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.

A099582 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k-1)*4^(n-k-1).

Original entry on oeis.org

0, 0, 1, 4, 24, 112, 560, 2688, 13056, 62976, 304384, 1469440, 7096320, 34263040, 165441536, 798818304, 3857055744, 18623496192, 89922273280, 434183077888, 2096421666816, 10122418978816, 48875363631104, 235991130439680
Offset: 0

Views

Author

Paul Barry, Oct 23 2004

Keywords

Comments

In general a(n) = Sum{k=0..floor(n/2)} binomial(n-k, k-1) * r^(n-k-1) has g.f. x^2/((1-r*x^2)*(1-r*x-r*x^2)) and satisfies a(n) = r*a(n-1) + 2*r*a(n-2) - r^2*a(n-3) - r^2*a(n-4).

Links

Crossrefs

Cf. A099177, A099581.

Programs

  • Magma
    I:=[0,0,1,4]; [n le 4 select I[n] else 4*(Self(n-1) +2*Self(n-2) -4*Self(n-3) -4*Self(n-4)): n in [1..41]]; // G. C. Greubel, Jul 22 2022
  • Mathematica
    Table[Sum[Binomial[n-k,k-1]*4^(n-k-1),{k,0,Floor[n/2]}],{n,0,30}] (* or *) LinearRecurrence[{4,8,-16,-16},{0,0,1,4},30] (* Harvey P. Dale, Jul 19 2012 *)
  • SageMath
    [2^(n-3)*(lucas_number1(n,2,-1) - (n%2)) for n in (0..40)] # G. C. Greubel, Jul 22 2022

Formula

G.f.: x^2/((1-4*x^2)*(1-4*x-4*x^2)).
a(n) = 4*a(n-1) + 8*a(n-2) - 16*a(n-3) - 16*a(n-4) with a(n) = (n^3-n)/6 for n<5.
From G. C. Greubel, Jul 22 2022: (Start)
a(n) = 2^(n-4)*(2*A000129(n) - (1 - (-1)^n)).
a(n) = (1/4)*(A057087(n-1) - 2^(n-2)*(1 - (-1)^n)).
E.g.f.: (exp(2*x)*sinh(2*sqrt(2)*x) - sqrt(2)*sinh(2*x))/(8*sqrt(2)). (End)

A099581 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k-1)*3^(n-k-1).

Original entry on oeis.org

0, 0, 1, 3, 15, 54, 216, 810, 3105, 11745, 44631, 169128, 641520, 2431944, 9221121, 34959195, 132543135, 502506990, 1905156936, 7222991778, 27384465825, 103822372809, 393620574951, 1492328843280, 5657848431840, 21450531825360
Offset: 0

Views

Author

Paul Barry, Oct 23 2004

Keywords

Comments

In general a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k-1)*r^(n-k-1) has g.f. x^2/((1-r*x^2)*(1-r*x-r*x^2)) and satisfies a(n) = r*a(n-1) + 2*r*a(n-2) - r^2*a(n-3) - r^2*a(n-4).

Links

Crossrefs

Cf. A030195, A099177, A099582.

Programs

  • Magma
    [n le 4 select Floor((n-1)^2/3) else 3*Self(n-1) +6*Self(n-2) -9*Self(n-3) -9*Self(n-4): n in [1..41]]; // G. C. Greubel, Jul 23 2022
  • Mathematica
    LinearRecurrence[{3,6,-9,-9},{0,0,1,3},40] (* Harvey P. Dale, Jun 07 2021 *)
  • SageMath
    @CachedFunction
def a(n):
    if (n<4): return floor(n^2/3)
    else: return 3*a(n-1) + 6*a(n-2) - 9*a(n-3) - 9*a(n-4)
[a(n) for n in (0..40)] # G. C. Greubel, Jul 23 2022

Formula

G.f.: x^2/((1-3*x^2)*(1-3*x-3*x^2)).
a(n) = 3*a(n-1) + 6*a(n-2) - 9*a(n-3) - 9*a(n-4).
From G. C. Greubel, Jul 23 2022: (Start)
a(n) = (2*(-i*sqrt(3))^(n-1)*ChebyshevU(n-1, i*sqrt(3)/2) - (1-(-1)^n)*3^((n - 1)/2))/6.
E.g.f.: (4*exp(3*x/2)*sinh(sqrt(21)*x/2) - 2*sqrt(7)*sinh(sqrt(3)*x))/(6*sqrt(21)). (End)

A099176 a(n) = 2*a(n-1) + 4*a(n-2) - 4*a(n-3) - 4*a(n-4).

Original entry on oeis.org

1, 1, 4, 8, 24, 60, 168, 448, 1232, 3344, 9152, 24960, 68224, 186304, 509056, 1390592, 3799296, 10379520, 28357632, 77473792, 211662848, 578272256, 1579870208, 4316282880, 11792306176, 32217174016, 88018960384, 240472260608, 656982441984, 1794909388800
Offset: 0

Views

Author

Paul Barry, Oct 02 2004

Keywords

Comments

Form the 6 node graph with matrix A=[1,1,1,1,0,0; 1,1,0,0,1,1; 1,0,0,0,0,0; 1,0,0,0,0,0; 0,1,0,0,0,0; 0,1,0,0,0,0]. Then a(n) counts closed walks of length n at either of the degree 5 vertices.

Links

Crossrefs

Cf. A002605, A099177.

Formula

G.f.: (1+x)*(1-2*x)/((1-2x^2)(1-2x-2x^2)).
a(n) = (3+sqrt(3))(1+sqrt(3))^n/12+(3-sqrt(3))(1-sqrt(3))^n/12+2^((n-4)/2)(1+(-1)^n).
a(n) = A002605(n)/2+2^((n-4)/2)(1+(-1)^n).
E.g.f.: (3*cosh(sqrt(2)*x) + exp(x)*(3*cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x)))/6. - Stefano Spezia, Jun 07 2025
Showing 1-3 of 3 results.

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