OFFSET
0,2
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).
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))).
MATHEMATICA
RecurrenceTable[{a[0] == 1, a[n] == Floor[GoldenRatio^7 a[n - 1]]}, a, {n, 17}]
LinearRecurrence[{30, -28, -1}, {1, 29, 841}, 18]
PROG
(PARI) Vec( (1 - x - x^2)/((1 - x)*(1 - 29*x - x^2)) + O(x^50) ) \\ G. C. Greubel, Nov 24 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Ilya Gutkovskiy, Nov 23 2016
STATUS
approved