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.

Previous Showing 21-24 of 24 results.

A192921 Constant term in the reduction by (x^2->x+1) of the polynomial p(n,x) defined below at Comments.

Original entry on oeis.org

1, 2, 2, 7, 16, 44, 113, 298, 778, 2039, 5336, 13972, 36577, 95762, 250706, 656359, 1718368, 4498748, 11777873, 30834874, 80726746, 211345367, 553309352, 1448582692, 3792438721, 9928733474, 25993761698, 68052551623, 178163893168, 466439127884, 1221153490481
Offset: 0

Views

Author

Clark Kimberling, Jul 12 2011

Keywords

Comments

The titular polynomial is defined by p(n,x) = x*p(n-1,x) +(x^2)*p(n-2,x), with p(0,x)=1, p(1,x)=1+x^2. For discussions of polynomial reduction, see A192232, A192744, and A192872.

Examples

			The coefficients in the polynomials p(n,x) are Fibonacci numbers.  The first seven and their reductions:
...
1 -> 1
1 + x^2 -> 2 + x
x + x^2 + x^3 -> 2 + 4*x
2*x^2 + x^3 + 2*x^4 -> 7 + 10*x
3*x^3 + 2*x^4 + 3*x^5 -> 16 + 27*x
5*x^4 + 3*x^5 + 5*x^6 -> 44 + 70*x
8*x^5 + 5*x^6 + 8*x^7 -> 113 + 184*x,
so that A192921=(1,2,2,7,16,44,113,...).

Links

Crossrefs

Cf. A192232, A192744, A192872.

Programs

  • GAP
    List([0..30], n -> Fibonacci(n-2)^2 +Fibonacci(n)*Fibonacci(n+1)); # G. C. Greubel, Feb 06 2019
  • Magma
    [Fibonacci(n-2)^2 + Fibonacci(n)*Fibonacci(n+1): n in [0..30]]; // G. C. Greubel, Feb 06 2019
  • Maple
    a:= n-> (<<0|1|0>, <0|0|1>, <-1|2|2>>^n. <<1,2,2>>)[1,1]:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 28 2016
  • Mathematica
    q = x^2; s = x + 1; z = 28;
p[0, x_] := 1; p[1, x_] := x^2 + 1;
p[n_, x_] := p[n - 1, x]*x + p[n - 2, x]*x^2;
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}] :=
FixedPoint[(s PolynomialQuotient @@ #1 +
       PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]
  (* A192921 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
  (* A192879 *)
LinearRecurrence[{2,2,-1}, {1,2,2}, 30] (* G. C. Greubel, Feb 06 2019 *)
  • PARI
    a(n) = round((2^(-n)*(-3*(-2)^n-(-4+sqrt(5))*(3+sqrt(5))^n+(3-sqrt(5))^n*(4+sqrt(5))))/5) \\ Colin Barker, Oct 01 2016
  • PARI
    Vec(-(2*x-1)*(1+2*x)/((1+x)*(x^2-3*x+1)) + O(x^40)) \\ Colin Barker, Oct 01 2016
  • PARI
    {a(n) = fibonacci(n-2)^2 +fibonacci(n)*fibonacci(n+1)}; \\ G. C. Greubel, Feb 06 2019
  • Sage
    [fibonacci(n-2)^2 +fibonacci(n)*fibonacci(n+1) for n in range(30)] # G. C. Greubel, Feb 06 2019

Formula

a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3).
G.f.: (1-2*x)*(1+2*x) / ( (1+x)*(1-3*x+x^2) ). - R. J. Mathar, May 08 2014
a(n) = A059929(n-1) + 2*A059929(n-2). - R. J. Mathar, May 08 2014
a(n) = F(n-4)*F(n) + F(n-1)*F(n+2), where F(-4)=-3, F(-3)=2, F(-2)=-1, F(-1)=1. - Bruno Berselli, Nov 03 2015
a(n) = (2^(-n)*(-3*(-2)^n-(-4+sqrt(5))*(3+sqrt(5))^n+(3-sqrt(5))^n*(4+sqrt(5))))/5. - Colin Barker, Oct 01 2016

A374259 a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3), with a(0)=4, a(1)=6, a(2)=20.

Original entry on oeis.org

4, 6, 20, 48, 130, 336, 884, 2310, 6052, 15840, 41474, 108576, 284260, 744198, 1948340, 5100816, 13354114, 34961520, 91530452, 239629830, 627359044, 1642447296, 4299982850, 11257501248, 29472520900, 77160061446, 202007663444, 528862928880, 1384581123202, 3624880440720, 9490060198964
Offset: 0

Views

Author

Bridget Rozema, Jul 01 2024

Keywords

Comments

a(n) is the number of edge covers of a rocket graph R_{3,n,n}.
A rocket graph R_{3,n,n} is cycle graph C_3 with two paths of n edges, where an end vertex of each path is identified with a distinct vertex in the C_3.
In other words, a rocket graph is a path with vertices -n-1, ..., -1, 0, 1, ..., n+1 with an additional edge (-1,1).

Examples

			For n=1, the R_{3,1,1} rocket graph is as follows and has a(1)=6 edge covers.
    *--*
   /|
  * |
   \|
    *--*

Links

Crossrefs

Equals twice A059929.

Programs

  • Mathematica
    LinearRecurrence[{2, 2, -1}, {4, 6, 20}, 50] (* Paolo Xausa, Jul 20 2024 *)

Formula

G.f.: (4-2*x)/(1-2*x-2*x^2+x^3).
a(n) = 2*A059929(n+1).
a(n) = Fibonacci(2n+2)+3*Fibonacci(n+1)*Fibonacci(n+1).

A336630 a(n) = 2*F(2*n+1) + 4*F(n+1)*F(n-1) for n > 0, with a(0) = 0 and F(n) = A000045(n).

Original entry on oeis.org

0, 4, 18, 38, 108, 274, 726, 1892, 4962, 12982, 33996, 88994, 232998, 609988, 1596978, 4180934, 10945836, 28656562, 75023862, 196415012, 514221186, 1346248534, 3524524428, 9227324738, 24157449798, 63245024644, 165577624146, 433487847782
Offset: 0

Views

Author

Michael Tulskikh, Jul 28 2020

Keywords

Links

Crossrefs

Cf. A000045, A001519, A059929.

Formula

a(n) = (F(4n+1) - F(n+1)^4)/F(n)^2 for n > 0 and a(0) = 0, where F(n) = A000045(n).
a(n) = 2*A001519(n+1) + 4*A059929(n-1) for n > 0.
From Stefano Spezia, Jul 28 2020: (Start)
O.g.f.: 2*x*(2 + 5*x - 3*x^2)/(1 - 2*x - 2*x^2 + x^3).
a(n) = 2*a(n-1) + 2*a(n-2) - a(n-2) for n > 3. (End)

A378277 Denominators in a harmonic triangle, based on products of Fibonacci numbers.

Original entry on oeis.org

1, 2, 2, 2, 3, 6, 2, 3, 10, 15, 2, 3, 10, 24, 40, 2, 3, 10, 24, 65, 104, 2, 3, 10, 24, 65, 168, 273, 2, 3, 10, 24, 65, 168, 442, 714, 2, 3, 10, 24, 65, 168, 442, 1155, 1870, 2, 3, 10, 24, 65, 168, 442, 1155, 3026, 4895, 2, 3, 10, 24, 65, 168, 442, 1155, 3026, 7920, 12816
Offset: 1

Views

Author

Werner Schulte, Nov 21 2024

Keywords

Comments

The harmonic triangle uses the terms of this sequence as denominators, numerators = 1.
The inverse of the harmonic triangle has entries -(Fibonacci(k+1))^2 for 1<=k
Row sums of the harmonic triangle are 1.
Conjecture: Alt. row sums of the harmonic triangle are Fibonacci(n-2) / Fibonacci(n+1), where Fibonacci(-1) = 1.

Examples

			Triangle T(n, k) for 1 <= k <= n starts:
n\ k :  1  2   3   4   5    6    7     8     9    10     11
===========================================================
   1 :  1
   2 :  2  2
   3 :  2  3   6
   4 :  2  3  10  15
   5 :  2  3  10  24  40
   6 :  2  3  10  24  65  104
   7 :  2  3  10  24  65  168  273
   8 :  2  3  10  24  65  168  442   714
   9 :  2  3  10  24  65  168  442  1155  1870
  10 :  2  3  10  24  65  168  442  1155  3026  4895
  11 :  2  3  10  24  65  168  442  1155  3026  7920  12816
  etc.

Crossrefs

Cf. A000045, A110034, A110035, A001654 (main diagonal), A059929 (subdiagonals).

Programs

  • PARI
    T(n,k)=if(k==n,Fibonacci(n)*Fibonacci(n+1),Fibonacci(k)*Fibonacci(k+2))

Formula

T(n, k) = Fibonacci(n) * Fibonacci(n+1) if k = n, and Fibonacci(k) * Fibonacci(k+2) if 1 <= k < n.
Row sums are A110035(n) - 1 = -A110034(n+1).
G.f.: A(t, x) = x*t*(1 + t - x*t^2) / ((1 - t) * (1 + x*t) * (1 - 3*x*t + x^2*t^2)).
Previous Showing 21-24 of 24 results.

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

Content is available under The OEIS End-User License Agreement.