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A243254 Number of compositions of n into parts {3,4,5} when all parts 3,4 and 5 are present.

%I A243254 #25 Jun 06 2025 11:09:19
%S A243254 6,0,0,12,12,12,20,30,50,60,80,120,162,225,305,401,560,763,1017,1365,
%T A243254 1834,2484,3328,4420,5936,7943,10593,14148,18828,25092,33468,44517,
%U A243254 59214,78734,104698,139232,184889,245532,326177,433052,574841,762856,1012219,1343160
%N A243254 Number of compositions of n into parts {3,4,5} when all parts 3,4 and 5 are present.
%C A243254 Compositions of n from the set {3,4,5} that can be partitioned into the equivalence classes [345][34][45][35][3][4][5], where each class is defined by the relation "all elements are present".
%H A243254 Robert Israel, <a href="/A243254/b243254.txt">Table of n, a(n) for n = 12..7366</a>
%H A243254 <a href="/index/Rec#order_27">Index entries for linear recurrences with constant coefficients</a>, signature (-2, -2, 2, 9, 16, 14, -2, -29, -52, -52, -20, 34, 82, 97, 67, 7, -53, -84, -77, -43, -4, 22, 29, 23, 13, 5, 1).
%F A243254 a(n) = A017818(n-1) -A245492(n) -A245487(n) -A245527(n) -A022003(n) -A011765(n) -A112765(n).
%F A243254 G.f.: -(x^15 +5*x^14 +13*x^13 +24*x^12 +34*x^11 +36*x^10 +24*x^9-26*x^7 -40*x^6 -36*x^5 -18*x^4 +12*x^2 +12*x +6) *x^12 /((x-1) *(x+1) *(x^2+1) *(x^3+x^2-1) *(x^4+x^3-1) *(x^5+x^3-1) *(x^2+x+1) *(x^5+x^4-1) *(x^4+x^3+x^2+x+1)). - _Alois P. Heinz_, Jul 30 2014
%F A243254 a(n) = A017818(n) - A017817(n) - A052920(n) - A017827(n) + A079978(n) + A121262(n) + A079998(n). - _Robert Israel_, Aug 18 2014
%e A243254 a(24) = 162 = 42 + 90 + 30: the tuples are (5433333) -> 7!/5! = 42, (554433) -> 6!/2!2!2! = 90, (544443) -> 6!/4! = 30.
%p A243254 N:= 100;
%p A243254 C34:= Vector(N):
%p A243254 C35:= Vector(N):
%p A243254 C45:= Vector(N):
%p A243254 C345:= Vector(N):
%p A243254 C1:= Vector(N,i -> numboccur([i mod 3, i mod 4, i mod 5],0)):
%p A243254 C34[3]:= 1: C34[4]:= 1:
%p A243254 C35[3]:= 1: C35[5]:= 1:
%p A243254 C45[4]:= 1: C45[5]:= 1:
%p A243254 C345[3]:= 1: C345[4]:= 1: C345[5]:= 1:
%p A243254 for n from 6 to N do
%p A243254   C34[n]:= C34[n-3] + C34[n-4];
%p A243254   C35[n]:= C35[n-3] + C35[n-5];
%p A243254   C45[n]:= C45[n-4] + C45[n-5];
%p A243254   C345[n]:= C345[n-3]+C345[n-4]+C345[n-5];
%p A243254 od:
%p A243254 A:= C345 - C34 - C35 - C45 + C1:
%p A243254 convert(A[12..N],list); # _Robert Israel_, Aug 18 2014
%t A243254 CoefficientList[Series[x^12*(x^15 + 5*x^14 + 13*x^13 + 24*x^12 + 34*x^11 + 36*x^10 + 24*x^9 - 26*x^7 - 40*x^6 - 36*x^5 - 18*x^4 + 12*x^2 + 12*x +6)/((1 - x)*(x + 1)*(x^2 + 1)*(x^3 + x^2 - 1)*(x^4 + x^3 - 1)*(x^5 + x^3 - 1)*(x^2 + x + 1)*(x^5 + x^4 - 1)*(x^4 + x^3 + x^2 + x + 1)), {x, 0, 50}], x] (* _Wesley Ivan Hurt_, Aug 02 2014 *)
%t A243254 Drop[LinearRecurrence[{-2,-2,2,9,16,14,-2,-29,-52,-52,-20,34,82,97,67,7,-53,-84,-77,-43,-4,22,29,23,13,5,1},{0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,12,12,12,20,30,50,60,80,120,162,225,305,401},60],12] (* _Harvey P. Dale_, Jun 06 2025 *)
%Y A243254 Cf. A245492, A245487, A245527, A022003, A011765, A112765, A017818.
%K A243254 nonn,easy
%O A243254 12,1
%A A243254 _David Neil McGrath_, Jul 30 2014