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.

A050935 a(n) = a(n-1) - a(n-3) with a(1)=0, a(2)=0, a(3)=1.

Original entry on oeis.org

0, 0, 1, 1, 1, 0, -1, -2, -2, -1, 1, 3, 4, 3, 0, -4, -7, -7, -3, 4, 11, 14, 10, -1, -15, -25, -24, -9, 16, 40, 49, 33, -7, -56, -89, -82, -26, 63, 145, 171, 108, -37, -208, -316, -279, -71, 245, 524, 595, 350, -174, -769, -1119, -945, -176, 943, 1888, 2064, 1121, -767, -2831, -3952
Offset: 1

Views

Author

Richard J. Palmaccio (palmacr(AT)pinecrest.edu), Dec 31 1999

Keywords

Comments

The Ze3 sums, see A180662, of triangle A108299 equal the terms of this sequence without the two leading zeros. - Johannes W. Meijer, Aug 14 2011

References

  • R. Palmaccio, "Average Temperatures Modeled with Complex Numbers", Mathematics and Informatics Quarterly, pp. 9-17 of Vol. 3, No. 1, March 1993.

Links

Crossrefs

When run backwards this gives a signed version of A000931.
Cf. A099529.
Apart from signs, essentially the same as A078013.
Cf. A203400 (partial sums).

Programs

  • Haskell
    a050935 n = a050935_list !! (n-1)
a050935_list = 0 : 0 : 1 : zipWith (-) (drop 2 a050935_list) a050935_list
-- Reinhard Zumkeller, Jan 01 2012
  • Maple
    A050935 := proc(n) option remember; if n <= 1 then 0 elif n = 2 then 1 else A050935(n-1)-A050935(n-3); fi; end: seq(A050935(n), n=0..61);
  • Mathematica
    LinearRecurrence[{1,0,-1},{0,0,1},70] (* Harvey P. Dale, Jan 30 2014 *)
  • PARI
    a(n)=([0,1,0; 0,0,1; -1,0,1]^(n-1)*[0;0;1])[1,1] \\ Charles R Greathouse IV, Feb 06 2017

Formula

From Paul Barry, Oct 20 2004: (Start)
G.f.: x^2/(1-x+x^3).
a(n+2) = Sum_{k=0..floor(n/3)} binomial(n-2*k, k)*(-1)^k. (End)
G.f.: Q(0)*x^2/2, where Q(k) = 1 + 1/(1 - x*(12*k-1 + x^2)/( x*(12*k+5 + x^2 ) - 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 12 2013

Extensions

Offset adjusted by Reinhard Zumkeller, Jan 01 2012

A104769 Expansion of g.f. -x/(1+x-x^3).

Original entry on oeis.org

0, -1, 1, -1, 0, 1, -2, 2, -1, -1, 3, -4, 3, 0, -4, 7, -7, 3, 4, -11, 14, -10, -1, 15, -25, 24, -9, -16, 40, -49, 33, 7, -56, 89, -82, 26, 63, -145, 171, -108, -37, 208, -316, 279, -71, -245, 524, -595, 350, 174, -769, 1119, -945, 176, 943, -1888, 2064, -1121, -767, 2831, -3952
Offset: 0

Views

Author

Creighton Dement, Mar 24 2005

Keywords

Comments

Generating floretion is "jesright".
Pisano period lengths: 1, 7, 13, 14, 24, 91, 48, 28, 39, 168, 120, 182, 183, 336, 312, 56, 288, 273, 180, 168,.. (which differs from A104217 for example at index 23). - R. J. Mathar, Aug 10 2012

Links

Crossrefs

Apart from signs, essentially the same as A050935 and A078013.
Cf. A247917 (negative).
Cf. A000931, A057597.

Programs

  • Mathematica
    LinearRecurrence[{-1, 0, 1}, {0, -1, 1}, 61] (* or *)
CoefficientList[Series[-x/(1 + x - x^3), {x, 0, 60}], x] (* Michael De Vlieger, Jul 02 2021 *)
  • PARI
    a(n)=([0,1,0;0,0,1;1,0,-1]^n*[0;-1;1])[1,1] \\ Charles R Greathouse IV, Jun 11 2015

Formula

a(n) = -A247917(n-1).
Recurrence: a(n+3) = a(n) - a(n+2); a(0) = 0, a(1) = -1, a(2) = 1.
a(n+1) - a(n) = ((-1)^(n+1))*a(n+5).
a(n) = ((-1)^n)*A050935(n+1) = ((-1)^n)*A078013(n+2).
a(n) = A104771(n) - A104770(n).

Extensions

Edited by Ralf Stephan, Apr 05 2009

A176971 Expansion of (1+x)/(1+x-x^3) in powers of x.

Original entry on oeis.org

1, 0, 0, 1, -1, 1, 0, -1, 2, -2, 1, 1, -3, 4, -3, 0, 4, -7, 7, -3, -4, 11, -14, 10, 1, -15, 25, -24, 9, 16, -40, 49, -33, -7, 56, -89, 82, -26, -63, 145, -171, 108, 37, -208, 316, -279, 71, 245, -524, 595
Offset: 0

Views

Author

Roger L. Bagula, Apr 29 2010

Keywords

Comments

Except for signs the sequence is the essentially same as A078013, A050935 and A104769.
Padovan sequence extended to negative indices. - Hugo Pfoertner, Jul 16 2017

Examples

			G.f. = 1 + x^3 - x^4 + x^5 - x^7 + 2*x^8 - 2*x^9 + x^10 + x^11 - 3*x^12 + ...

Links

Crossrefs

Cf. A000931, A005251, A050935, A078013, A104769, A182097.

Programs

  • Magma
    m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x)/(1+x-x^3))); // G. C. Greubel, Sep 25 2018
  • Mathematica
    a[0] := 1; a[1] = 0; a[2] = 0;
a[n_] := a[n] = a[n - 2] + a[n - 3];
b = Table[a[n], {n, 0, 50}];
Table[b[[n]]^2 - b[[n - 1]]*b[[n + 1]], {n, 1, Length[b] - 1}]
a[ n_] := If[ n >= 0, SeriesCoefficient[ (1 + x) / (1 + x - x^3), {x, 0, n}], SeriesCoefficient[ 1 / (1 - x^2 - x^3), {x, 0, Abs@n}]]; (* Michael Somos, Dec 13 2013 *)
  • PARI
    {a(n) = if( n>=0, polcoeff( (1 + x) / (1 + x - x^3) + x * O(x^n), n), polcoeff( 1 / (1 - x^2 - x^3) + x * O(x^-n), -n))}; /* Michael Somos, Dec 13 2013 */

Formula

a(n) = A000931(n)^2 -A000931(n-1)*A000931(n+1).
a(n) = -a(n-1) +a(n-3). - R. J. Mathar, Apr 30 2010
a(n) = -A104769(n) - A104769(n+1). - Ralf Stephan, Aug 18 2013
G.f.: 1 / (1 - x^3 / (1 + x)). - Michael Somos, Dec 13 2013
a(n) = A182097(-n) for all n in Z. - Michael Somos, Dec 13 2013
A000931(n) = a(n)^2 - a(n-1) * a(n+1). - Michael Somos, Dec 13 2013
Binomial transform is A005251(n+1). - Michael Somos, Dec 13 2013

Extensions

Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021

A247917 Expansion of 1 / (1 + x - x^3) in powers of x.

Original entry on oeis.org

1, -1, 1, 0, -1, 2, -2, 1, 1, -3, 4, -3, 0, 4, -7, 7, -3, -4, 11, -14, 10, 1, -15, 25, -24, 9, 16, -40, 49, -33, -7, 56, -89, 82, -26, -63, 145, -171, 108, 37, -208, 316, -279, 71, 245, -524, 595, -350, -174, 769, -1119, 945, -176, -943, 1888, -2064, 1121
Offset: 0

Views

Author

Michael Somos, Sep 26 2014

Keywords

Examples

			G.f. = 1 - x + x^2 - x^4 + 2*x^5 - 2*x^6 + x^7 + x^8 - 3*x^9 + 4*x^10 + ...

Links

Crossrefs

Cf. A000931, A050935, A078013, A104769, A176971.

Programs

  • Magma
    m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1 + x - x^3)));  // G. C. Greubel, Aug 04 2018
  • Mathematica
    CoefficientList[Series[1/(1 + x - x^3), {x, 0, 100}], x] (* Vincenzo Librandi, Sep 27 2014 *)
LinearRecurrence[{-1,0,1},{1,-1,1},60] (* Harvey P. Dale, Apr 10 2025 *)
  • PARI
    {a(n) = if( n<0, n = -3-n; polcoeff( 1 / (1 - x^2 - x^3) + x * O(x^n), n), polcoeff( 1 / (1 + x - x^3) + x * O(x^n), n))};

Formula

G.f.: 1 / (1 + x - x^3).
0 = a(n) - a(n+2) - a(n+3) for all n in Z.
a(-n) = A000931(n) for all n in Z.
a(n) = A176971(n+3) for all n in Z.
-a(n) = A104769(n+1) for all n in Z.
(-1)^n * a(n) = A050935(n+3) for all n in Z.
-(-1)^n * a(n) = A078013(n+3) for all n in Z.
Showing 1-4 of 4 results.

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