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(PARI) Vec( (1 - x - x^2)/((1 - x)*(1 - 29*x - x^2)) + O(x^50) ) \\ G. C. Greubel, Nov 24 2016

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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.

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))).

allocated a(n) = floor(phi^7*a(n-1)) for Ilya Gutkovskiyn>0, a(0) = 1, where phi is the golden ratio (A001622).

1, 29, 841, 24417, 708933, 20583473, 597629649, 17351843293, 503801085145, 14627583312497, 424703717147557, 12331035380591649, 358024729754305377, 10395048198255447581, 301814422479162285225, 8763013300093961719105, 254429200125204052139269, 7387209816931011473757905

0,2

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.

<a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (30,-28,-1).

G.f.: (1 - x - x^2)/((1 - x)*(1 - 29*x - x^2)).

a(n) = 30*a(n-1) - 28*a(n-2) - a(n-3).

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))).

RecurrenceTable[{a[0] == 1, a[n] == Floor[GoldenRatio^7 a[n - 1]]}, a, {n, 17}]

LinearRecurrence[{30, -28, -1}, {1, 29, 841}, 18]

Cf. A001622.

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).

allocated

nonn,easy

Ilya Gutkovskiy, Nov 23 2016

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allocated for Ilya Gutkovskiy

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