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 A325168 #20 Aug 04 2025 11:20:55
%S A325168 0,0,0,1,3,5,8,9,12,13,16,17,20,21,24,25,28,29,32,33,36,37,40,41,44,
%T A325168 45,48,49,52,53,56,57,60,61,64,65,68,69,72,73,76,77,80,81,84,85,88,89,
%U A325168 92,93,96,97,100,101,104,105,108,109,112,113,116,117,120,121
%N A325168 Number of integer partitions of n with origin-to-boundary graph-distance equal to 2.
%C A325168 The origin-to-boundary graph-distance of a Young diagram is the minimum number of unit steps left or down from the upper-left square to a nonsquare in the lower-right quadrant. It is also the side-length of the maximum triangular partition contained inside it.
%H A325168 Colin Barker, <a href="/A325168/b325168.txt">Table of n, a(n) for n = 0..1000</a>
%H A325168 N. Guru Sharan, <a href="https://arxiv.org/abs/2507.20260">Rook decomposition of the Partition function</a>, arXiv:2507.20260 [math.CO], 2025. See p. 5.
%H A325168 N. Guru Sharan and Armin Straub, <a href="https://arxiv.org/abs/2507.19047">Partitions with Durfee triangles of fixed size</a>, arXiv:2507.19047 [math.CO], 2025. See p. 10.
%H A325168 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%F A325168 From _Colin Barker_, Apr 08 2019: (Start)
%F A325168 G.f.: x^3*(1 + 2*x + x^2 + x^3 - x^4) / ((1 - x)^2*(1 + x)).
%F A325168 a(n) = a(n-1) + a(n-2) - a(n-3) for n>7.
%F A325168 a(n) = 2*n - 4 for n>4 and even.
%F A325168 a(n) = 2*n - 5 for n>4 and odd.
%F A325168 (End)
%e A325168 The a(3) = 1 through a(10) = 16 partitions:
%e A325168 (21) (22) (32) (33) (43) (44) (54) (55)
%e A325168 (31) (41) (42) (52) (53) (63) (64)
%e A325168 (211) (221) (51) (61) (62) (72) (73)
%e A325168 (311) (222) (511) (71) (81) (82)
%e A325168 (2111) (411) (2221) (611) (711) (91)
%e A325168 (2211) (4111) (2222) (6111) (811)
%e A325168 (3111) (22111) (5111) (22221) (7111)
%e A325168 (21111) (31111) (22211) (51111) (22222)
%e A325168 (211111) (41111) (222111) (61111)
%e A325168 (221111) (411111) (222211)
%e A325168 (311111) (2211111) (511111)
%e A325168 (2111111) (3111111) (2221111)
%e A325168 (21111111) (4111111)
%e A325168 (22111111)
%e A325168 (31111111)
%e A325168 (211111111)
%t A325168 otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
%t A325168 Table[Length[Select[IntegerPartitions[n],otb[#]==2&]],{n,0,30}]
%o A325168 (PARI) concat([0,0,0], Vec(x^3*(1 + 2*x + x^2 + x^3 - x^4) / ((1 - x)^2*(1 + x)) + O(x^80))) \\ _Colin Barker_, Apr 08 2019
%Y A325168 Cf. A006918, A065770, A115994, A117485, A257990, A297113, A325135, A325166, A325169, A325170.
%K A325168 nonn,easy
%O A325168 0,5
%A A325168 _Gus Wiseman_, Apr 05 2019