cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A306713 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. 1/(1-x^k-x^(k+1)).

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, 0, 1, 3, 1, 0, 0, 1, 5, 1, 0, 0, 1, 1, 8, 1, 0, 0, 0, 1, 2, 13, 1, 0, 0, 0, 1, 0, 2, 21, 1, 0, 0, 0, 0, 1, 1, 3, 34, 1, 0, 0, 0, 0, 1, 0, 2, 4, 55, 1, 0, 0, 0, 0, 0, 1, 0, 1, 5, 89, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 7, 144, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 3, 9, 233
Offset: 0

Views

Author

Seiichi Manyama, Mar 05 2019

Keywords

Comments

A(n,k) is the number of compositions of n into parts k and k+1.

Examples

			Square array begins:
    1, 1, 1, 1, 1, 1, 1, 1, 1, ...
    1, 0, 0, 0, 0, 0, 0, 0, 0, ...
    2, 1, 0, 0, 0, 0, 0, 0, 0, ...
    3, 1, 1, 0, 0, 0, 0, 0, 0, ...
    5, 1, 1, 1, 0, 0, 0, 0, 0, ...
    8, 2, 0, 1, 1, 0, 0, 0, 0, ...
   13, 2, 1, 0, 1, 1, 0, 0, 0, ...
   21, 3, 2, 0, 0, 1, 1, 0, 0, ...
   34, 4, 1, 1, 0, 0, 1, 1, 0, ...
   55, 5, 1, 2, 0, 0, 0, 1, 1, ...

Links

Crossrefs

Columns 1-10 give A000045(n+1), A182097, A017817, A017827, A017837, A017847, A017857, A017867, A017877, A017887.
Cf. A306646, A306680.

Programs

  • Mathematica
    T[n_, k_] := Sum[Binomial[j, n-k*j], {j, 0, Floor[n/k]}]; Table[T[k, n - k + 1], {n, 0, 12}, {k, 0, n}] // Flatten (* Amiram Eldar, Jun 21 2021 *)

Formula

A(n,k) = Sum_{j=0..floor(n/k)} binomial(j,n-k*j).

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

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Comments

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

Links

Crossrefs

Column k=8 of A306713.
Cf. A017817, A017827, A017857, A017877.

Programs

  • Magma
    m:=80; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-x^8-x^9))); // Vincenzo Librandi, Jun 28 2013
  • Magma
    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
  • Mathematica
    CoefficientList[Series[1 / (1 - Total[x^Range[8, 9]]), {x, 0, 80}], x] (* Vincenzo Librandi, Jun 28 2013 *)
  • PARI
    x='x+O('x^66); Vec(1/(1-x^8-x^9)) \\ Altug Alkan, Oct 07 2018

Formula

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

A306721 a(n) = Sum_{k=0..n} binomial(k, 7*(n-k)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 9, 37, 121, 331, 793, 1717, 3433, 6437, 11456, 19569, 32505, 53449, 89149, 155041, 286825, 564929, 1163317, 2442210, 5117225, 10558381, 21308121, 41973391, 80778601, 152344397, 282855561, 520060249, 953217792, 1753553441, 3256528177, 6127896977, 11694334137
Offset: 0

Views

Author

Seiichi Manyama, Mar 06 2019

Keywords

Links

Crossrefs

Column 7 of A306680.
Cf. A017857.

Programs

  • Mathematica
    a[n_] := Sum[Binomial[k, 7*(n-k)], {k, 0, n}]; Array[a, 40, 0] (* Amiram Eldar, Jun 21 2021 *)
  • PARI
    {a(n) = sum(k=0, n, binomial(k, 7*(n-k)))}
  • PARI
    N=66; x='x+O('x^N); Vec((1-x)^6/((1-x)^7-x^8))

Formula

G.f.: (1-x)^6/((1-x)^7-x^8).
a(n) = 7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7)+a(n-8) for n > 7.
a(n) = A017857(7*n).

A127842 a(1)=1, a(2)=...=a(8)=0, a(n) = a(n-8)+a(n-7) for n>8.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 3, 3, 1, 0, 0, 0, 1, 4, 6, 4, 1, 0, 0, 1, 5, 10, 10, 5, 1, 0, 1, 6, 15, 20, 15, 6, 1, 1, 7, 21, 35, 35, 21, 7, 2, 8, 28, 56, 70, 56, 28, 9, 10, 36, 84, 126, 126, 84, 37
Offset: 1

Views

Author

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

Keywords

Comments

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 A017857. - Georg Fischer, Oct 07 2018

References

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

Links

Programs

  • GAP
    a:=[1,0,0,0,0,0,0,0];;  for n in [9..80] do a[n]:=a[n-7]+a[n-8]; od; a; # Muniru A Asiru, Oct 07 2018
  • Magma
    I:=[1,0,0,0,0,0,0,0]; [n le 8 select I[n] else Self(n-7)+Self(n-8): n in [1..100]]; // Vincenzo Librandi, Oct 08 2018
  • Mathematica
    LinearRecurrence[{0, 0, 0, 0, 0, 0, 1, 1}, {1, 0, 0, 0, 0, 0, 0, 0}, 100] (* Vincenzo Librandi, Oct 08 2018 *)
  • PARI
    Vec(x*(1-x)*(1+x+x^2+x^3+x^4+x^5+x^6)/(1-x^7-x^8) + O(x^100)) \\ Colin Barker, May 30 2016

Formula

Binet-like formula: a(n) = Sum_{i=1..8} (r_i^n)/(7(r_i)^2+8(r_i)) where r_i is a root of x^8=x+1.
G.f.: x*(1-x)*(1+x+x^2+x^3+x^4+x^5+x^6) / (1-x^7-x^8). - Colin Barker, May 30 2016

A373913 Number of compositions of 8*n into parts 7 and 8.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 10, 46, 166, 496, 1288, 3004, 6437, 12888, 24464, 44728, 80428, 146320, 278104, 564929, 1225811, 2778772, 6396236, 14620646, 32760586, 71565796, 152344397, 316911454, 647536777, 1308456096, 2635130392, 5330198752, 10896635912
Offset: 0

Views

Author

Seiichi Manyama, Jun 22 2024

Keywords

Links

Crossrefs

Cf. A099099, A099131, A107025, A373912.
Cf. A017857.

Programs

  • Mathematica
    CoefficientList[Series[1/(1-x-x^7/(1-x)^7),{x,0,40}],x] (* or *) LinearRecurrence[ {8,-28,56,-70,56,-28,9,-1},{1,1,1,1,1,1,1,2},40] (* Harvey P. Dale, Jul 29 2024 *)
  • PARI
    a(n) = sum(k=0, n\7, binomial(n+k, n-7*k));

Formula

a(n) = A017857(8*n).
a(n) = Sum_{k=0..floor(n/7)} binomial(n+k,n-7*k).
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 9*a(n-7) - a(n-8).
G.f.: 1/(1 - x - x^7/(1 - x)^7).
Showing 1-5 of 5 results.

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