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a(0)=1, a(1)=5; a(n) = 6*a(n-1) + 5*a(n-2) for n > 1.
5

%I #25 Oct 28 2024 04:52:38

%S 1,5,35,235,1585,10685,72035,485635,3273985,22072085,148802435,

%T 1003175035,6763062385,45594249485,307380808835,2072256100435,

%U 13970440646785,94183924382885,634955749531235,4280654119101835

%N a(0)=1, a(1)=5; a(n) = 6*a(n-1) + 5*a(n-2) for n > 1.

%C Let the generator matrix for the ternary Golay G_12 code be [I|B], where the elements of B are taken from the set {0,1,2}. Then a(n)=sum of first row of B^n.

%H G. C. Greubel, <a href="/A091928/b091928.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,5).

%F G.f.: (1-x)/(1-6*x-5*x^2).

%F a(n) = (1/2 +1/sqrt(14))*(3 +sqrt(14))^n + (1/2 -1/sqrt(14))*(3 -sqrt(14))^n.

%F From _Philippe Deléham_, Sep 22 2006: (Start)

%F a(n) = Sum_{k=0..n} 5^k*A122542(n,k).

%F Lim_{n->infinity} a(n+1)/a(n) = 3 + sqrt(14) = 6.741657386773... . (End)

%t LinearRecurrence[{6,5},{1,5},30] (* _Harvey P. Dale_, Apr 09 2022 *)

%o (Magma)

%o [n le 2 select 5^(n-1) else 6*Self(n-1) +5*Self(n-2): n in [1..41]]; // _G. C. Greubel_, Oct 27 2024

%o (SageMath)

%o A091928= BinaryRecurrenceSequence(6,5,1,5)

%o [A091928(n) for n in range(41)] # _G. C. Greubel_, Oct 27 2024

%Y Cf. A015551.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Feb 13 2004

%E Edited by _N. J. A. Sloane_ at the suggestion of _Andrew S. Plewe_, Jun 05 2007