cp's OEIS Frontend

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

%I A064825 #24 Sep 08 2022 08:45:04
%S A064825 12,17,19,21,22,23,26,27,28,31,32,33,34,35,39,41,42,43,44,46,47,51,54,
%T A064825 56,62,64,65,68,74,80,88,89,90,92,95,106,113,123,128,137,141,146,153,
%U A064825 164,179,194,200,218,245,281,326,335
%N A064825 Numbers which are the sums of three positive triangular numbers in exactly two different ways.
%C A064825 Checked up to 1000000 but haven't found any other values.
%C A064825 A063993(a(n)) = 2. - _Reinhard Zumkeller_, Jul 20 2012
%C A064825 Without the qualifier "positive" in the Name, sequence A071530 would result. - _Jon E. Schoenfield_, Jan 01 2020
%C A064825 No further terms <= 3*10^7. - _Michael S. Branicky_, Dec 17 2021
%e A064825 19 = 10 + 6 + 3 = 15 + 3 + 1.
%o A064825 (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
%o A064825 (Python)
%o A064825 from collections import Counter
%o A064825 from itertools import count, takewhile, combinations_with_replacement as mc
%o A064825 def aupto(N):
%o A064825     tris = takewhile(lambda x: x <= N, (i*(i+1)//2 for i in count(1)))
%o A064825     sum3 = filter(lambda x: x <= N, (sum(c) for c in mc(tris, 3)))
%o A064825     sum3counts = Counter(sum3)
%o A064825     return sorted(k for k in sum3counts if sum3counts[k] == 2)
%o A064825 print(aupto(1000)) # _Michael S. Branicky_, Dec 17 2021
%Y A064825 Cf. A063993, A064816, A002097, A071530, A111638.
%K A064825 nonn
%O A064825 1,1
%A A064825 Jonathan Ayres (jonathan.ayres(AT)btinternet.com), Oct 23 2001
%E A064825 Offset changed to 1 by _Michel Marcus_, Jan 14 2014