|
%I #26 Feb 18 2024 01:59:24
%S 1,16,17,33,50,83,133,216,349,565,914,1479,2393,3872,6265,10137,16402,
%T 26539,42941,69480,112421,181901,294322,476223,770545,1246768,2017313,
%U 3264081,5281394,8545475,13826869
%N Fibonacci sequence beginning 1, 16.
%C a(n-1)=sum(P(16;n-1-k,k),k=0..ceiling((n-1)/2)), n>=1, with a(-1)=15. These are the SW-NE diagonals in P(16;n,k), the (16,1) Pascal triangle. Cf. A093645 for the (10,1) Pascal triangle. Observation by _Paul Barry_, Apr 29 2004. Proof via recursion relations and comparison of inputs.
%H Harvey P. Dale, <a href="/A022106/b022106.txt">Table of n, a(n) for n = 0..1000</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,1).
%F a(n)= a(n-1)+a(n-2), n>=2, a(0)=1, a(1)=16. a(-1):=15.
%F G.f.: (1+15*x)/(1-x-x^2).
%t a={};b=1;c=16;AppendTo[a,b];AppendTo[a,c];Do[b=b+c;AppendTo[a,b];c=b+c;AppendTo[a,c],{n,1,12,1}];a (* _Vladimir Joseph Stephan Orlovsky_, Jul 23 2008 *)
%t LinearRecurrence[{1,1},{1,16},40] (* _Harvey P. Dale_, Jun 22 2016 *)
%o (Magma) a0:=1; a1:=16; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..30]]; // _Bruno Berselli_, Feb 12 2013
%Y a(n) = A109754(15, n+1) = A101220(15, 0, n+1).
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
|
