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A278476 a(n) = floor((1 + sqrt(2))^3*a(n-1)) for n>0, a(0) = 1. 0
1, 14, 196, 2757, 38793, 545858, 7680804, 108077113, 1520760385, 21398722502, 301102875412, 4236838978269, 59616848571177, 838872718974746, 11803834914217620, 166092561518021425, 2337099696166517569, 32885488307849267390, 462733936006056261028 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=0..18.

Index entries for linear recurrences with constant coefficients, signature (15,-13,-1).

FORMULA

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

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

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

MATHEMATICA

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

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

PROG

(PARI) Vec((1 - x - x^2)/((1 - x)*(1 - 14*x - x^2)) + O(x^50)) \\ G. C. Greubel, Nov 24 2016

CROSSREFS

Cf. A014176.

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).

Sequence in context: A207720 A171288 A001023 * A067221 A072533 A041085

Adjacent sequences:  A278473 A278474 A278475 * A278477 A278478 A278479

KEYWORD

nonn,easy

AUTHOR

Ilya Gutkovskiy, Nov 23 2016

STATUS

approved

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Last modified February 23 23:56 EST 2018. Contains 299595 sequences. (Running on oeis4.)