This site is supported by donations to The OEIS Foundation. Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing. Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A015442 Generalized Fibonacci numbers: a(n) = a(n-1) + 7*a(n-2), a(0)=0, a(1)=1. 16

%I

%S 0,1,1,8,15,71,176,673,1905,6616,19951,66263,205920,669761,2111201,

%T 6799528,21577935,69174631,220220176,704442593,2245983825,7177081976,

%U 22898968751,73138542583,233431323840,745401121921,2379420388801

%N Generalized Fibonacci numbers: a(n) = a(n-1) + 7*a(n-2), a(0)=0, a(1)=1.

%C One obtains A015523 through a binomial transform, and A197189 by shifting one place left (starting 1,1,8 with offset 0) followed by a binomial transform. - _R. J. Mathar_, Oct 11 2011

%C The compositions of n in which each positive integer is colored by one of p different colors are called p-colored compositions of n. For n>=2, 8*a(n-1) equals the number of 8-colored compositions of n, with all parts >=2, such that no adjacent parts have the same color. - _Milan Janjic_, Nov 26 2011

%C Pisano period lengths: 1, 3, 8, 6, 4, 24, 1, 6, 24, 12, 60, 24, 12, 3, 8, 6, 288, 24, 120, 12, ... - _R. J. Mathar_, Aug 10 2012

%H Vincenzo Librandi, <a href="/A015442/b015442.txt">Table of n, a(n) for n = 0..1000</a>

%H Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, section 14.8 "Strings with no two consecutive nonzero digits", pp.317-318

%H M. Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Janjic/janjic63.html">On Linear Recurrence Equations Arising from Compositions of Positive Integers</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,7).

%F O.g.f.: x/(1-x-7x^2). - _R. J. Mathar_, May 06 2008

%F a(n) = ( ((1+sqrt(29))/2)^(n+1) - ((1-sqrt(29))/2)^(n+1) )/sqrt(29).

%F a(n) = 8*a(n-2) + 7*a(n-3) with characteristic polynomial x^3 - 8*x - 7. - _Roger L. Bagula_, May 30 2007

%F a(n+1) = Sum_{k=0..n} A109466(n,k)*(-7)^(n-k). - _Philippe Deléham_, Oct 26 2008

%F a(n) = (Sum_{1<=k<=n, k odd} C(n,k)*29^((k-1)/2))/2^(n-1). - _Vladimir Shevelev_, Feb 05 2014

%t (* recursion *) A015442 := 0; A015442 := 1; A015442[ 2] := 1; A015442[n_] := A015442[n] = 8*A015442[n - 2] + 7*A015442[n - 3]; Table[A015442[n], {n, 0, 25}] (* _Roger L. Bagula_ *)

%t (* matrix representation *) mat15442 = {{0, 1, 0}, {0, 0, 1}, {7, 8, 0}}; w15442[ 0] = {0, 1, 1}; w15442[n_] := w15442[n] = mat15442.w15442[n - 1]; Table[w15442[n][], {n, 0, 25}] (* _Roger L. Bagula_ *)

%t LinearRecurrence[{1, 7}, {0, 1}, 30] (* _Vincenzo Librandi_, Oct 17 2012 *)

%o (Sage) [lucas_number1(n,1,-7) for n in range(0, 27)] # _Zerinvary Lajos_, Apr 22 2009

%o (MAGMA) I:=[0, 1]; [n le 2 select I[n] else Self(n-1) + 7*Self(n-2): n in [1..40]]; // _Vincenzo Librandi_, Oct 17 2012

%o (PARI) concat(0,Vec(1/(1-x-7*x^2)+O(x^99))) \\ _Charles R Greathouse IV_, Mar 12 2014

%Y Cf. A015440, A015441.

%K nonn,easy,changed

%O 0,4

%A _Olivier Gérard_

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 8 01:54 EST 2019. Contains 329850 sequences. (Running on oeis4.)