A245527 -id:A245527 - cp's OEIS frontend

A245738 Number of compositions of n into parts 1 and 2 with both parts present.

2, 3, 7, 11, 20, 32, 54, 87, 143, 231, 376, 608, 986, 1595, 2583, 4179, 6764, 10944, 17710, 28655, 46367, 75023, 121392, 196416, 317810, 514227, 832039, 1346267, 2178308, 3524576, 5702886, 9227463, 14930351, 24157815, 39088168, 63245984, 102334154, 165580139, 267914295, 433494435, 701408732, 1134903168, 1836311902

Offset: 3

David Neil McGrath, Jul 31 2014

			a(9) = 54. The tuples are (22221) = 5!/4! = 5, (222111) = 6!/3!/3! = 20, (2211111) = 7!/5!/2! = 21, (21111111) = 8!/7! = 8.

Colin Barker, Table of n, a(n) for n = 3..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-1).

Cf. A245332, A245492, A245487, A245527.

Column k=2 of A373118.

LinearRecurrence[{1,2,-1,-1},{2,3,7,11},50] (* Harvey P. Dale, Dec 20 2014 *)

Vec(1+1/(1-x-x^2)-1/(1-x)-1/(1-x^2)+O(x^66)) \\ Joerg Arndt, Aug 04 2014

G.f.: 1+1/(1-x-x^2)-1/(1-x)-1/(1-x^2).
a(n) = A052952(n-4)+2*A052952(n-3). - R. J. Mathar, Aug 05 2014
From Colin Barker, Jul 13 2017: (Start)
a(n) = (-20 + sqrt(5)*(-(1-sqrt(5))^(1+n) + (1+sqrt(5))^(1+n))/2^n) / 10 for n even.
a(n) = (-10 + sqrt(5)*(-(1-sqrt(5))^(1+n) + (1+sqrt(5))^(1+n))/2^n) / 10 for n odd.
a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4) for n>6. (End)
a(n) = Sum_{i=1..floor((n-1)/2)} C(n-i,i). - Wesley Ivan Hurt, Sep 19 2017
a(n) = A000045(n+1) - A000034(n+1). - J. M. Bergot and Robert Israel, Oct 11 2021

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

Original entry on oeis.org

6, 0, 0, 12, 12, 12, 20, 30, 50, 60, 80, 120, 162, 225, 305, 401, 560, 763, 1017, 1365, 1834, 2484, 3328, 4420, 5936, 7943, 10593, 14148, 18828, 25092, 33468, 44517, 59214, 78734, 104698, 139232, 184889, 245532, 326177, 433052, 574841, 762856, 1012219, 1343160

Offset: 12

David Neil McGrath, Jul 30 2014

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".

			a(24) = 162 = 42 + 90 + 30: the tuples are (5433333) -> 7!/5! = 42, (554433) -> 6!/2!2!2! = 90, (544443) -> 6!/4! = 30.

Robert Israel, Table of n, a(n) for n = 12..7366
Index entries for linear recurrences with constant coefficients, 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).

Cf. A245492, A245487, A245527, A022003, A011765, A112765, A017818.

N:= 100;
C34:= Vector(N):
C35:= Vector(N):
C45:= Vector(N):
C345:= Vector(N):
C1:= Vector(N,i -> numboccur([i mod 3, i mod 4, i mod 5],0)):
C34[3]:= 1: C34[4]:= 1:
C35[3]:= 1: C35[5]:= 1:
C45[4]:= 1: C45[5]:= 1:
C345[3]:= 1: C345[4]:= 1: C345[5]:= 1:
for n from 6 to N do
  C34[n]:= C34[n-3] + C34[n-4];
  C35[n]:= C35[n-3] + C35[n-5];
  C45[n]:= C45[n-4] + C45[n-5];
  C345[n]:= C345[n-3]+C345[n-4]+C345[n-5];
od:
A:= C345 - C34 - C35 - C45 + C1:
convert(A[12..N],list); # Robert Israel, Aug 18 2014

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 *)
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 *)

a(n) = A017818(n-1) -A245492(n) -A245487(n) -A245527(n) -A022003(n) -A011765(n) -A112765(n).
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
a(n) = A017818(n) - A017817(n) - A052920(n) - A017827(n) + A079978(n) + A121262(n) + A079998(n). - Robert Israel, Aug 18 2014

cp's OEIS Frontend

A245738 Number of compositions of n into parts 1 and 2 with both parts present.

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