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%I #12 Jan 08 2021 21:16:59
%S 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 45,48,49,52,53,56,57,60,61,64,65,68,69,72,73,76,77,80,81,84,85,88,89,
%U 92,93,96,97,100,101,104,105,108,109,112,113,116,117,120,121
%N Number of integer partitions of n with origin-to-boundary graph-distance equal to 2.
%C 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 Colin Barker, <a href="/A325168/b325168.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%F From _Colin Barker_, Apr 08 2019: (Start)
%F G.f.: x^3*(1 + 2*x + x^2 + x^3 - x^4) / ((1 - x)^2*(1 + x)).
%F a(n) = a(n-1) + a(n-2) - a(n-3) for n>7.
%F a(n) = 2*n - 4 for n>4 and even.
%F a(n) = 2*n - 5 for n>4 and odd.
%F (End)
%e The a(3) = 1 through a(10) = 16 partitions:
%e (21) (22) (32) (33) (43) (44) (54) (55)
%e (31) (41) (42) (52) (53) (63) (64)
%e (211) (221) (51) (61) (62) (72) (73)
%e (311) (222) (511) (71) (81) (82)
%e (2111) (411) (2221) (611) (711) (91)
%e (2211) (4111) (2222) (6111) (811)
%e (3111) (22111) (5111) (22221) (7111)
%e (21111) (31111) (22211) (51111) (22222)
%e (211111) (41111) (222111) (61111)
%e (221111) (411111) (222211)
%e (311111) (2211111) (511111)
%e (2111111) (3111111) (2221111)
%e (21111111) (4111111)
%e (22111111)
%e (31111111)
%e (211111111)
%t otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
%t Table[Length[Select[IntegerPartitions[n],otb[#]==2&]],{n,0,30}]
%o (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 Cf. A006918, A065770, A115994, A117485, A257990, A297113, A325135, A325166, A325169, A325170.
%K nonn,easy
%O 0,5
%A _Gus Wiseman_, Apr 05 2019
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