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.

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A177346 Decimal expansion of (1+sqrt(10))/3.

Original entry on oeis.org

1, 3, 8, 7, 4, 2, 5, 8, 8, 6, 7, 2, 2, 7, 9, 3, 1, 1, 0, 6, 6, 6, 2, 9, 7, 8, 4, 8, 1, 4, 4, 2, 3, 9, 5, 1, 1, 2, 3, 9, 8, 5, 1, 7, 1, 3, 1, 0, 8, 4, 0, 5, 6, 0, 8, 9, 5, 2, 5, 0, 1, 6, 1, 7, 5, 9, 7, 5, 3, 1, 4, 7, 9, 5, 4, 6, 4, 1, 2, 7, 4, 0, 4, 4, 8, 0, 8, 2, 7, 0, 2, 7, 9, 3, 1, 0, 0, 0, 9, 8, 3, 9, 5, 7, 8
Offset: 1

Views

Author

Klaus Brockhaus, May 07 2010

Keywords

Comments

Continued fraction expansion of (1+sqrt(10))/3 is A131534.
A root of x - 1/x = 2/3. - Gary W. Adamson, May 18 2023

Examples

			(1+sqrt(10))/3 = 1.38742588672279311066...

Crossrefs

Cf. A010467 (decimal expansion of sqrt(10)), A131534 (repeat 1, 2, 1).

Programs

  • Mathematica
    RealDigits[(1+Sqrt[10])/3,10,120][[1]] (* Harvey P. Dale, Jul 07 2011 *)
RealDigits[Exp@ ArcSinh[1/3], 10, 111][[1]] (* Robert G. Wilson v, Aug 17 2011 *)

Formula

Equals exp(arcsinh(1/3)). - Gary W. Adamson, Mar 16 2023

A201864 a(n) = ((F(n-1)+F(n-2))-1)/2 if F(n) is odd, otherwise a(n) = ((F(n-1)+F(n-2))-2)/2, where F(n) = A000045(n) is the n-th Fibonacci number.

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 6, 10, 16, 27, 44, 71, 116, 188, 304, 493, 798, 1291, 2090, 3382, 5472, 8855, 14328, 23183, 37512, 60696, 98208, 158905, 257114, 416019, 673134, 1089154, 1762288, 2851443, 4613732, 7465175, 12078908, 19544084, 31622992, 51167077, 82790070
Offset: 1

Views

Author

Giovanni Teofilatto, Dec 06 2011

Keywords

Comments

See also similar sequence A130578.
Numbers whose Zeckendorf representation is a prefix of 100100100... . - Jeffrey Shallit, Jun 29 2024

Links

Crossrefs

Cf. A000045, A131534.

Programs

  • Magma
    [IsOdd(Fibonacci(n)) select (Fibonacci(n)-1)/2 else Fibonacci(n)/2-1: n in [1..41]];  // Bruno Berselli, Dec 14 2011
  • Maple
    a:= n-> (Matrix(5, (i, j)-> `if`(i=j-1, 1, `if`(i=5,
        [-1, -1, 1, 1, 1][j], 0)))^n. <<-1, 0, 0, 0, 1>>)[1, 1]:
seq(a(n), n=1..50);  # Alois P. Heinz, Dec 13 2011
  • Mathematica
    CoefficientList[Series[x^3*(1+x)/((1-x)(1+x+x^2)(1-x-x^2)),{x,0,30}],x] (* Vincenzo Librandi, Mar 20 2012 *)

Formula

G.f.: x^4*(1+x)/((1-x)(1+x+x^2)(1-x-x^2)). - Alois P. Heinz, Dec 13 2011
a(n) = (1/2)*(A000045(n)-A131534(n+1)). - Bruno Berselli, Dec 14 2011
a(n) = F(n) - ceiling(F(n-1)/2) - ceiling(F(n-2)/2). - Chunqing Liu, Aug 21 2023

A052579 E.g.f. (2+x+x^2)/((1-x)(1+x+x^2)).

Original entry on oeis.org

2, 1, 2, 12, 24, 120, 1440, 5040, 40320, 725760, 3628800, 39916800, 958003200, 6227020800, 87178291200, 2615348736000, 20922789888000, 355687428096000, 12804747411456000, 121645100408832000, 2432902008176640000
Offset: 0

Views

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

Links

Programs

  • Maple
    spec := [S,{S=Union(Sequence(Prod(Z,Z,Z)), Sequence(Z))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
  • Mathematica
    With[{nn=20},CoefficientList[Series[(2+x+x^2)/((1-x)(1+x+x^2)),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 10 2019 *)

Formula

E.g.f.: -(x^2+x+2)/(-1+x)/(1+x+x^2)
Recurrence: {a(1)=1, a(2)=2, a(0)=2, (-11*n-6-n^3-6*n^2)*a(n)+a(n+3)=0}
(4/3+Sum(1/3*_alpha^(-n), _alpha=RootOf(_Z^2+_Z+1)))*n!
a(n) = n!*A131534(n+1). - R. J. Mathar, Nov 27 2011

A226122 Expansion of (1+2*x+x^2+x^3+2*x^4+x^5)/(1-2*x^3+x^6).

Original entry on oeis.org

1, 2, 1, 3, 6, 3, 5, 10, 5, 7, 14, 7, 9, 18, 9, 11, 22, 11, 13, 26, 13, 15, 30, 15, 17, 34, 17, 19, 38, 19, 21, 42, 21, 23, 46, 23, 25, 50, 25, 27, 54, 27, 29, 58, 29, 31, 62, 31, 33, 66, 33, 35, 70, 35, 37, 74, 37, 39, 78, 39
Offset: 0

Views

Author

Paul Curtz, May 27 2013

Keywords

Comments

A226023 (starting from A226023(-2)=0) and successive differences:
0, -1, 0, 2, 3, 6, 12, 15, 20, 30,...
-1, 1, 2, 1, 3, 6, 3, 5, 10, 5,... = a(n-1)
2, 1, -1, 2, 3, -3, 2, 5, -5, 2,...
-1, -2, 3, 1, -6, 5, 3, -10, 7, 5,...
-1, 5, -2, -7, 11, -2, -13, 17, -2, -19,...
6, -7, -5, 18, -13, -11, 30, -19, -17, 42,...
-13, 2, 23, -31, 2, 41, -49, 2, 59, 67,...
15, 21, -54, 33, 39, -90, 51, 57, -126, 69,... multiples of 3
6, -75, 87, 6, -129, 141, 6, -183, 195, 6,... multiples of 3
-81, 162, -81, -135, 270, -135, -189, 378, -189, -243,... multiples of 27
The last line is -27*a(n+3)*A131561(n+1).
The recurrences in the Formula field hold for the array.

Examples

			Given A130823 = 1,1,1,3,3,3,5,5,5,7,7,7,... and A131534 = 1,2,1,1,2,1,1,2,1,1,2,1,..., then a(0)=1*1=1, a(1)=1*2=2, a(2)=1*1=1, a(3)=3*1=3, a(4)=3*2=6, etc.
Given A226023(n) from A226023(-1)=-1, then a(0)=0-(-1)=1, a(1)=2-0=2, a(2)=3-2=1, a(3)=6-3=3, a(4)=12-6=6, etc.

Links

Crossrefs

Cf. A005408, A016754, A130823, A131534, A226023.

Programs

  • Mathematica
    repeat=20; Table[{1, 2, 1}, {repeat}]*(2*Range[repeat]-1) // Flatten
(* or *) Table[Floor[(2*n+1)/3]*Floor[(2*n+5)/3], {n, -1, 59}] // Differences (* Jean-François Alcover, May 29 2013 *)

Formula

a(n) = A130823(n-1) * A131534(n).
a(n) = A226023(n) - A226023(n-1) with A226023(-1)=-1.
a(n) = 3*a(n-3) -3*a(n-6) +a(n-9) = a(n-1) +2*a(n-3) -2*a(n-4) -a(n-6) +a(n-7). [Ralf Stephan]
From Bruno Berselli, May 29 2013: (Start)
G.f.: (1+x)^3*(1-x+x^2)/((1-x)^2*(1+x+x^2)^2).
a(n) = 2*a(n-3)-a(n-6).
a(3n)*a(3n-1)-a(3n-2) = A016754(n-1), n>0. (End)
Previous Showing 21-24 of 24 results.

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