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%I #27 Feb 18 2024 01:59:20
%S 1,15,16,31,47,78,125,203,328,531,859,1390,2249,3639,5888,9527,15415,
%T 24942,40357,65299,105656,170955,276611,447566,724177,1171743,1895920,
%U 3067663,4963583,8031246,12994829
%N Fibonacci sequence beginning 1, 15.
%C a(n-1)=sum(P(15;n-1-k,k),k=0..ceiling((n-1)/2)), n>=1, with a(-1)=14. These are the SW-NE diagonals in P(15;n,k), the (15,1) 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 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)=15. a(-1):=14.
%F G.f.: (1+14*x)/(1-x-x^2).
%F a(n) = A101220(14,0,n+1). - _Ross La Haye_, May 02 2006
%t a={};b=1;c=15;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},{1,15},40] (* _Harvey P. Dale_, Oct 11 2015 *)
%o (Magma) a0:=1; a1:=15; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..30]]; // _Bruno Berselli_, Feb 12 2013
%Y a(n) = A109754(14, n+1).
%Y a(k) = A118654(4, k).
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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