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 1-2 of 2 results.

A090591 Expansion of g.f.: 1/(1 - 2*x + 8*x^2).

Original entry on oeis.org

1, 2, -4, -24, -16, 160, 448, -384, -4352, -5632, 23552, 92160, -4096, -745472, -1458176, 3047424, 17760256, 11141120, -119799808, -328728576, 300941312, 3231711232, 4055891968, -17741905920, -67930947584
Offset: 0

Views

Author

Simone Severini, Dec 04 2003

Keywords

Comments

(1,2) entry of powers of the orthogonal design shown below:
+1 +1 +1 +1 +1 +1 +1 +1
-1 +1 +1 -1 +1 -1 -1 +1
-1 -1 +1 +1 +1 +1 -1 -1
-1 +1 -1 +1 +1 -1 +1 -1
-1 -1 -1 -1 +1 +1 +1 +1
-1 +1 -1 +1 -1 +1 -1 +1
-1 +1 +1 -1 -1 +1 +1 -1
-1 -1 +1 +1 -1 -1 +1 +1
Pisano period lengths: 1, 1, 8, 1, 24, 8, 7, 1, 24, 24, 10, 8, 56, 7, 24, 1, 144, 24, 120, 24, ... - R. J. Mathar, Aug 10 2012

Links

Crossrefs

Cf. A087621, A090590.

Programs

  • GAP
    a:=[1,2];; for n in [3..30] do a[n]:=2*a[n-1]-8*a[n-2]; od; a; # Muniru A Asiru, Oct 23 2018
  • Magma
    [n le 2 select n else 2*Self(n-1) - 8*Self(n-2): n in [1..30]]; // G. C. Greubel, Oct 22 2018
  • Maple
    seq(coeff(series(1/(1-2*x+8*x^2),x,n+1), x, n), n = 0 .. 25); # Muniru A Asiru, Oct 23 2018
  • Mathematica
    LinearRecurrence[{2,-8}, {1,2}, 30] (* G. C. Greubel, Oct 22 2018 *)
CoefficientList[Series[1/(1-2x+8x^2),{x,0,60}],x] (* Harvey P. Dale, Jan 17 2021 *)
  • PARI
    x='x+O('x^30); Vec(1/(1 - 2*x + 8*x^2)) \\ G. C. Greubel, Oct 22 2018
  • Sage
    [lucas_number1(n,2,8) for n in range(1, 21)] # Zerinvary Lajos, Apr 23 2009

Formula

Binomial transform of (1+x)/(1+7*x^2).
a(0)=1, a(1)=2, a(n) = 2*a(n-1) - 8*a(n-2) for n>1. - Philippe Deléham, Sep 19 2009

Extensions

Formulae from Paul Barry, Dec 05 2003
Corrected by T. D. Noe, Dec 11 2006

A090590 (1,1) entry of powers of the orthogonal design shown below.

Original entry on oeis.org

1, -6, -20, 8, 176, 288, -832, -3968, -1280, 29184, 68608, -96256, -741376, -712704, 4505600, 14712832, -6619136, -130940928, -208928768, 629669888, 2930769920, 824180736, -21797797888, -50189041664, 74004299776, 549520932864
Offset: 1

Views

Author

Simone Severini, Dec 04 2003

Keywords

Comments

+1 +1 +1 +1 +1 +1 +1 +1
-1 +1 +1 -1 +1 -1 -1 +1
-1 -1 +1 +1 +1 +1 -1 -1
-1 +1 -1 +1 +1 -1 +1 -1
-1 -1 -1 -1 +1 +1 +1 +1
-1 +1 -1 +1 -1 +1 -1 +1
-1 +1 +1 -1 -1 +1 +1 -1
-1 -1 +1 +1 -1 -1 +1 +1
Also real part of (1 +- i*sqrt(7))^n. - Bruno Berselli, Jun 24-25 2011

Links

Crossrefs

Cf. A087621, A090591.

Programs

  • Magma
    m:=27; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1-8*x)/(1-2*x+8*x^2))); // Bruno Berselli, Jun 24-25 2011
  • Maple
    a := proc(n) option remember: if(n=1)then return 1:elif(n=2)then return -6:fi: return 2*a(n-1)-8*a(n-2): end: seq(a(n),n=1..26); # Nathaniel Johnston, Jun 25 2011
  • Mathematica
    LinearRecurrence[{2,-8},{1,-6},30] (* Harvey P. Dale, Mar 30 2019 *)
  • Maxima
    makelist(expand(((1+sqrt(-1)*sqrt(7))^n+(1-sqrt(-1)*sqrt(7))^n)/2),n,1,26); /* Bruno Berselli, Jun 24-25 2011 */
  • PARI
    a=vector(26); a[1]=1; a[2]=-6; for(i=3, #a, a[i]=2*a[i-1]-8*a[i-2]); a \\ Bruno Berselli, Jun 24-25 2011

Formula

G.f.: x*(1-8*x)/(1-2*x+8*x^2). - T. D. Noe, Dec 11 2006
From Bruno Berselli, Jun 24-25 2011: (Start)
a(n) = (1/2)*((1+i*sqrt(7))^n + (1-i*sqrt(7))^n), where i=sqrt(-1).
a(n) = cos(n*arctan(sqrt(7)))*sqrt(8)^n.
a(n) = 2*a(n-1) - 8*a(n-2) (n > 2). (End)

Extensions

Corrected by T. D. Noe, Dec 11 2006
More terms from Bruno Berselli, Jun 24 2011
Showing 1-2 of 2 results.

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