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A048655 Generalized Pellian with second term equal to 5.

%I A048655 #59 Aug 30 2025 20:26:36
%S A048655 1,5,11,27,65,157,379,915,2209,5333,12875,31083,75041,181165,437371,
%T A048655 1055907,2549185,6154277,14857739,35869755,86597249,209064253,
%U A048655 504725755,1218515763,2941757281,7102030325,17145817931,41393666187,99933150305,241259966797,582453083899
%N A048655 Generalized Pellian with second term equal to 5.
%C A048655 Equals binomial transform of A143095: (1, 4, 2, 8, 4, 16, 8, 32, ...). - _Gary W. Adamson_, Jul 23 2008
%H A048655 T. D. Noe, <a href="/A048655/b048655.txt">Table of n, a(n) for n = 0..300</a>
%H A048655 M. Bicknell, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/13-4/bicknell.pdf">A primer on the Pell sequence and related sequences</a>, Fibonacci Quarterly, Vol. 13, No. 4, 1975, pp. 345-349.
%H A048655 A. F. Horadam, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/3-3/horadam.pdf">Basic properties of a certain generalized sequence of numbers</a>, Fibonacci Quarterly, Vol. 3, No. 3, 1965, pp. 161-176.
%H A048655 A. F. Horadam, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/5-5/horadam.pdf">Special properties of the sequence W_n(a,b; p,q)</a>, Fib. Quart., 5.5 (1967), 424-434.
%H A048655 A. F. Horadam, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/9-3/horadam-a.pdf">Pell identities</a>, Fib. Quart., Vol. 9, No. 3, 1971, pp. 245-252.
%H A048655 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive sequences</a>
%H A048655 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,1)
%F A048655 a(n) = 2*a(n-1) + a(n-2); a(0)=1, a(1)=5.
%F A048655 a(n) = ((4+sqrt(2))(1+sqrt(2))^n - (4-sqrt(2))(1-sqrt(2))^n)/2*sqrt(2).
%F A048655 a(n) = P(n) - 3*P(n+1) + 2*P(n+2). - _Creighton Dement_, Jan 18 2005
%F A048655 G.f.: (1+3*x)/(1 - 2*x - x^2). - _Philippe Deléham_, Nov 03 2008
%F A048655 E.g.f.: exp(x)*(cosh(sqrt(2)*x) + 2*sqrt(2)*sinh(sqrt(2)*x)). - _Vaclav Kotesovec_, Feb 16 2015
%F A048655 a(n) = 3*Pell(n) + Pell(n+1), where Pell = A000129. - _Vladimir Reshetnikov_, Sep 27 2016
%p A048655 with(combinat): a:=n->3*fibonacci(n, 2)+fibonacci(n+1, 2): seq(a(n), n=0..26); # _Zerinvary Lajos_, Apr 04 2008
%t A048655 a[n_]:=(MatrixPower[{{1,2},{1,1}},n].{{4},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* _Vladimir Joseph Stephan Orlovsky_, Feb 20 2010 *)
%t A048655 LinearRecurrence[{2,1},{1,5},30] (* _Harvey P. Dale_, Nov 05 2011 *)
%o A048655 (Maxima)
%o A048655 a[0]:1$
%o A048655 a[1]:5$
%o A048655 a[n]:=2*a[n-1]+a[n-2]$
%o A048655 A048655(n):=a[n]$
%o A048655 makelist(A048655(n),n,0,30); /* _Martin Ettl_, Nov 03 2012 */
%o A048655 (PARI) a(n)=([0,1; 1,2]^n*[1;5])[1,1] \\ _Charles R Greathouse IV_, Feb 09 2017
%o A048655 (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+3*x)/(1-2*x-x^2))); // _G. C. Greubel_, Jul 26 2018
%Y A048655 Cf. A001333, A000129, A048654, A143095.
%K A048655 easy,nice,nonn,changed
%O A048655 0,2
%A A048655 _Barry E. Williams_