A278475 a(n) = floor(phi^7*a(n-1)) for n>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

Offset: 0

Author

Ilya Gutkovskiy, Nov 23 2016

Keywords

• nonn
• easy

Comments

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.

Links

• Index entries for linear recurrences with constant coefficients, signature (30,-28,-1).

Crossrefs

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).

Programs

Mathematica

RecurrenceTable[{a[0] == 1, a[n] == Floor[GoldenRatio^7 a[n - 1]]}, a, {n, 17}]
LinearRecurrence[{30, -28, -1}, {1, 29, 841}, 18]

PARI

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

Formula

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))).