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 A278476 a(n) = floor((1 + sqrt(2))^3*a(n-1)) for n>0, a(0) = 1. 1

%I

%S 1,14,196,2757,38793,545858,7680804,108077113,1520760385,21398722502,

%T 301102875412,4236838978269,59616848571177,838872718974746,

%U 11803834914217620,166092561518021425,2337099696166517569,32885488307849267390,462733936006056261028

%N a(n) = floor((1 + sqrt(2))^3*a(n-1)) for n>0, a(0) = 1.

%C In general, the ordinary generating function for the recurrence relation b(n) = floor((1 + sqrt(2))^k*b(n - 1)) with n>0 and b(0) = 1, is (1 - x)/(1 - round((1 + sqrt(2))^k)*x + x^2) if k is nonzero even, and (1 - x - x^2)/((1 - x)*(1 - round((1 + sqrt(2))^k)*x - x^2)) if k is odd or k = 0.

%H G. C. Greubel, <a href="/A278476/b278476.txt">Table of n, a(n) for n = 0..869</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (15,-13,-1).

%F G.f.: (1 - x - x^2)/((1 - x)*(1 - 14*x - x^2)).

%F a(n) = 15*a(n-1) - 13*a(n-2) - a(n-3).

%F a(n) = ((65 - 52*sqrt(2))*(7 - 5*sqrt(2))^n + 13*(5 + 4*sqrt(2))*(7 + 5*sqrt(2))^n + 10)/140.

%p seq(coeff(series((1-x-x^2)/((1-x)*(1-14*x-x^2)),x,n+1), x, n), n = 0 .. 20); # _Muniru A Asiru_, Oct 11 2018

%t RecurrenceTable[{a[0] == 1, a[n] == Floor[(1 + Sqrt[2])^3 a[n - 1]]}, a, {n, 18}]

%t LinearRecurrence[{15, -13, -1}, {1, 14, 196}, 19]

%t CoefficientList[Series[(1-x-x^2)/((1-x)*(1-14*x-x^2)), {x,0,50}], x] (* _G. C. Greubel_, Oct 10 2018 *)

%o (PARI) Vec((1 - x - x^2)/((1 - x)*(1 - 14*x - x^2)) + O(x^50)) \\ _G. C. Greubel_, Nov 24 2016

%o (MAGMA) m:=25; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x-x^2)/((1-x)*(1-14*x-x^2)))); // _G. C. Greubel_, Oct 10 2018

%Y Cf. A014176.

%Y Cf. similar sequences with recurrence relation b(n) = floor((1 + sqrt(2))^k*b(n-1)) for n>0, b(0) = 1: A024537 (k = 1), A001653 (k = 2), this sequence (k = 3), A077420 (k = 4), A097733 (k = 6).

%K nonn,easy,changed

%O 0,2

%A _Ilya Gutkovskiy_, Nov 23 2016

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Last modified October 20 17:47 EDT 2018. Contains 316395 sequences. (Running on oeis4.)