A017827 -id:A017827 - cp's OEIS frontend

A127839 a(1)=1, a(2)=...=a(5)=0, a(n) = a(n-5) + a(n-4) for n > 5.

1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 1, 3, 3, 1, 1, 4, 6, 4, 2, 5, 10, 10, 6, 7, 15, 20, 16, 13, 22, 35, 36, 29, 35, 57, 71, 65, 64, 92, 128, 136, 129, 156, 220, 264, 265, 285, 376, 484, 529, 550, 661, 860, 1013, 1079, 1211

Offset: 1

Stephen Suter (sms5064(AT)psu.edu), Apr 02 2007

Part of the phi_k family of sequences defined by a(1)=1, a(2)=...=a(k)=0, a(n) = a(n-k) + a(n-k+1) for n > k. phi_2 is a shift of the Fibonacci sequence and phi_3 is a shift of the Padovan sequence.

S. Suter, Binet-like formulas for recurrent sequences with characteristic equation x^k=x+1, preprint, 2007

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Sadjia Abbad and Hacène Belbachir, The r-Fibonacci polynomial and its companion sequences linked with some classical sequences, Integers (2025), Vol. 25, Art. No. A38. See p. 17.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1,1)

LinearRecurrence[{0,0,0,1,1},{1,0,0,0,0},70] (* Harvey P. Dale, Mar 19 2012 *)

Binet-like formula: a(n) = Sum_{i=1...5} (r_i^n)/(4(r_i)^2+5(r_i)) where r_i is a root of x^5=x+1.
G.f.: x*(x^4-1)/(x^5+x^4-1). - Harvey P. Dale, Mar 19 2012
a(n) = A017827(n-6) for n >= 6. - R. J. Mathar, May 09 2013

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

A375691 G.f. A(x) satisfies A(x) = 1 + x^4(1+x)A(x)^3.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 0, 0, 3, 6, 3, 0, 12, 36, 36, 12, 55, 220, 330, 220, 328, 1365, 2730, 2730, 2793, 8841, 21420, 28560, 29172, 62832, 164220, 271320, 314583, 508896, 1265628, 2430480, 3275085, 4642803, 10091664, 21066804, 32555028, 45388200, 85102875

Offset: 0

Seiichi Manyama, Sep 27 2024

nonn

Cf. A017827, A366589, A376487, A376546.

a(n) = sum(k=0, n\4, binomial(k, n-4*k)*binomial(3*k, k)/(2*k+1));

a(n) = Sum_{k=0..floor(n/4)} binomial(k,n-4*k) * binomial(3*k,k)/(2*k+1).

A385142 a(n) = 4a(n-1) - 6a(n-2) + 4*a(n-3) - a(n-4) + a(n-5) with a(1) = a(2) = a(3) = 0, a(4) = 1, and a(5) = 3.

Original entry on oeis.org

0, 0, 0, 1, 3, 6, 10, 15, 22, 35, 64, 129, 265, 529, 1013, 1873, 3394, 6126, 11148, 20552, 38303, 71760, 134408, 250880, 466361, 864339, 1600062, 2963186, 5494247, 10200142, 18952107, 35221440, 65442625, 121544393, 225655617, 418857277, 777451793, 1443184210, 2679343966

Offset: 1

Hung Viet Chu, Jun 19 2025

a(n) is the number of subsets of {4, 8, 12,.., 4*n} that are maximal Schreier and contain 4*n.

Paolo Xausa, Table of n, a(n) for n = 1..1000
Hung Viet Chu and Zachary Louis Vasseur, Schreier sets of multiples of an integer, linear recurrence, and Pascal triangle, arXiv:2506.14312 [math.CO], 2025. See Table 2 p. 2.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1,1).

Cf. A017827.

LinearRecurrence[{4, -6, 4, -1, 1}, {0, 0, 0, 1, 3}, 50] (* Paolo Xausa, Jun 27 2025 *)

a(n) = Sum_{i=1..floor((n+1)/5)} binomial(n-i-1, 4i-2).

a(n) = A017827(4*n-6), n > 1.

cp's OEIS Frontend

A127839 a(1)=1, a(2)=...=a(5)=0, a(n) = a(n-5) + a(n-4) for n > 5.

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

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A375691 G.f. A(x) satisfies A(x) = 1 + x^4*(1+x)*A(x)^3.

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A385142 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + a(n-5) with a(1) = a(2) = a(3) = 0, a(4) = 1, and a(5) = 3.

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A375691 G.f. A(x) satisfies A(x) = 1 + x^4(1+x)A(x)^3.

A385142 a(n) = 4a(n-1) - 6a(n-2) + 4*a(n-3) - a(n-4) + a(n-5) with a(1) = a(2) = a(3) = 0, a(4) = 1, and a(5) = 3.