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%I #38 May 04 2023 01:56:47
%S 1,13,64,307,1471,7048,33769,161797,775216,3714283,17796199,85266712,
%T 408537361,1957420093,9378563104,44935395427,215298414031,
%U 1031556674728,4942484959609,23680868123317,113461855656976,543628410161563,2604680195150839,12479772565592632
%N A Pellian-related sequence.
%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, p. 256.
%H Reinhard Zumkeller, <a href="/A054477/b054477.txt">Table of n, a(n) for n = 0..1000</a>
%H A. F. Horadam, <a href="http://www.fq.math.ca/Scanned/9-3/horadam-a.pdf">Pell Identities</a>, Fib. Quart., Vol. 9, No. 3, 1971, pp. 245-252.
%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 (5,-1)
%F a(n) = 5a(n-1)-a(n-2); a(0)=1, a(1)=13.
%F (A054477)=sqrt{21*(A002320)^2-20}; where the algebraic operations on (A------) are performed from the inside - out; that is, first squared, then multiplied by 21, then 20 is subtracted and finally the square root is performed term by term.
%F G.f.: (1+8*x)/(1-5*x+x^2). - _Alois P. Heinz_, Aug 07 2008
%F a(n) = (2^(-1-n)*((5-sqrt(21))^n*(-21+sqrt(21))+(5+sqrt(21))^n*(21+sqrt(21))))/sqrt(21). - _Colin Barker_, May 26 2016
%F E.g.f.: (sqrt(21)*sinh(sqrt(21)*x/2) + cosh(sqrt(21)*x/2))*exp(5*x/2). - _Ilya Gutkovskiy_, May 26 2016
%p a:= n-> (Matrix([[1,-8]]). Matrix([[5,1],[ -1,0]])^(n))[1,1]:
%p seq(a(n), n=0..20); # _Alois P. Heinz_, Aug 07 2008
%t LinearRecurrence[{5, -1}, {1, 13}, 20] (* _Jean-François Alcover_, Jan 09 2016 *)
%o (Haskell)
%o a054477 n = a054477_list !! n
%o a054477_list = 1 : 13 :
%o (zipWith (-) (map (* 5) (tail a054477_list)) a054477_list)
%o -- _Reinhard Zumkeller_, Oct 16 2011
%Y Cf. A002320.
%K easy,nonn
%O 0,2
%A _Barry E. Williams_, Apr 16 2000
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