A017877 -id:A017877 - cp's OEIS frontend

A017867 Expansion of 1/(1 - x^8 - x^9).

1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 1, 3, 3, 1, 0, 0, 0, 0, 1, 4, 6, 4, 1, 0, 0, 0, 1, 5, 10, 10, 5, 1, 0, 0, 1, 6, 15, 20, 15, 6, 1, 0, 1, 7, 21, 35, 35, 21, 7, 1, 1, 8, 28, 56, 70, 56, 28, 8, 2, 9

Offset: 0

N. J. A. Sloane

Number of compositions of n into parts 8 and 9. - Joerg Arndt, Jun 29 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,1,1).

Column k=8 of A306713.

Cf. A017817, A017827, A017857, A017877.

m:=80; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-x^8-x^9))); // Vincenzo Librandi, Jun 28 2013

I:=[1,0,0,0,0,0,0,0,1]; [n le 9 select I[n] else Self(n-8)+Self(n-9): n in [1..80]]; // Vincenzo Librandi, Jun 28 2013

CoefficientList[Series[1 / (1 - Total[x^Range[8, 9]]), {x, 0, 80}], x] (* Vincenzo Librandi, Jun 28 2013 *)

x='x+O('x^66); Vec(1/(1-x^8-x^9)) \\ Altug Alkan, Oct 07 2018

a(n) = a(n-8) + a(n-9) for n>8. - Vincenzo Librandi, Jun 28 2013

a(n) = Sum_{k=0..floor(n/8)} binomial(k,n-8*k). - Seiichi Manyama, Oct 01 2024

A306753 a(n) = Sum_{k=0..n} binomial(k, 9*(n-k)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 11, 56, 221, 716, 2003, 5006, 11441, 24311, 48621, 92380, 167980, 294121, 498751, 824506, 1341154, 2177572, 3605251, 6249101, 11593726, 23138117, 48904469, 106653707, 234305936, 510034166, 1089810953, 2275676459, 4637090547

Offset: 0

Seiichi Manyama, Mar 07 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..3506
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1,1).

Column 9 of A306680.

Cf. A017877.

a[n_] := Sum[Binomial[k, 9*(n-k)], {k, 0, n}]; Array[a, 38, 0] (* Amiram Eldar, Jun 21 2021 *)

{a(n) = sum(k=0, n, binomial(k, 9*(n-k)))}

N=66; x='x+O('x^N); Vec((1-x)^8/((1-x)^9-x^10))

G.f.: (1-x)^8/((1-x)^9 - x^10).
a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9) + a(n-10) for n > 9.
a(n) = A017877(9*n).

A376649 a(n) = Sum_{k=0..floor(n/3)} binomial(floor(k/3),n-3*k).

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 3, 3, 2, 3, 3, 2, 3, 3, 2, 4, 6, 5, 5, 6, 5, 5, 6, 5, 6, 10, 11, 10, 11, 11, 10, 11, 11, 11, 16, 21, 21, 21, 22, 21, 21, 22, 22, 27, 37, 42, 42, 43, 43, 42, 43, 44, 49, 64, 79, 84, 85, 86, 85, 85, 87, 93, 113, 143

Offset: 0

Seiichi Manyama, Oct 01 2024

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,1,1).

Cf. A079398, A376650.

Cf. A017877.

a(n) = sum(k=0, n\3, binomial(k\3, n-3*k));

my(N=90, x='x+O('x^N)); Vec((1+x^3+x^6)/(1-x^9-x^10))

G.f.: (1-x^9)/((1-x^3) * (1-x^9-x^10)) = (1+x^3+x^6)/(1-x^9-x^10).
a(n) = a(n-9) + a(n-10).
a(n) = A017877(n) + A017877(n-3) + A017877(n-6).

A127844 a(1) = 1, a(2) = ... = a(10) = 0, a(n) = a(n-10)+a(n-9) for n>10.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 0, 1, 3, 3, 1, 0, 0, 0, 0, 0, 1, 4, 6, 4, 1, 0, 0, 0, 0, 1, 5, 10, 10, 5, 1, 0, 0, 0, 1, 6, 15, 20, 15, 6, 1, 0, 0, 1, 7, 21, 35, 35, 21, 7, 1, 0, 1, 8, 28

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.

Apart from offset same as A017877. - Georg Fischer, Oct 07 2018

S. Suter, Binet-like formulas for recurrent sequences with characteristic equation x^k=x+1, preprint, 2007. [Apparently unpublished as of May 2016]

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,1,1).

a:=[1,0,0,0,0,0,0,0,0,0];;  for n in [11..90] do a[n]:=a[n-9]+a[n-10]; od; a; # Muniru A Asiru, Oct 07 2018

LinearRecurrence[{0,0,0,0,0,0,0,0,1,1},{1,0,0,0,0,0,0,0,0,0},100] (* Harvey P. Dale, Jun 18 2017 *)

Vec(x*(1-x)*(1+x+x^2)*(1+x^3+x^6)/(1-x^9-x^10) + O(x^100)) \\ Colin Barker, May 30 2016

Binet-like formula: a(n) = Sum_{i=1..10} (r_i^n)/(9(r_i)^2+10(r_i)) where r_i is a root of x^10=x+1.

G.f.: x*(1-x)*(1+x+x^2)*(1+x^3+x^6) / (1-x^9-x^10). - Colin Barker, May 30 2016

cp's OEIS Frontend

A017867 Expansion of 1/(1 - x^8 - x^9).

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A306753 a(n) = Sum_{k=0..n} binomial(k, 9*(n-k)).

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A376649 a(n) = Sum_{k=0..floor(n/3)} binomial(floor(k/3),n-3*k).

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A127844 a(1) = 1, a(2) = ... = a(10) = 0, a(n) = a(n-10)+a(n-9) for n>10.

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