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 1-4 of 4 results.

A189315 Expansion of g.f. 5*(1-3*x+x^2)/(1-5*x+5*x^2).

Original entry on oeis.org

5, 10, 30, 100, 350, 1250, 4500, 16250, 58750, 212500, 768750, 2781250, 10062500, 36406250, 131718750, 476562500, 1724218750, 6238281250, 22570312500, 81660156250, 295449218750, 1068945312500, 3867480468750, 13992675781250, 50625976562500, 183166503906250, 662702636718750
Offset: 0

Views

Author

L. Edson Jeffery, Apr 20 2011

Keywords

Comments

Let A be the unit-primitive matrix (see [Jeffery])
A=A_(10,1)=
(0 1 0 0 0)
(1 0 1 0 0)
(0 1 0 1 0)
(0 0 1 0 1)
(0 0 0 2 0).
Then a(n) = Trace(A^(2*n)).
Evidently one of a class of accelerator sequences for Catalan's constant based on traces of successive powers (here they are A^(2*n)) of a unit-primitive matrix A_(N,r) (0
From Tom Copeland, Dec 08 2015: (Start)
These are also the non-vanishing traces for the adjacency matrices of the simple Lie algebras B_5 and C_5. See links for B_4, A265185, and B_3, A025192.
a(n+1) = 10 * A081567(n), and, ignoring a(0), a G.F. is 10 *(1-2*x)/(1-5*x+5*x^2) whose denominator is y^5 * A127672(5,1/y) with y = sqrt(x).
-log(1 - 5x^2 + 5x^4) = 10 x^2/2 + 30 x^4/4 + ... provides a logarithmic series for the traces of both the odd and even powers of the matrix beginning with the first power. (End)

Links

Crossrefs

Cf. A147748, A189316, A189317, A189318.
Cf. A025192, A081567, A127672, A265185.

Programs

  • Magma
    I:=[5,10,30]; [n le 3 select I[n] else 5*Self(n-1)-5*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Dec 09 2015
  • Mathematica
    CoefficientList[Series[5(1-3x+x^2)/(1-5x+5x^2),{x,0,40}],x] (* or *)
Join[{5},LinearRecurrence[{5,-5},{10,30},40]]  (* Harvey P. Dale, Apr 25 2011 *)
  • PARI
    Vec(5*(1-3*x+x^2)/(1-5*x+5*x^2)+O(x^99)) \\ Charles R Greathouse IV, Sep 24 2012

Formula

a(n) = 5*a(n-1)-5*a(n-2), n>2, a(0)=5, a(1)=10, a(2)=30.
a(n) = Sum_{k=1..5} (w_k)^(2*n), w_k=2*cos((2*k-1)*Pi/10).
a(n) = 2^(1-n)*((5-Sqrt(5))^n+(5+Sqrt(5))^n), for n>0, with a(0)=5.
a(n) = 5*A147748(n).
E.g.f.: 1 + 4*exp(5*x/2)*cosh(sqrt(5)*x/2). - Stefano Spezia, Jul 09 2024

A189316 Expansion of g.f. 5*(1-x-x^2)/((1+x)*(1-3*x+x^2)).

Original entry on oeis.org

5, 5, 15, 35, 95, 245, 645, 1685, 4415, 11555, 30255, 79205, 207365, 542885, 1421295, 3720995, 9741695, 25504085, 66770565, 174807605, 457652255, 1198149155, 3136795215, 8212236485, 21499914245, 56287506245, 147362604495, 385800307235, 1010038317215
Offset: 0

Views

Author

L. Edson Jeffery, Apr 20 2011

Keywords

Comments

(Start) Let A be the unit-primitive matrix (see [Jeffery])
A=A_(10,2)=
(0 0 1 0 0)
(0 1 0 1 0)
(1 0 1 0 1)
(0 1 0 2 0)
(0 0 2 0 1).
Then a(n)=Trace(A^n). For m=1,2,..., A^(m) can also be written
A^(m)=
[ F(m-1)^2 0 F(m)^2 0 F(m-1)*F(m) ]
[ 0 F(2*m-1) 0 F(2*m) 0 ]
[ F(m)^2 0 F(m+1)^2 0 F(m)*F(m+1) ]
[ 0 F(2*m) 0 F(2*m+1) 0 ]
[ 2*F(m-1)*F(m) 0 2*F(m)*F(m+1) 0 F(2*m+1)-F(m)*F(m+1) ],
where F(m-1)=A000045(n) are the Fibonacci numbers and m=n+1. Hence also a(n+1)=Trace(A^(n+1))=F(m-1)^2+F(2*m-1)+F(m+1)^2+2*F(2*m+1)-F(m)*F(m+1). (End)
Evidently one of a class of accelerator sequences for Catalan's constant based on traces of successive powers of a unit-primitive matrix A_(N,r), 0

Links

Crossrefs

Cf. A000045, A061646, A189315, A189317, A189318.

Programs

  • Mathematica
    CoefficientList[Series[5 (1-x-x^2)/((1+x)(1-3x+x^2)),{x,0,40}],x] (* or *) LinearRecurrence[{2,2,-1},{5,5,15},40] (* Harvey P. Dale, Nov 26 2016 *)

Formula

G.f.: 5*(1-x-x^2)/((1+x)*(1-3*x+x^2)).
a(n) = 2*a(n-1)+2*a(n-2)-a(n-3), n>2, a(0)=5, a(1)=5, a(2)=15.
a(n) = Sum_{k=1..5} ((w_k)^2-1)^n, w_k = 2*cos((2*k-1)*Pi/10).
a(n) = (-1)^n+2*(1/tau^(2*n)+tau^(2*n)), tau = (1+sqrt(5))/2=1.618033....
a(n) = 5*A061646(n), n>=0 (offset for A061646 is -1).
E.g.f.: cosh(x) + 4*exp(3*x/2)*cosh(sqrt(5)*x/2) - sinh(x). - Stefano Spezia, Jul 09 2024

A189318 Expansion of 5*(1-2*x)/(1-3*x-2*x^2+4*x^3).

Original entry on oeis.org

5, 5, 25, 65, 225, 705, 2305, 7425, 24065, 77825, 251905, 815105, 2637825, 8536065, 27623425, 89391105, 289275905, 936116225, 3029336065, 9803137025, 31723618305, 102659784705, 332214042625, 1075067224065, 3478990618625, 11258250133505
Offset: 0

Views

Author

L. Edson Jeffery, Apr 20 2011

Keywords

Comments

(Start) Let A be the unit-primitive matrix (see [Jeffery])
A=A_(10,4)=
(0 0 0 0 1)
(0 0 0 2 0)
(0 0 2 0 1)
(0 2 0 2 0)
(2 0 2 0 1).
Then a(n)=Trace(A^n). (End)
Evidently one of a class of accelerator sequences for Catalan's constant based on traces of successive powers of a unit-primitive matrix A_(N,r) (0

Links

Crossrefs

Cf. A052899.
A189315, A189316, A189317.

Programs

  • Mathematica
    CoefficientList[Series[5(1-2x)/(1-3x-2x^2+4x^3),{x,0,30}],x] (* or *) LinearRecurrence[{3,2,-4},{5,5,25},30] (* Harvey P. Dale, Jun 02 2014 *)
  • PARI
    Vec(5*(1-2*x)/(1-3*x-2*x^2+4*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 25 2012

Formula

G.f.: 5*(1-2*x)/(1-3*x-2*x^2+4*x^3).
a(n)=3*a(n-1)+2*a(n-2)-4*a(n-3), n>3, a(0)=5, a(1)=5, a(2)=25, a(3)=65.
a(n)=Sum_{k=1..5} ((w_k)^4-3*(w_k)^2+1)^n, w_k=2*cos((2*k-1)*Pi/10).
a(n)=1+2*(1-Sqrt(5))^n+2*(1+Sqrt(5))^n.
a(n)=5*A052899(n).

A189334 Expansion of g.f. (1-6*x+x^2)/(1-10*x+5*x^2).

Original entry on oeis.org

1, 4, 36, 340, 3220, 30500, 288900, 2736500, 25920500, 245522500, 2325622500, 22028612500, 208658012500, 1976437062500, 18721080562500, 177328620312500, 1679680800312500, 15910164901562500, 150703245014062500, 1427481625632812500, 13521300031257812500, 128075592184414062500
Offset: 0

Views

Author

L. Edson Jeffery, Apr 20 2011

Keywords

Comments

(Start) Let A be the unit-primitive matrix (see [Jeffery])
A=A_(10,3)=
(0 0 0 1 0)
(0 0 1 0 1)
(0 1 0 2 0)
(1 0 2 0 1)
(0 2 0 2 0).
Then a(n)=(1/5)*Trace(A^(2*n)). (See also A189317.) (End)
Evidently one of a class of accelerator sequences for Catalan's constant based on traces of successive powers (here they are A^(2*n)) of a unit-primitive matrix A_(N,r) (0

Links

Crossrefs

Cf. A189315, A189316, A189317, A189318.

Programs

  • Mathematica
    LinearRecurrence[{10,-5},{1,4,36},22] (* Stefano Spezia, Jul 09 2024 *)

Formula

a(n) = 10*a(n-1) - 5*a(n-2), n>2, a(0)=1, a(1)=4, a(2)=36.
a(n) = (1/5)*Sum_{k=1..5} ((w_k)^3-2*w_k)^(2*n), w_k = 2*cos((2*k-1)*Pi/10).
From Stefano Spezia, Jul 09 2024: (Start)
a(n) = 2*((5 - 2*sqrt(5))^n + (5 + 2*sqrt(5))^n)/5 for n > 0.
E.g.f.: (1 + 4*exp(5*x)*cosh(2*sqrt(5)*x))/5. (End)

Extensions

a(20)-a(21) from Stefano Spezia, Jul 09 2024
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