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%I #24 Feb 18 2024 01:59:16
%S 1,14,15,29,44,73,117,190,307,497,804,1301,2105,3406,5511,8917,14428,
%T 23345,37773,61118,98891,160009,258900,418909,677809,1096718,1774527,
%U 2871245,4645772,7517017,12162789
%N Fibonacci sequence beginning 1, 14.
%C a(n-1)=sum(P(14;n-1-k,k),k=0..ceiling((n-1)/2)), n>=1, with a(-1)=13. These are the SW-NE diagonals in P(14;n,k), the (14,1) 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)=14. a(-1):=13.
%F G.f.: (1+13*x)/(1-x-x^2).
%t a={};b=1;c=14;AppendTo[a,b];AppendTo[a,c];Do[b=b+c;AppendTo[a,b];c=b+c;AppendTo[a,c],{n,1,9,1}];a (* _Vladimir Joseph Stephan Orlovsky_, Jul 22 2008 *)
%t LinearRecurrence[{1,1},{1,14},40] (* _Harvey P. Dale_, Jun 12 2017 *)
%o (Magma) a0:=1; a1:=14; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..30]]; // _Bruno Berselli_, Feb 12 2013
%Y a(n) = A109754(13, n+1) = A101220(13, 0, n+1).
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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