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 1-3 of 3 results.

A111672 Array T(n,k) = A153277(n-1,k) = A144150(n,k-1) read by downwards antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 12, 15, 1, 1, 5, 22, 60, 52, 1, 1, 6, 35, 154, 358, 203, 1, 1, 7, 51, 315, 1304, 2471, 877, 1, 1, 8, 70, 561, 3455, 12915, 19302, 4140, 1
Offset: 1

Views

Author

Gary W. Adamson, Aug 14 2005

Keywords

Comments

Column k is obtained by taking the k-th matrix power of the triangle A008277 and multiplying from the right with the column vector [1,0,0,0,....].

Examples

			The array starts
1,  1,   1,    1,    1,    1,  ...
1,  2,   3,    4,    5,    6,  ...
1,  5,  12,   22,   35,   51,  ...
1, 15,  60,  154,  315,  561,  ...
1, 52, 358, 1304, 3455, 7556,  ...

Links

Crossrefs

Cf. A008277, A144150, A153277.
Cf. A000326 (row 3), A005945 (row 4), A000110 (column 2), A000258 (column 3), A000307 (column 4), A000357 (column 5), A000405 (column 6), A111669 (column 7), A081624.

Extensions

a(44) and definition corrected by Georg Fischer, May 18 2022

A135934 O.g.f.: A(x) = Sum_{n>=0} x^n / Product_{k=0..n} (1 - Fibonacci(k)*x).

Original entry on oeis.org

1, 1, 2, 4, 9, 24, 77, 299, 1419, 8312, 60452, 547939, 6213566, 88468601, 1585646789, 35846274127, 1023893974778, 37005881297226, 1694206791508891, 98335493373334998, 7241161595237290969, 676871453643079089963, 80351261743964014059133, 12117563014768206457325416
Offset: 0

Views

Author

Paul D. Hanna, Dec 07 2007

Keywords

Comments

After the first term, row sums of triangle A111669. - Emanuele Munarini, Dec 05 2017

Examples

			A(x) = 1 + x/(1-x) + x^2/((1-x)*(1-x)) + x^3/((1-x)*(1-x)*(1-2*x)) +
x^4/((1-x)*(1-x)*(1-2*x)(1-3*x)) + x^5/((1-x)*(1-x)*(1-2*x)*(1-3*x)*(1-5*x)) + x^6/((1-x)*(1-x)*(1-2*x)*(1-3*x)*(1-5*x)*(1-8*x)) +...

Links

Crossrefs

Cf. A000045, A006116, A111669.

Programs

  • Maple
    b:= proc(n, m) option remember; `if`(n=0, 1,
      (<<0|1>, <1|1>>^m)[1, 2]*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 08 2021
  • Mathematica
    b[n_, m_] := b[n, m] = If[n == 0, 1,
   MatrixPower[{{0, 1}, {1, 1}}, m][[1, 2]]*b[n-1, m]+b[n-1, m+1]];
a[n_] :=  b[n, 0];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Sep 07 2022, after Alois P. Heinz *)
  • PARI
    {a(n)=polcoeff(sum(k=0, n, x^k/prod(j=0, k, 1-fibonacci(j)*x+x*O(x^n))), n)}

Formula

G.f.: (1 - G(0) )/(1-x) where G(k) = 1 - 1/(1-Fibonacci(k)*x)/(1-x/(x-1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 17 2013
G.f.: 1/(x*(1-x)*G(0)) - 1/x where G(k) = 1 - x/(x - 1/(1 + 1/(x*Fibonacci(k)-1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Feb 13 2013

A114163 Triangle read by rows, based on a simple Jacobsthal number recursion rule.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 18, 10, 1, 1, 5, 58, 68, 21, 1, 1, 6, 179, 398, 299, 42, 1, 1, 7, 543, 2169, 3687, 1181, 85, 1, 1, 8, 1636, 11388, 42726, 28488, 4836, 170, 1, 1, 9, 4916, 58576, 481374, 640974, 236436, 19286, 341, 1, 1, 10, 14757, 297796, 5353690
Offset: 0

Views

Author

Paul Barry, Nov 14 2005

Keywords

Comments

Subdiagonal S(n+1,n) is A000975(n+1). Row sums of inverse are 0^n.

Examples

			Triangle begins
1....1....3....5...11...21...43....J(k+1)
1
1....1
1....2....1
1....3....5....1
1....4...18...10....1
1....5...58...68...21....1
1....6..179..398..299...42....1
For example, T(6,3)=398=58+5*68=T(5,2)+J(4)*T(5,3).

Crossrefs

Cf. A111669.

Formula

Number triangle T(n, k)=T(n-1, k-1)+J(k+1)*T(n-1, k) where J(n)=A001045(n); Column k has g.f. x^k/Product(1-J(i+1)x, i, 0, k).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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