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%I #44 Mar 20 2023 14:24:52
%S 1,1,8,22,92,316,1184,4264,15632,56848,207488,756064,2757056,10050496,
%T 36643328,133589632,487039232,1775616256,6473467904,23600633344,
%U 86042074112,313687948288,1143628341248,4169384372224,15200538791936
%N a(n) = 2*a(n-1) + 6*a(n-2).
%C a(n+1) = a(n) + 7*A083099(n-1); a(n+1)/A083099(n) converges to sqrt(7).
%C Binomial transform of expansion of cosh(sqrt(7)x) (A000420 with interpolated zeros: 1, 0, 7, 0, 49, 0, 343, 0, ...).
%C The same sequence may be obtained by the following process. Starting a priori with the fraction 1/1, the numerators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 7 times the bottom to get the new top. The limit of the sequence of fractions is sqrt(7). - _Cino Hilliard_, Sep 25 2005
%C a(n) is the number of compositions of n when there are 1 type of 1 and 7 types of other natural numbers. - _Milan Janjic_, Aug 13 2010
%D John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.
%H G. C. Greubel, <a href="/A083098/b083098.txt">Table of n, a(n) for n = 0..1000</a>
%H Project Euler, <a href="https://projecteuler.net/problem=752">Problem 752</a>, sequence alpha(n).
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,6).
%F G.f.: (1-x)/(1-2*x-6*x^2).
%F a(n) = (1+sqrt(7))^n/2 + (1-sqrt(7))^n/2.
%F E.g.f.: exp(x)*cosh(sqrt(7)x).
%F a(n) = Sum_{k=0..n} A098158(n,k)*7^(n-k). - _Philippe Deléham_, Dec 26 2007
%F If p[1]=1, and p[i]=7, (i>1), and if A is Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n) = det A. - _Milan Janjic_, Apr 29 2010
%F G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(7*k-1)/(x*(7*k+6) - 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 26 2013
%t CoefficientList[Series[(1+6x)/(1-2x-6x^2), {x, 0, 25}], x]
%t LinearRecurrence[{2, 6}, {1, 1}, 25] (* _Sture Sjöstedt_, Dec 06 2011 *)
%t a[n_] := Simplify[((1 + Sqrt[7])^n + (1 - Sqrt[7])^n)/2]; Array[a, 25, 0] (* _Robert G. Wilson v_, Sep 18 2013 *)
%o (Sage) [lucas_number2(n,2,-6)/2 for n in range(0, 25)] # _Zerinvary Lajos_, Apr 30 2009
%o (PARI) x='x+O('x^30); Vec((1-x)/(1-2*x-6*x^2)) \\ _G. C. Greubel_, Jan 08 2018
%o (Magma) I:=[1,1]; [n le 2 select I[n] else 2*Self(n-1) + 6*Self(n-2): n in [1..30]]; // _G. C. Greubel_, Jan 08 2018
%Y The following sequences (and others) belong to the same family: A001333, A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533, A002532, A083098, A083099, A083100, A015519.
%K easy,nonn
%O 0,3
%A Mario Catalani (mario.catalani(AT)unito.it), Apr 22 2003
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