A278476 a(n) = floor((1 + sqrt(2))^3*a(n-1)) for n>0, a(0) = 1.

1, 14, 196, 2757, 38793, 545858, 7680804, 108077113, 1520760385, 21398722502, 301102875412, 4236838978269, 59616848571177, 838872718974746, 11803834914217620, 166092561518021425, 2337099696166517569, 32885488307849267390, 462733936006056261028

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((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

• G. C. Greubel, Table of n, a(n) for n = 0..869
• Index entries for linear recurrences with constant coefficients, signature (15,-13,-1).

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).

Programs

Magma

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x-x^2)/((1-x)*(1-14*x-x^2)))); // G. C. Greubel, Oct 10 2018

Maple

seq(coeff(series((1-x-x^2)/((1-x)*(1-14*x-x^2)),x,n+1), x, n), n = 0 .. 20); # Muniru A Asiru, Oct 11 2018

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]
CoefficientList[Series[(1-x-x^2)/((1-x)*(1-14*x-x^2)), {x,0,50}], x] (* G. C. Greubel, Oct 10 2018 *)

PARI

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

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.