A135265 -id:A135265 - cp's OEIS frontend

A135264 a(n) = 3*A132357(n).

3, 12, 42, 123, 366, 1092, 3279, 9840, 29526, 88575, 265722, 797160, 2391483, 7174452, 21523362, 64570083, 193710246, 581130732, 1743392199, 5230176600, 15690529806, 47071589415, 141214768242, 423644304720, 1270932914163

Offset: 0

Paul Curtz, Dec 02 2007

Digital roots yield a hexaperiodic sequence A010888(a(n))= 3*A135265(n+1).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,0,-1,3).

a:=[3,12,42,123];; for n in [5..30] do a[n]:=3*a[n-1]-a[n-3]+ 3*a[n-4]; od; a; # G. C. Greubel, Nov 21 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( 3*(1+x+2*x^2)/(1-3*x+x^3-3*x^4) )); // G. C. Greubel, Nov 21 2019

seq(coeff(series(3*(1+x+2*x^2)/(1-3*x+x^3-3*x^4), x, n+1), x, n), n = 0..30); # G. C. Greubel, Nov 21 2019

LinearRecurrence[{3,0,-1,3}, {3,12,42,123}, 25] (* G. C. Greubel, Oct 07 2016 *)

my(x='x+O('x^30)); Vec(3*(1+x+2*x^2)/(1-3*x+x^3-3*x^4)) \\ G. C. Greubel, Nov 21 2019

def A135264_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(3*(1+x+2*x^2)/(1-3*x+x^3-3*x^4)).list()
A135264_list(30) # G. C. Greubel, Nov 21 2019

a(n) = 3*a(n-1) - a(n-3) + 3*a(n-4).

G.f.: 3*(1 + x + 2*x^2)/(1 - 3*x + x^3 - 3*x^4). - G. C. Greubel, Oct 07 2016 [corrected by Georg Fischer, May 10 2019]

Edited, corrected and extended by R. J. Mathar, Jul 28 2008

A144110 Period 6: repeat [2, 2, 2, 1, 1, 1].

Original entry on oeis.org

2, 2, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1

Offset: 0

Philippe Deléham, Sep 11 2008, Sep 15 2008

a(n) = 2 for n = 0,1,2 modulo 6; a(n) = 1 for n = 3,4,5 modulo 6.

Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1).

Cf. A057079, A088911, A135265.

[1+(Floor((-n-1)/3) mod 2) : n in [0..100]]; // Wesley Ivan Hurt, Sep 04 2014

A144110:=n->1+(floor((-n-1)/3) mod 2): seq(A144110(n), n=0..100); # Wesley Ivan Hurt, Sep 04 2014

Table[1 + Mod[Floor[(-n - 1)/3], 2], {n, 0, 100}] (* Wesley Ivan Hurt, Sep 04 2014 *)

a(n)=[2,2,2,1,1,1][n%6+1] \\ Edward Jiang, Sep 04 2014

G.f.: (1+2*x^3)/((1-x)*(1+x)*(1-x+x^2)); a(n) = 3/2-(-1)^n/6-A057079(n)/3. [R. J. Mathar, Sep 17 2008]
a(n) = a(n-1) - a(n-3) + a(n-4) for n>3; a(n) = 1 + mod(floor((-n-1)/3), 2); a(n) = A088911(n) + 1. - Wesley Ivan Hurt, Sep 04 2014
a(n) = (9 + cos(n*Pi) + 2*cos(n*Pi/3) + 2*sqrt(3)*sin(n*Pi/3))/6. - Wesley Ivan Hurt, Jun 23 2016

A178331 Decimal expansion of (17+2*sqrt(210))/29.

Original entry on oeis.org

1, 5, 8, 5, 6, 1, 2, 1, 8, 9, 3, 9, 2, 3, 7, 5, 0, 7, 4, 0, 4, 9, 5, 6, 3, 0, 4, 5, 3, 2, 2, 0, 5, 3, 2, 2, 2, 2, 8, 5, 5, 2, 6, 4, 0, 5, 1, 5, 5, 7, 9, 1, 2, 4, 7, 4, 9, 6, 2, 7, 0, 6, 7, 6, 3, 5, 2, 9, 7, 5, 4, 7, 3, 9, 7, 6, 1, 1, 5, 8, 9, 1, 1, 5, 2, 5, 9, 0, 5, 6, 4, 9, 6, 5, 7, 6, 6, 7, 2, 3, 9, 8, 2, 9, 6

Offset: 1

Klaus Brockhaus, May 25 2010

Continued fraction expansion of (17+2*sqrt(210))/29 is A135265.

			(17+2*sqrt(210))/29 = 1.58561218939237507404...

Cf. A176441 (decimal expansion of sqrt(210)), A135265 (repeat 1, 1, 1, 2, 2, 2).

RealDigits[(17+2Sqrt[210])/29,10,120][[1]] (* Harvey P. Dale, Apr 05 2019 *)

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A135264 a(n) = 3*A132357(n).

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A144110 Period 6: repeat [2, 2, 2, 1, 1, 1].

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A178331 Decimal expansion of (17+2*sqrt(210))/29.

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