This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A053614 #30 Mar 12 2025 10:39:07
%S A053614 2,5,8,12,23,33
%N A053614 Numbers that are not the sum of distinct triangular numbers.
%C A053614 The Mathematica program first computes A024940, the number of partitions of n into distinct triangular numbers. Then it finds those n having zero such partitions. It appears that A024940 grows exponentially, which would preclude additional terms in this sequence. - _T. D. Noe_, Jul 24 2006, Jan 05 2009
%D A053614 Joe Roberts, Lure of the Integers, The Mathematical Association of America, 1992, page 184, entry 33.
%D A053614 David Wells in "The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, page 94, states that "33 is the largest number that is not the sum of distinct triangular numbers".
%H A053614 Shyam Sunder Gupta, <a href="https://doi.org/10.1007/978-981-97-2465-9_3">Triangular Numbers</a>, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 3, 83-125.
%F A053614 Complement of A061208.
%e A053614 a(2) = 5: the 7 partitions of 5 are 5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, 1+1+1+1+1. Among those the distinct ones are 5, 4+1, 3+2. None contains all distinct triangular numbers.
%e A053614 12 is a term as it is not a sum of 1, 3, 6 or 10 taken at most once.
%t A053614 nn=100; t=Rest[CoefficientList[Series[Product[(1+x^(k*(k+1)/2)), {k,nn}], {x,0,nn(nn+1)/2}], x]]; Flatten[Position[t,0]] (* _T. D. Noe_, Jul 24 2006 *)
%Y A053614 Cf. A025524 (number of numbers not the sum of distinct n-th-order polygonal numbers)
%Y A053614 Cf. A007419 (largest number not the sum of distinct n-th-order polygonal numbers)
%Y A053614 Cf. A001422, A121405 (corresponding sequences for square and pentagonal numbers)
%Y A053614 Cf. A000217, A002243, A002244, A014134, A014156, A014158, A020757, A050941, A050942, A051611, A007294, A051533, A060773.
%K A053614 fini,full,nonn
%O A053614 1,1
%A A053614 _Jud McCranie_, Mar 19 2000
%E A053614 Entry revised by _N. J. A. Sloane_, Jul 23 2006