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%I A278475 #8 Nov 27 2016 21:46:17
%S A278475 1,29,841,24417,708933,20583473,597629649,17351843293,503801085145,
%T A278475 14627583312497,424703717147557,12331035380591649,358024729754305377,
%U A278475 10395048198255447581,301814422479162285225,8763013300093961719105,254429200125204052139269,7387209816931011473757905
%N A278475 a(n) = floor(phi^7*a(n-1)) for n>0, a(0) = 1, where phi is the golden ratio (A001622).
%C A278475 In general, the ordinary generating function for the recurrence relation b(n) = floor(phi^k*b(n - 1)) with n>0 and b(0) = 1, is (1 - x)/(1 - (phi^k + (-phi)^(-k))*x + x^2) if k is even, and (1 - x - x^2)/((1 - x)*(1 - (phi^k + (-phi)^(-k))*x - x^2)) if k is odd.
%H A278475 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (30,-28,-1).
%F A278475 G.f.: (1 - x - x^2)/((1 - x)*(1 - 29*x - x^2)).
%F A278475 a(n) = 30*a(n-1) - 28*a(n-2) - a(n-3).
%F A278475 a(n) = ((-29 - 13*sqrt(5))^(-n)*(-7*(407 + 182*sqrt(5))*2^(n+3) + 13*(1885 + 843*sqrt(5))*(-29 - 13*sqrt(5))^n + 28*(25319 + 11323*sqrt(5))*(-843 - 377*sqrt(5))^n))/(377*(1885 + 843*sqrt(5))).
%t A278475 RecurrenceTable[{a[0] == 1, a[n] == Floor[GoldenRatio^7 a[n - 1]]}, a, {n, 17}]
%t A278475 LinearRecurrence[{30, -28, -1}, {1, 29, 841}, 18]
%o A278475 (PARI) Vec( (1 - x - x^2)/((1 - x)*(1 - 29*x - x^2)) + O(x^50) ) \\ _G. C. Greubel_, Nov 24 2016
%Y A278475 Cf. A001622.
%Y A278475 Cf. similar sequences with recurrence relation b(n) = floor(phi^k*b(n-1)) for n>0, b(0) = 1: A000012 (k = 1), A001519 (k = 2), A024551 (k = 3), A049685 (k = 4), A214993 (k = 5), A007805 (k = 6), this sequence (k = 7).
%K A278475 nonn,easy
%O A278475 0,2
%A A278475 _Ilya Gutkovskiy_, Nov 23 2016

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