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.

A099484 A Fibonacci convolution.

Original entry on oeis.org

1, 1, 2, 7, 19, 48, 125, 329, 862, 2255, 5903, 15456, 40465, 105937, 277346, 726103, 1900963, 4976784, 13029389, 34111385, 89304766, 233802911, 612103967, 1602508992, 4195423009, 10983760033, 28755857090, 75283811239, 197095576627
Offset: 0

Views

Author

Paul Barry, Oct 18 2004

Keywords

Comments

A Chebyshev transform of the sequence 1,1,3,9,27 with g.f. (1-2x)/(1-3x). The image of G(x) under the Chebyshev transform is (1/(1+x^2))G(x/(1+x^2)).

Links

Crossrefs

Cf. A000045, A099483, A099485.

Programs

  • Mathematica
    LinearRecurrence[{3,-2,3,-1},{1,1,2,7},40] (* Harvey P. Dale, Mar 25 2020 *)

Formula

G.f.: (1-x)^2/((1+x^2)*(1-3*x+x^2)), convolution of A176742 and A001906.
a(n)=3a(n-1)-2a(n-2)+3a(n-3);
a(n)=sum{k=0..floor(n/2), binomial(n-k, k)(-1)^n*(3^(n-2k)+2*0^(n-2k))/3};
a(n)=sum{k=0..n, (0^k-2sin(pi*k/2))F(2(n-k)+2)}.
(1/3) [Fib(2n+2) + I^n + (-I)^n ]. - Ralf Stephan, Dec 04 2004
3*a(n) = A001906(n+1) +2*A056594(n). - R. J. Mathar, Jun 17 2020

A099485 A Fibonacci convolution.

Original entry on oeis.org

1, 2, 5, 14, 37, 96, 251, 658, 1723, 4510, 11807, 30912, 80929, 211874, 554693, 1452206, 3801925, 9953568, 26058779, 68222770, 178609531, 467605822, 1224207935, 3205017984, 8390846017, 21967520066, 57511714181, 150567622478
Offset: 0

Views

Author

Paul Barry, Oct 18 2004

Keywords

Comments

A Chebyshev transform of A025192 with g.f. (1-x)/(1-3*x). The image of G(x) under the Chebyshev transform is (1/(1+x^2))*G(x/(1+x^2)).

Links

Crossrefs

Cf. A000045, A099483, A099484, A001045, A128834.

Programs

  • Mathematica
    LinearRecurrence[{3,-2,3,-1},{1,2,5,14},30] (* Harvey P. Dale, Jul 06 2017 *)

Formula

G.f.: (1-x+x^2)/((1+x^2)*(1-3*x+x^2)).
a(n) = 3*a(n-1)-2*a(n-2)+3*a(n-3).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-1)^n*(2*3^(n-2*k)+0^(n-2*k))/3.
a(n) = Sum_{k=0..n} (0^k-sin(Pi*k/2))*Fibonacci(2*(n-k)+2).
a(n) = (1/6) * (4*Fibonacci(2*n+2) + I^n + (-I)^n). - Ralf Stephan, Dec 04 2004
Also a transformation of the Jacobsthal numbers A001045(n+1) under the mapping G(x)-> (1/(1-x+x^2))*G(x/(1-x+x^2)). - Paul Barry, Dec 11 2004
G.f.: g(f(x))/x, where g is g.f. of A001045 and f is g.f. of A128834. - Oboifeng Dira, Jun 21 2020

A049678 a(n) = F(8*n+4)/3, where F=A000045 (the Fibonacci sequence).

Original entry on oeis.org

1, 48, 2255, 105937, 4976784, 233802911, 10983760033, 516002918640, 24241153416047, 1138818207635569, 53500214605455696, 2513371268248782143, 118074949393087305025, 5547009250206854554032, 260591359810329076734479, 12242246901835259751966481
Offset: 0

Views

Author

Clark Kimberling

Keywords

Examples

			a(2) = F(8 * 2 + 4) / 3 = F(20) / 3 = 6765 / 3 = 2255. - _Indranil Ghosh_, Feb 04 2017

Links

Crossrefs

Cf. A004187, A099483, A100047.

Programs

  • Magma
    [Fibonacci(8*n+4)/3: n in [0..30]]; // G. C. Greubel, Dec 02 2017
  • Mathematica
    CoefficientList[Series[(1+x)/(1-47x+x^2),{x,0,20}],x]  (* Harvey P. Dale, Feb 18 2011 *)
Table[Fibonacci[8*n+4]/3, {n,0,30}] (* G. C. Greubel, Dec 02 2017 *)
  • PARI
    for(n=0,30, print1(fibonacci(8*n+4)/3, ", ")) \\ G. C. Greubel, Dec 02 2017

Formula

a(n) = 47*a(n-1) - a(n-2), n>1. a(0)=1, a(1)=48.
G.f.: (1+x)/(1-47*x+x^2).
From Peter Bala, Mar 23 2015: (Start)
a(n) = A004187(2*n + 1); a(n) = A099483(4*n + 1).
a(n) = ( Fibonacci(8*n + 8 - 2*k) + Fibonacci(8*n + 2*k) )/( Fibonacci(8 - 2*k) + Fibonacci(2*k) ), for k an arbitrary integer.
a(n) = ( Fibonacci(8*n + 8 - 2*k - 1) - Fibonacci(8*n + 2*k + 1) )/( Fibonacci(8 - 2*k - 1) - Fibonacci(2*k + 1) ), for k an arbitrary integer.
The aerated sequence (b(n))n>=1 = [1, 0, 48, 0, 2255, 0, 105937, 0, ...] is a fourth-order linear divisibility sequence; that is, if n | m then b(n) | b(m). It is the case P1 = 0, P2 = -45, Q = -1 of the 3-parameter family of divisibility sequences found by Williams and Guy. See A100047 for the connection with Chebyshev polynomials. (End)

Extensions

Better description and more terms from Michael Somos
2 more terms from Indranil Ghosh, Feb 04 2017

A140824 Expansion of (x-x^3)/(1-3*x+2*x^2-3*x^3+x^4).

Original entry on oeis.org

0, 1, 3, 6, 15, 41, 108, 281, 735, 1926, 5043, 13201, 34560, 90481, 236883, 620166, 1623615, 4250681, 11128428, 29134601, 76275375, 199691526, 522799203, 1368706081, 3583319040, 9381251041, 24560434083, 64300051206, 168339719535, 440719107401, 1153817602668
Offset: 0

Views

Author

N. J. A. Sloane, Sep 07 2009, based on email from R. K. Guy, Mar 09 2009

Keywords

Comments

Case P1 = 3, P2 = 0, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Mar 25 2014

Links

Crossrefs

Cf. A006238, A005248, A054493, A078070, A092184, A098306, A100047, A100048, A108196, A138573, A152090, A218134.

Programs

  • Mathematica
    LinearRecurrence[{3, -2, 3, -1}, {0, 1, 3, 6}, 50] (* G. C. Greubel, Aug 08 2017 *)
  • PARI
    x='x+O('x^50); concat([0], Vec((x-x^3)/(1-3*x+2*x^2-3*x^3+x^4))) \\ G. C. Greubel, Aug 08 2017

Formula

a(0) = 0, a(1) = 1, a(2) = 3, a(3) = 6, a(n) - 3 a(n + 1) + 2 a(n + 2) - 3 a(n + 3) + a(n + 4) = 0.
From Peter Bala, Mar 25 2014: (Start)
a(n) = 2/3*( T(n,3/2) - T(n,0) ), where T(n,x) is a Chebyshev polynomial of the first kind.
a(n) = 1/3 * (A005248(n) - (i^n + (-i)^n)) = 1/3 * (Fibonacci(2*n-1) + Fibonacci(2*n+1) - (i^n + (-i)^n)).
a(n) = bottom left entry of the 2 X 2 matrix 2*T(n, 1/2*M), where M is the 2 X 2 matrix [0, 0; 1, 3].
The o.g.f. is the Hadamard product of the rational functions x/(1 - 1/sqrt(2)*(sqrt(5) + i)*x + x^2) and x/(1 - 1/sqrt(2)*(sqrt(5) - i)*x + x^2). See the remarks in A100047 for the general connection between Chebyshev polynomials and 4th-order linear divisibility sequences. (End)
a(n) = A099483(n) - A099483(n-2). - R. J. Mathar, Feb 10 2016
Showing 1-4 of 4 results.

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

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