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 1-9 of 9 results.

A156331 a(n)=8*A154811(n).

Original entry on oeis.org

8, 16, 40, 32, 56, 64, 64, 56, 32, 40, 16, 8, 8, 16, 40, 32, 56, 64, 64, 56, 32, 40, 16, 8, 8, 16, 40, 32, 56, 64, 64, 56, 32, 40, 16, 8, 8, 16, 40, 32, 56, 64, 64, 56, 32, 40, 16, 8, 8, 16, 40, 32, 56, 64, 64, 56, 32, 40, 16, 8, 8, 16, 40, 32, 56, 64, 64, 56, 32, 40, 16, 8, 8, 16, 40
Offset: 0

Views

Author

Paul Curtz, Feb 08 2009

Keywords

Links

Programs

  • Mathematica
    8*Mod[Fibonacci[Range[1,151,2]],9] (* or *) PadRight[{},80,{8,16,40,32,56,64,64,56,32,40,16,8}] (* Harvey P. Dale, Jul 10 2018 *)

Formula

Period length 12: a(n)=a(n-12).
a(n) = A154811(n+6) = A155110(n) (mod 9).

Extensions

Edited and extended by R. J. Mathar, Apr 10 2009

A153990 Period 6: repeat [1, 2, 5, 4, 7, 8].

Original entry on oeis.org

1, 2, 5, 4, 7, 8, 1, 2, 5, 4, 7, 8, 1, 2, 5, 4, 7, 8, 1, 2, 5, 4, 7, 8, 1, 2, 5, 4, 7, 8, 1, 2, 5, 4, 7, 8, 1, 2, 5, 4, 7, 8, 1, 2, 5, 4, 7, 8, 1, 2, 5, 4, 7, 8, 1, 2, 5, 4, 7, 8, 1, 2, 5, 4, 7, 8, 1, 2, 5, 4, 7, 8, 1, 2, 5, 4, 7, 8, 1, 2, 5, 4, 7, 8, 1, 2
Offset: 0

Views

Author

Paul Curtz, Jan 04 2009

Keywords

Comments

Shares digits with other 6-periodic sequences, see the list in A153130.
Also the decimal expansion of the constant 13942/111111. [R. J. Mathar, Jan 23 2009]

Links

Crossrefs

Cf. A131533, A141425, A153130, A154811.

Programs

Formula

a(n) - A141425(n) = A131533(n+2).
a(6n+0) + a(6n+5) = a(6n+1) + a(6n+4) = a(6n+2) + a(6n+3) = 9.
G.f.: (1+2*x+5*x^2+4*x^3+7*x^4+8*x^5)/((1-x)*(1+x)*(1+x+x^2)*(x^2-x+1)). [R. J. Mathar, Jan 23 2009]
From Wesley Ivan Hurt, Jun 17 2016: (Start)
a(n) = (27-cos(n*Pi)-8*sqrt(3)*cos((1-4*n)*Pi/6)-16*sin((1+2*n)*Pi/6))/6.
a(n) = a(n-6) for n>5. (End)

Extensions

Edited by R. J. Mathar, Jan 23 2009

A155110 a(n) = 8*Fibonacci(2n+1).

Original entry on oeis.org

8, 16, 40, 104, 272, 712, 1864, 4880, 12776, 33448, 87568, 229256, 600200, 1571344, 4113832, 10770152, 28196624, 73819720, 193262536, 505967888, 1324641128, 3467955496, 9079225360, 23769720584, 62229936392, 162920088592, 426530329384, 1116670899560
Offset: 0

Views

Author

Paul Curtz, Jan 20 2009

Keywords

Links

Crossrefs

Cf. A001519, A027941, A099496, A122367, A153873, A154811, A156551.

Programs

Formula

a(n) = 8*A001519(n+1) = 8*A122367(n) = 8 *|A099496(n)|.
a(n) == A154811(n+6) (mod 9).
a(n) == A156551(n) (mod 10).
a(n) = A153873(n) - A027941(n).
G.f.: 8*(1 - x)/(1 - 3*x + x^2). - G. C. Greubel, Apr 21 2021

Extensions

Comments converted to formulas by R. J. Mathar, Oct 06 2009

A154815 Period 6: repeat [8, 7, 4, 5, 2, 1].

Original entry on oeis.org

8, 7, 4, 5, 2, 1, 8, 7, 4, 5, 2, 1, 8, 7, 4, 5, 2, 1, 8, 7, 4, 5, 2, 1, 8, 7, 4, 5, 2, 1, 8, 7, 4, 5, 2, 1, 8, 7, 4, 5, 2, 1, 8, 7, 4, 5, 2, 1, 8, 7, 4, 5, 2, 1, 8, 7, 4, 5, 2, 1, 8, 7, 4, 5, 2, 1, 8, 7, 4, 5, 2, 1, 8, 7, 4, 5, 2, 1, 8, 7, 4, 5, 2, 1, 8, 7
Offset: 0

Views

Author

Paul Curtz, Jan 15 2009

Keywords

Comments

Obtained through reversion of the period in A153990, or by taking a half period of A154811.
Shares digits with other 6-periodic sequences, see the list in A153130.
Also the decimal expansion of the constant 97169/111111. [R. J. Mathar, Jan 23 2009]

Links

Crossrefs

Cf. A153130, A153990, A154811.

Programs

Formula

a(n) = (8*A153990(n)) mod 9.
G.f.: (8+7*x+4*x^2+5*x^3+2*x^4+x^5)/((1-x)*(1+x)*(1+x+x^2)(x^2-x+1)). [R. J. Mathar, Jan 23 2009]
From Wesley Ivan Hurt, Jun 23 2016: (Start)
a(n) = a(n-6) for n>5.
a(n) = (27 + cos(n*Pi) + 8*cos(n*Pi/3) + 12*cos(2*n*Pi/3) + 8*sqrt(3)*sin(n*Pi/3) + 4*sqrt(3)*sin(2*n*Pi/3))/6. (End)

Extensions

Edited by R. J. Mathar, Jan 23 2009

A156346 Palindromic period of length 12: repeat 1,2,-4,4,-2,-1,-1,-2,4,-4,2,1.

Original entry on oeis.org

1, 2, -4, 4, -2, -1, -1, -2, 4, -4, 2, 1, 1, 2, -4, 4, -2, -1, -1, -2, 4, -4, 2, 1, 1, 2, -4, 4, -2, -1, -1, -2, 4, -4, 2, 1, 1, 2, -4, 4, -2, -1, -1, -2, 4, -4, 2, 1
Offset: 0

Views

Author

Paul Curtz, Feb 08 2009

Keywords

Links

Crossrefs

Cf. A156283.

Programs

  • Mathematica
    PadRight[{},50,{1,2,-4,4,-2,-1,-1,-2,4,-4,2,1}] (* Harvey P. Dale, May 29 2021 *)

Formula

a(n) == A154811(n) (mod 9).
a(n) = ( (2*A154811(n)) mod 9 ) - A154811(n).
G.f.: -(x-1)*(x^4+3*x^3-x^2+3*x+1) / ( (1+x^2)*(x^4-x^2+1) ). - R. J. Mathar, Mar 08 2011

A156536 Period 12: repeat 7,5,-1,1,-5,-7,-7,-5,1,-1,5,7.

Original entry on oeis.org

7, 5, -1, 1, -5, -7, -7, -5, 1, -1, 5, 7, 7, 5, -1, 1, -5, -7, -7, -5, 1, -1, 5, 7, 7, 5, -1, 1, -5, -7, -7, -5, 1, -1, 5, 7, 7, 5, -1, 1, -5, -7, -7, -5, 1, -1, 5, 7, 7, 5, -1, 1, -5, -7, -7, -5, 1, -1, 5, 7, 7, 5, -1, 1, -5, -7, -7, -5, 1, -1, 5, 7, 7, 5, -1, 1, -5, -7, -7, -5, 1, -1, 5, 7, 7, 5
Offset: 0

Views

Author

Paul Curtz, Feb 09 2009

Keywords

Links

Crossrefs

Cf. A154870.

Programs

  • Mathematica
    PadRight[{},120,{7,5,-1,1,-5,-7,-7,-5,1,-1,5,7}] (* or *) LinearRecurrence[{0,0,0,0,0,-1},{7,5,-1,1,-5,-7},120] (* Harvey P. Dale, Oct 15 2023 *)

Formula

a(n) = A154811(n+6) - A154811(n).
From R. J. Mathar, Apr 10 2009: (Start)
a(n) = -a(n-6).
G.f.: -(x-1)*(7*x^4 + 12*x^3 + 11*x^2 + 12*x + 7)/((1+x^2)*(x^4 - x^2 + 1)). (End)
G.f.: (-7*x^5 - 5*x^4 + x^3 - x^2 + 5*x + 7)/(x^6 + 1). - Chai Wah Wu, Feb 16 2021

Extensions

Edited by R. J. Mathar, Apr 10 2009

A156561 Floor(Fibonacci(2n+1)/9).

Original entry on oeis.org

0, 0, 0, 1, 3, 9, 25, 67, 177, 464, 1216, 3184, 8336, 21824, 57136, 149585, 391619, 1025273, 2684201, 7027331, 18397793, 48166048, 126100352, 330135008, 864304672, 2262779008, 5924032352, 15509318049, 40603921795, 106302447337, 278303420217
Offset: 0

Views

Author

Paul Curtz, Feb 10 2009

Keywords

Links

Crossrefs

Cf. A069403.

Programs

  • Mathematica
    Floor[Fibonacci[2*Range[0,30]+1]/9] (* or *) LinearRecurrence[{4,-4,1,0,0,-1,4,-4,1},{0,0,0,1,3,9,25,67,177},31] (* Harvey P. Dale, Jun 06 2016 *)

Formula

a(n) = ( A000045(2n+1)-A154811(n) )/9 = floor(A122367(n)/9) = floor(A001519(n+1)/9) = floor( |A099496(n)|/9).
a(n)=3a(n-1)-a(n-2)+|A112690(n+10)|, i.e., a(n)-3a(n-1)+a(n-2) is a sequence of period 12 containing 0's and 1's. - R. J. Mathar, Feb 23 2009
G.f.: (1-x+x^2)/((1-x)(1+x^2)(1-3x+x^2)(1-x^2+x^4)). - R. J. Mathar, Feb 23 2009

Extensions

Edited and extended by R. J. Mathar, Jan 23 2009, Feb 23 2009

A156872 Period 12: 1,3,-1,3,1,0,-1,-3,1,-3,-1,0 repeated.

Original entry on oeis.org

1, 3, -1, 3, 1, 0, -1, -3, 1, -3, -1, 0, 1, 3, -1, 3, 1, 0, -1, -3, 1, -3, -1, 0, 1, 3, -1, 3, 1, 0, -1, -3, 1, -3, -1, 0, 1, 3, -1, 3, 1, 0, -1, -3, 1, -3, -1, 0, 1, 3, -1, 3, 1, 0, -1, -3, 1, -3, -1, 0
Offset: 0

Views

Author

Paul Curtz, Feb 17 2009

Keywords

Comments

First differences of A154811.

Links

Programs

  • Mathematica
    PadRight[{},70,{1,3,-1,3,1,0,-1,-3,1,-3,-1,0}] (* Harvey P. Dale, Sep 23 2012 *)

Formula

Palindromic properties: a(n+6)= -a(n). a(12k+i)=a(12k+4-i), i=0..2. a(12k+5+i)=a(12k+11-i), i=0..3.
a(n) = A156194(n+1)-A156194(n+7) = A156194(n+1)-A156199(n+1).
a(n) = A156227(n+1) (mod 9).
a(n+1) -a(n)= A156346(n+1).
a(n)=A056594(n)+3*A014021(n-1). G.f.: (1+3*x-x^2+3*x^3+x^4)/((1+x^2)*(x^4-x^2+1)). - R. J. Mathar, Feb 23 2009

Extensions

Edited, formulas commenting other sequences removed, by R. J. Mathar, Feb 23 2009

A168037 Period length 18: repeat 0,1,2,0,8,7,0,4,5,0,5,4,0,7,8,0,2,1.

Original entry on oeis.org

0, 1, 2, 0, 8, 7, 0, 4, 5, 0, 5, 4, 0, 7, 8, 0, 2, 1, 0, 1, 2, 0, 8, 7, 0, 4, 5, 0, 5, 4, 0, 7, 8, 0, 2, 1, 0, 1, 2, 0, 8, 7, 0, 4, 5, 0, 5, 4, 0, 7, 8, 0, 2, 1
Offset: 0

Views

Author

Paul Curtz, Nov 17 2009

Keywords

Comments

Represents also the decimal expansion of 447668335336223/37037037037037037.

Links

Crossrefs

Cf. A154811.

Programs

  • Mathematica
    PadRight[{},60,{0,1,2,0,8,7,0,4,5,0,5,4,0,7,8,0,2,1}] (* Harvey P. Dale, Jan 17 2021 *)

Formula

a(n) = A129194(n) mod 9.
a(n) = a(n-18).
a(n+1) = a(17-n), 0<=n<= 16, palindromic.
G.f. ( -x*(1+2*x+8*x^3+7*x^4+4*x^6+5*x^7+5*x^9+4*x^10+7*x^12+8*x^13+2*x^15+x^16) ) / ( (x-1)*(1+x+x^2)*(1+x^3+x^6)*(1+x)*(1-x+x^2)*(1-x^3+x^6) ). - R. J. Mathar, Jan 22 2011
Showing 1-9 of 9 results.

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

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