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 1-3 of 3 results.

A105371 Expansion of (1-x)*(1-x+x^2)/(1-3*x+4*x^2-2*x^3+x^4).

Original entry on oeis.org

1, 1, 1, 0, -3, -8, -13, -13, 0, 34, 89, 144, 144, 0, -377, -987, -1597, -1597, 0, 4181, 10946, 17711, 17711, 0, -46368, -121393, -196418, -196418, 0, 514229, 1346269, 2178309, 2178309, 0, -5702887, -14930352, -24157817, -24157817, 0, 63245986, 165580141
Offset: 0

Views

Author

Paul Barry, Apr 01 2005

Keywords

Comments

Binomial transform of A105367.

Links

Crossrefs

Cf. A000045, A105367.

Programs

  • Mathematica
    CoefficientList[Series[(1-x)(1-x+x^2)/(1-3x+4x^2-2x^3+x^4),{x,0,60}],x] (* or *) LinearRecurrence[{3,-4,2,-1},{1,1,1,0},60] (* Harvey P. Dale, Dec 21 2013 *)
  • PARI
    a(n)=local(m); m=n%5+1; [1,-1,-1,0,1][m]*fibonacci(-n-(m<3)) /* Michael Somos, Apr 09 2005 */

Formula

G.f.: (1-2x+2x^2-x^3)/(1-3x+4x^2-2x^3+x^4).
a(n) = 3*a(n-1)-4*a(n-2)+2*a(n-3)-a(n-4).
a(n) = (1/2+sqrt(5)/2)^n((1/2+sqrt(5)/10)cos(Pi*n/5)+sqrt(1/10-sqrt(5)/50)sin(Pi*n/5))- (sqrt(5)/2-1/2)^n((sqrt(5)/10-1/2)cos(2*Pi*n/5)+sqrt(1/10+sqrt(5)/50)sin(2*Pi*n/5)).
a(5n) = -F(-5n-1), a(5n+1) = a(5n+2) = -F(-5n-2), a(5n+3) = 0, a(5n+4) = F(-5n-4). - Michael Somos, Apr 09 2005

A105368 Expansion of (1-x-x^3+x^4)/(1-x^5).

Original entry on oeis.org

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

Views

Author

Paul Barry, Apr 01 2005

Keywords

Comments

Period 5: repeat [1,-1,0,-1,1]. Partial sums are A105367.
Binomial transform of A105369. - R. J. Mathar, Jul 29 2010

Examples

			G.f. = 1 - x - x^3 + x^4 + x^5 - x^6 - x^8 + x^9 + x^10 - x^11 - x^13 + ...
G.f. = q - q^3 - q^7 + q^9 + q^11 - q^13 - q^17 + q^19 + q^21 - q^23 + ...

Links

Programs

  • Magma
    &cat [[1,-1,0,-1,1]: n in [1..20]]; // Vincenzo Librandi, Apr 29 2015
  • Magma
    I:=[1,-1, 0,-1]; [n le 4 select I[n] else -Self(n-1)-Self(n-2)-Self(n-3)-Self(n-4): n in [1..100]]; // Vincenzo Librandi, Apr 29 2015
  • Maple
    A105368:=n->2-(n^2+n-4 mod 5): seq(A105368(n), n=0..100);
  • Mathematica
    CoefficientList[Series[(1 - x - x^3 + x^4)/(1 - x^5), {x, 0, 100}], x] (* Vincenzo Librandi, Apr 29 2015 *)
a[ n_] := {-1, 0, -1, 1, 1}[[Mod[n, 5, 1]]]; (* Michael Somos, Jun 16 2015 *)
  • PARI
    {a(n) = [ 1, -1, 0, -1, 1][n%5 + 1]}; /* Michael Somos, Jun 16 2015 */

Formula

G.f.: (1-x)*(1-x^3)/(1-x^5).
a(n) = A105367(n)-A105367(n-1).
a(n) = (1/2-sqrt(5)/10)*cos(4*Pi*n/5)-sqrt(1/2+sqrt(5)/10)*sin(4*Pi*n/5)+(1/2+sqrt(5)/10)*cos(2*Pi*n/5)-sqrt(1/2-sqrt(5)/10)*sin(2*Pi*n/5).
From R. J. Mathar, Jul 29 2010: (Start)
a(n) = -a(n-1) -a(n-2) -a(n-3) -a(n-4), n>4.
G.f.: (1-x)*(1+x+x^2)/(1+x+x^2+x^3+x^4). (End)
a(n) = -sign( (n+3) mod 5)*(-1)^floor(-2n/5-4/5). - Wesley Ivan Hurt, Apr 28 2015
a(n) = 2 - ( (n^2+n+1) mod 5 ). - Wesley Ivan Hurt, May 31 2015
Euler transform of length 5 sequence [ -1, 0, -1, 0, 1]. - Michael Somos, Jun 16 2015
a(n) = a(-1-n) for all n in Z. - Michael Somos, Jun 16 2015
a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = b(5^e) = 0^e, b(p^e) = 1 if p == 1, 9 (mod 10), b(p^e) = (-1)^e if p == 3, 7 (mod 10). - Michael Somos, Jun 16 2015

A105370 Expansion of ((1+x)^4-(1+x)x^3)/((1+x)^5-x^5).

Original entry on oeis.org

1, -1, 1, -2, 5, -10, 15, -15, 0, 50, -175, 450, -1000, 2000, -3625, 5875, -8125, 8125, 0, -29375, 106250, -278125, 621875, -1243750, 2250000, -3640625, 5031250, -5031250, 0, 18203125, -65859375, 172421875, -385546875, 771093750, -1394921875, 2257031250, -3119140625, 3119140625, 0
Offset: 0

Views

Author

Paul Barry, Apr 01 2005

Keywords

Comments

Binomial transform is A105367. Consecutive pair sums of 105369.

Links

Programs

  • Mathematica
    CoefficientList[Series[((1+x)^4-(1+x)x^3)/((1+x)^5-x^5),{x,0,40}],x] (* or *) LinearRecurrence[{-5,-10,-10,-5},{1,-1,1,-2},41]  (* Harvey P. Dale, May 23 2012 *)

Formula

G.f.: (1+x)(1+3x+3x^2)/(1+5x+10x^2+10x^3+5x^4).
a(n) = (5/2-sqrt(5)/2)^(n/2)((1/2+sqrt(5)/10)cos(7*Pi*n/10)+ sqrt(1/10-sqrt(5)/50)sin(7*Pi*n/10))- (5/2+sqrt(5)/2)^(n/2)((sqrt(5)/10-1/2)cos(9*Pi*n/10)+sqrt(1/10+sqrt(5)/50)sin(9*Pi*n/10)).
a(0)=1, a(1)=-1, a(2)=1, a(3)=-2, a(n)=-5*a(n-1)-10*a(n-2)- 10*a(n-3)- 5*a(n-4). - Harvey P. Dale, May 23 2012
Showing 1-3 of 3 results.

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