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 11-17 of 17 results.

A138112 a(n)=3a(n-1)-4a(n-2)+2a(n-3)-a(n-4), a(0)=a(1)=a(2)=0, a(3)=1, a(4)=3.

Original entry on oeis.org

0, 0, 0, 1, 3, 5, 5, 0, -13, -34, -55, -55, 0, 144, 377, 610, 610, 0, -1597, -4181, -6765, -6765, 0, 17711, 46368, 75025, 75025, 0, -196418, -514229, -832040, -832040, 0, 2178309, 5702887, 9227465, 9227465, 0, -24157817, -63245986, -102334155, -102334155
Offset: 0

Views

Author

Paul Curtz, May 04 2008

Keywords

Comments

Obeys also the recurrence a(n)=5a(n-1)-10a(n-2)+10a(n-3)-5a(n-4)+2a(n-5), so the sequence is identical to its fifth differences (cf. A135356). a(n) = A138110(0,n): if A138110 is interpreted as an array with five rows, this is the top row.
The first differences are represented by A100334(n-1).
The 2nd differences are represented by A103311(n).
The 3rd differences are essentially represented by -A138003(n-2).
The 4th differences are represented by -A105371(n).
A102312 contains the absolute values of the terms which occur in pairs, for example a(5)=a(6)=5=A102312(1), a(10)=a(11)= -55 = -A102312(2).
Inverse BINOMIAL transform yields two zeros followed by A105384. - R. J. Mathar, Jul 04 2008

Links

Crossrefs

Cf. A138003, A103311, A105371.

Programs

  • Mathematica
    CoefficientList[Series[x^3/(1-3x+4x^2-2x^3+x^4),{x,0,45}],x] (* or *) LinearRecurrence[{3,-4,2,-1},{0,0,0,1},45] (* Harvey P. Dale, Jun 22 2011 *)

Formula

O.g.f.: x^3/(1-3x+4x^2-2x^3+x^4). - R. J. Mathar, Jul 04 2008

Extensions

Edited and extended by R. J. Mathar, Jul 04 2008

A134491 a(n) = Fibonacci(5n+4).

Original entry on oeis.org

3, 34, 377, 4181, 46368, 514229, 5702887, 63245986, 701408733, 7778742049, 86267571272, 956722026041, 10610209857723, 117669030460994, 1304969544928657, 14472334024676221, 160500643816367088
Offset: 0

Views

Author

Artur Jasinski, Oct 28 2007

Keywords

Links

Crossrefs

Cf. A000045, A016897, A102312, A099100, A134490.

Programs

Formula

G.f.: ( -3-x ) / ( -1+11*x+x^2 ). - R. J. Mathar, Apr 17 2011
a(n) = A000045(A016897(n)). - Michel Marcus, Nov 07 2013

A134489 a(n) = Fibonacci(5*n + 2).

Original entry on oeis.org

1, 13, 144, 1597, 17711, 196418, 2178309, 24157817, 267914296, 2971215073, 32951280099, 365435296162, 4052739537881, 44945570212853, 498454011879264, 5527939700884757, 61305790721611591, 679891637638612258
Offset: 0

Views

Author

Artur Jasinski, Oct 28 2007

Keywords

Comments

The o.g.f. of {F(m*n + 2)}_{n>=0}, for m = 1, 2, ..., is
G(m,x) = (1 + F(m - 2)*x) / (1 - L(m)*x + (-1)^m*x^2), with F = A000045 and F(-1) = 1, and L = A000032. - Wolfdieter Lang, Feb 06 2023

Links

Crossrefs

Cf. A000045, A001906, A001519, A033887, A015448, A014445, A033888-A033891, A102312, A099100, A134490-A134495, A103134, A134497-A134504.

Programs

  • Magma
    [Fibonacci(5*n+2): n in [0..50]]; // Vincenzo Librandi, Apr 20 2011
  • Mathematica
    Table[Fibonacci[5n + 2], {n, 0, 30}]
LinearRecurrence[{11,1},{1,13},20] (* Harvey P. Dale, May 05 2022 *)

Formula

From R. J. Mathar, Jul 04 2011: (Start)
G.f.: (-1-2*x) / (-1 + 11*x + x^2).
a(n) = 2*A049666(n) + A049666(n+1). (End)
a(n) = A000045(A016873(n)). - Michel Marcus, Nov 05 2013

A138110 Table T(d,n) read column by column: the n-th term in the sequence of the d-th differences of A138112, d=0..4.

Original entry on oeis.org

0, 0, 0, 1, -1, 0, 0, 1, 0, -1, 0, 1, 1, -1, -1, 1, 2, 0, -2, -1, 3, 2, -2, -3, 0, 5, 0, -5, -3, 3, 5, -5, -8, 0, 8, 0, -13, -8, 8, 13, -13, -21, 0, 21, 13, -34, -21, 21, 34, 0, -55, 0, 55, 34, -34, -55, 55, 89, 0, -89, 0, 144, 89, -89, -144, 144, 233, 0, -233, -144, 377, 233, -233, -377, 0, 610, 0, -610, -377, 377
Offset: 0

Views

Author

Paul Curtz, May 04 2008

Keywords

Comments

Ignoring signs, the sequence contains A000045(2)=1 ten times and each of the following Fibonacci numbers A000045(i>2) four times.

Examples

			All 5 rows of the table T(d,n) are:
.0,.0,.0,.1,.3,.5,.5,..0,-13,-34,-55,-55,...0,.144,...
.0,.0,.1,.2,.2,.0,-5,-13,-21,-21,..0,.55,.144,.233,...
.0,.1,.1,.0,-2,-5,-8,.-8,..0,.21,.55,.89,..89,...0,...
.1,.0,-1,-2,-3,-3,.0,..8,.21,.34,.34,..0,.-89,-233,...
-1,-1,-1,-1,.0,.3,.8,.13,.13,..0,-34,-89,-144,-144,...

Crossrefs

Cf. A102312 (A049666), A099100, A134489, A134490, A134491.

Formula

T(0,n)=A138112(n). T(d,n)= T(d-1,n+1)-T(d-1,n), d=1..4.
T(1,n)=A100334(n-1). T(2,n)=A103311(n). T(3,n) = -A138003(n-2). T(4,n)= -A105371(n).
sum_(d=0..4) T(d,n)=0 (columns sum to zero).

Extensions

Edited by R. J. Mathar, Jul 04 2008

A138384 Reverse groups of five Fibonacci numbers.

Original entry on oeis.org

3, 2, 1, 1, 0, 34, 21, 13, 8, 5, 377, 233, 144, 89, 55, 4181, 2584, 1597, 987, 610, 46368, 28657, 17711, 10946, 6765, 514229, 317811, 196418, 121393, 75025, 5702887, 3524578, 2178309, 1346269, 832040, 63245986, 39088169, 24157817, 14930352, 9227465, 701408733
Offset: 0

Views

Author

Paul Curtz, May 08 2008

Keywords

Links

Crossrefs

Cf. A102312 (and A049666), A099100, A134889, A134890, A134891, A000045.

Programs

  • Mathematica
    With[{nn=10},Flatten[Reverse/@Partition[Fibonacci[Range[0,5*nn-1]],5]]] (* or *) LinearRecurrence[{0,0,0,0,11,0,0,0,0,1},{3,2,1,1,0,34,21,13,8,5},50] (* Harvey P. Dale, Jan 20 2015 *)

Formula

a(n) = 11*a(n-5) + a(n-10).
G.f.: (5*x^9 - 3*x^8 + 2*x^7 - x^6 + x^5 + x^3 + x^2 + 2*x + 3)/(-x^10 - 11*x^5 + 1). - Alexander R. Povolotsky, May 08 2008

Extensions

Extended beyond a(24) by R. J. Mathar, Nov 30 2008

A269500 a(n) = Fibonacci(10*n).

Original entry on oeis.org

0, 55, 6765, 832040, 102334155, 12586269025, 1548008755920, 190392490709135, 23416728348467685, 2880067194370816120, 354224848179261915075, 43566776258854844738105, 5358359254990966640871840, 659034621587630041982498215, 81055900096023504197206408605
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 03 2016

Keywords

Comments

More generally, the ordinary generating function for the Fibonacci(k*n) is F(k)*x/(1 - L(k)*x + (-1)^k*x^2), where F(k) is the k-th Fibonacci number (A000045), L(k) is the k-th Lucas number (A000032), or (phi^k - (-1/phi)^k)*x/(sqrt(5)*(1 - (phi^k + (-1/phi)^k)*x + (-1)^k*x^2)), where phi is the golden ratio (A001622).

Links

Crossrefs

Cf. similar sequences of the form Fibonacci(k*n): A000045 (k = 1), A001906 (k = 2), A014445 (k = 3), A033888 (k = 4), A102312 (k = 5), A134492 (k = 6), A134498 (k = 7), A138473 (k = 8), A138590 (k = 9), this sequence (k = 10), A167398 (k = 11), A214855 (k = 15).
Cf. A000032 (Lucas numbers), A001622 (golden ratio).

Programs

  • Mathematica
    Fibonacci[10Range[0, 14]]
FullSimplify[Table[(((1 + Sqrt[5])/2)^(10 n) - (2/(1 + Sqrt[5]))^(10 n))/Sqrt[5], {n, 0, 12}]]
LinearRecurrence[{123, -1}, {0, 55}, 15]
  • PARI
    a(n) = fibonacci(10*n); \\ Michel Marcus, Mar 03 2016
  • PARI
    concat(0, Vec(55*x/(1-123*x+x^2) + O(x^100))) \\ Altug Alkan, Mar 04 2016

Formula

G.f.: 55*x/(1 - 123*x + x^2).
a(n) = 123*a(n-1) - a(n-2).
a(n) = A000045(10*n).
Lim_{n -> infinity} a(n + 1)/a(n) = phi^10 = 122.9918693812442…

A138298 First differences of A137976 after having added two leading ones.

Original entry on oeis.org

0, 2, 10, 21, 110, 233, 1220, 2584, 13530, 28657, 150050, 317811, 1664080, 3524578, 18454930, 39088169, 204668310, 433494437, 2269806340, 4807526976, 25172538050, 53316291173, 279167724890, 591286729879, 3096017511840
Offset: 0

Views

Author

Paul Curtz, May 07 2008

Keywords

Formula

a(n)= A137986(n)-A137986(n-1), n>=2.
a(2n)= 2*A102312(n)= 10*A049666(n). a(2n+1)= A134490(n).

Extensions

Edited and extended by R. J. Mathar, Jul 28 2008
Previous Showing 11-17 of 17 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.