%I #12 Apr 18 2024 11:51:43

%S 1,3,4,6,7,9,11,13,15,18,20,23,26,29,32,36,39,43,47,51,55,60,64,69,74,

%T 79,84,90,95,101,107,113,119,126,132,139,146,153,160,168,175,183,191,

%U 199,207,216,224,233,242,251,260,270,279,289,299,309,319,330,340

%N Number of partitions of n into 3 parts such that every i-th smallest part (counted with multiplicity) is different from i.

%H Alois P. Heinz, <a href="/A244239/b244239.txt">Table of n, a(n) for n = 9..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,-1,-1,1).

%F From _Chai Wah Wu_, Apr 18 2024: (Start)

%F a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6) for n > 15.

%F G.f.: x^9*(-x^6 + 2*x^4 + x^3 - 2*x - 1)/((x - 1)^3*(x + 1)*(x^2 + x + 1)). (End)

%p a:= proc(n) option remember; `if`(n<14, [1, 3, 4, 6, 7][n-8],

%p ((-4*n+56)*a(n-5) +(3*n-16)*a(n-4) +(7*n-66)*a(n-3)

%p +(4*n-44)*a(n-2) +(28-3*n)*a(n-1)) / (7*n-78))

%p end:

%p seq(a(n), n=9..80);

%t b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, Function[p, Expand[x*(p-Coefficient[p, x, i-1]*x^(i-1))]][b[n-i, i]]]]];

%t a[n_] := Coefficient[b[n, n], x, 3];

%t Table[a[n], {n, 9, 80}] (* _Jean-François Alcover_, May 01 2018, after _Alois P. Heinz_ *)

%Y Column k=3 of A238406.

%K nonn

%O 9,2

%A _Alois P. Heinz_, Jun 23 2014