from itertools import count, takewhile, combinations as combs
def aupto(limit):
    tris = takewhile(lambda x: x <= limit, (k*(k+1)//2 for k in count(1)))
    trilst = list(tris); triset = set(trilst)
    tri2ct = Counter(sum(c) for c in combs(trilst, 2) if sum(c) in triset)
    return sorted(t for t in tri2ct if t <= limit and tri2ct[t] > 1)
print(aupto(16000)) # Michael S. Branicky, Dec 27 2021

cp's OEIS Frontend

A112353 Triangular numbers that are the sum of three distinct positive triangular numbers.

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