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%I #34 Sep 08 2022 08:44:57
%S 1,9,19,47,113,273,659,1591,3841,9273,22387,54047,130481,315009,
%T 760499,1836007,4432513,10701033,25834579,62370191,150574961,
%U 363520113,877615187,2118750487,5115116161,12348982809,29813081779,71975146367
%N Generalized Pellian with second term equal to 9.
%C Binomial transform of 5,6,10,12,20,24,40. - Al Hakanson (hawkuu(AT)gmail.com), Aug 12 2009
%C Binomial transform of A164587. Inverse binomial transform of A164298. - _Klaus Brockhaus_, Aug 17 2009
%C For n > 0: a(n) = A105082(n) - A105082(n-1). - _Reinhard Zumkeller_, Dec 15 2013
%H Reinhard Zumkeller, <a href="/A048696/b048696.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 (2,1).
%F a(n) = 2*a(n-1) + a(n-2); a(0)=1, a(1)=9.
%F a(n) = ((4*sqrt(2)+1)(1+sqrt(2))^n - (4*sqrt(2)-1)(1-sqrt(2))^n)/2.
%F G.f.: (1+7*x)/(1 - 2*x - x^2). - _Philippe Deléham_, Nov 03 2008
%p with(combinat): a:=n->7*fibonacci(n, 2)+fibonacci(n+1, 2): seq(a(n), n=0..25); # _Zerinvary Lajos_, Apr 04 2008
%t a[n_]:=(MatrixPower[{{1,2},{1,1}},n].{{8},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* _Vladimir Joseph Stephan Orlovsky_, Feb 20 2010 *)
%t LinearRecurrence[{2,1},{1,9},30] (* _Harvey P. Dale_, Apr 20 2012 *)
%o (Magma) [ n le 2 select 8*n-7 else 2*Self(n-1)+Self(n-2): n in [1..28] ]; // _Klaus Brockhaus_, Aug 17 2009
%o (Maxima) a[0]:1$
%o a[1]:9$
%o a[n]:=2*a[n-1]+a[n-2]$
%o A048696(n):=a[n]$
%o makelist(A048696(n),n,0,30); /* _Martin Ettl_, Nov 03 2012 */
%o (Haskell)
%o a048696 n = a048696_list !! n
%o a048696_list = 1 : 9 : zipWith (+)
%o a048696_list (map (2 *) $ tail a048696_list)
%o -- _Reinhard Zumkeller_, Dec 15 2013
%Y Cf. A001333, A000129, A048654, A048655.
%K easy,nonn
%O 0,2
%A _Barry E. Williams_
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