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A022391 Fibonacci sequence beginning 1, 21.

%I A022391 #38 Feb 18 2024 02:00:58
%S A022391 1,21,22,43,65,108,173,281,454,735,1189,1924,3113,5037,8150,13187,
%T A022391 21337,34524,55861,90385,146246,236631,382877,619508,1002385,1621893,
%U A022391 2624278,4246171,6870449,11116620,17987069,29103689,47090758,76194447,123285205,199479652,322764857,522244509
%N A022391 Fibonacci sequence beginning 1, 21.
%C A022391 a(n-1) = Sum_{k=0..ceiling((n-1)/2)} P(21;n-1-k,k), n>=1, with a(-1)=20. These are the SW-NE diagonals in P(21;n,k), the (21,1) 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 A022391 G. C. Greubel, <a href="/A022391/b022391.txt">Table of n, a(n) for n = 0..1000</a>
%H A022391 S. Kak, <a href="https://arxiv.org/abs/physics/0411195">The Golden Mean and the Physics of Aesthetics</a>, arXiv:physics/0411195 [physics.hist-ph], 2004.
%H A022391 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A022391 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1, 1).
%F A022391 a(n) = a(n-1) + a(n-2), n>=2, a(0)=1, a(1)=21. a(-1):=20.
%F A022391 G.f.: (1+20*x)/(1-x-x^2).
%t A022391 LinearRecurrence[{1, 1}, {1, 21}, 30] (* _Jean-François Alcover_, Feb 25 2018 *)
%t A022391 Table[Fibonacci[n + 2] + 19*Fibonacci[n], {n, 0, 50}] (* _G. C. Greubel_, Mar 02 2018 *)
%o A022391 (PARI) for(n=0, 50, print1(fibonacci(n+2) + 19*fibonacci(n), ", ")) \\ _G. C. Greubel_, Mar 02 2018
%o A022391 (Magma) [Fibonacci(n+2) + 19*Fibonacci(n): n in [0..50]]; // _G. C. Greubel_, Mar 02 2018
%K A022391 nonn
%O A022391 0,2
%A A022391 _N. J. A. Sloane_
%E A022391 Terms a(30) onward added by _G. C. Greubel_, Mar 02 2018