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.

A163147 a(n) = 14*a(n-1) - 44*a(n-2) for n > 1; a(0) = 1, a(1) = 12.

Original entry on oeis.org

1, 12, 124, 1208, 11456, 107232, 997184, 9242368, 85517056, 790574592, 7305293824, 67488831488, 623410712576, 5758241390592, 53185308114944, 491231692423168, 4537090136866816, 41905067449516032, 387038978271084544
Offset: 0

Views

Author

Klaus Brockhaus, Jul 21 2009

Keywords

Comments

Binomial transform of A163146. Inverse binomial transform of A163148.

Crossrefs

Programs

  • Magma
    [ n le 2 select 11*n-10 else 14*Self(n-1)-44*Self(n-2): n in [1..19] ];
    
  • PARI
    Vec((1-2*x)/(1-14*x+44*x^2) + O(x^30)) \\ Jinyuan Wang, Mar 23 2020

Formula

a(n) = ((1+sqrt(5))*(7+sqrt(5))^n+(1-sqrt(5))*(7-sqrt(5))^n)/2.
G.f.: (1-2*x)/(1-14*x+44*x^2).

A153882 a(n) = ((6 + sqrt(5))^n - (6 - sqrt(5))^n)/(2*sqrt(5)).

Original entry on oeis.org

1, 12, 113, 984, 8305, 69156, 572417, 4725168, 38957089, 321004860, 2644388561, 21781512072, 179402099473, 1477598319444, 12169714749665, 100231029093216, 825511191878977, 6798972400658028, 55996821859648049, 461193717895377720
Offset: 1

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Jan 03 2009

Keywords

Comments

Fourth binomial transform of A048877.
First differences are in A163146.
lim_{n -> infinity} a(n)/a(n-1) = 6 + sqrt(5) = 8.236067977499789696....

References

  • S. Falcon, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014.

Crossrefs

Cf. A002163 (decimal expansion of sqrt(5)), A048877, A163146.

Programs

  • Magma
    Z:= PolynomialRing(Integers()); N:=NumberField(x^2-5); S:=[ ((6+r)^n-(6-r)^n)/(2*r): n in [1..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jan 04 2009
    
  • Magma
    I:=[1,12]; [n le 2 select I[n] else 12*Self(n-1)-31*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 01 2016
  • Mathematica
    LinearRecurrence[{12, -31}, {1, 12}, 25] (* or *) Table[((6 + sqrt(5))^n - (6 - sqrt(5))^n)/(2*sqrt(5)) , {n,0,25}] (* G. C. Greubel, Aug 31 2016 *)
  • Sage
    [lucas_number1(n,12,31) for n in range(1, 21)] # Zerinvary Lajos, Apr 27 2009
    

Formula

From Philippe Deléham, Jan 03 2009: (Start)
a(n) = 12*a(n-1) - 31*a(n-2) for n>1, with a(0)=0, a(1)=1.
G.f.: x/(1 - 12*x + 31*x^2). (End)
a(n) = 12*a(n-1) - 31*a(n-2). - G. C. Greubel, Aug 31 2016

Extensions

Extended beyond a(7) by Klaus Brockhaus, Jan 04 2009
Edited by Klaus Brockhaus, Oct 11 2009
Showing 1-2 of 2 results.