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%I #29 Feb 22 2024 10:42:23
%S 1,20,21,41,62,103,165,268,433,701,1134,1835,2969,4804,7773,12577,
%T 20350,32927,53277,86204,139481,225685,365166,590851,956017,1546868,
%U 2502885,4049753,6552638,10602391,17155029,27757420,44912449,72669869,117582318,190252187
%N Fibonacci sequence beginning 1, 20.
%C a(n-1) = Sum(P(20;n-1-k,k),k=0..ceiling((n-1)/2)), n>=1, with a(-1) = 19. These are the SW-NE diagonals in P(20;n,k), the (20,1) 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 Paolo Xausa, <a href="/A022110/b022110.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) = 20.
%F G.f.: (1+19*x)/(1-x-x^2).
%t a={};b=1;c=20;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, 20}, 35] (* _Paolo Xausa_, Feb 22 2024 *)
%o (Magma) a0:=1; a1:=20; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..30]]; // _Bruno Berselli_, Feb 12 2013
%Y a(n) = A109754(19, n+1) = A101220(19, 0, n+1).
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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