Original entry on oeis.org
0, 1, 5, 19, 60, 182, 546, 1639, 4919, 14761, 44286, 132860, 398580, 1195741, 3587225, 10761679, 32285040, 96855122, 290565366, 871696099, 2615088299, 7845264901, 23535794706, 70607384120, 211822152360, 635466457081
Offset: 0
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Join[{0}, Table[(1/4)*3^(n + 1) - (1/12)*(-1)^n + (1/3)*Cos[Pi*n/3] - (Sqrt[3]/3)*Sin[Pi*n/3] - 1, {n, 1, 25}]] (* G. C. Greubel, Oct 07 2016 *)
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a(n)=([0,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; -3,4,-1,-3,4]^n*[0;1;5;19;60])[1,1] \\ Charles R Greathouse IV, Oct 08 2016
Original entry on oeis.org
2, 8, 28, 82, 244, 728, 2186, 6560, 19684, 59050, 177148, 531440, 1594322, 4782968, 14348908, 43046722, 129140164, 387420488, 1162261466, 3486784400, 10460353204, 31381059610, 94143178828, 282429536480, 847288609442
Offset: 0
Cf.
A133448 (hexaperiodic sequence of digital roots).
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a:=[2,8,28,82];; 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
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R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( 2*(1+x+2*x^2)/((1+x)*(1-3*x)*(1-x+x^2)) )); // G. C. Greubel, Nov 21 2019
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seq(coeff(series(2*(1+x+2*x^2)/((1+x)*(1-3*x)*(1-x+x^2)), x, n+1), x, n), n = 0..30); # G. C. Greubel, Nov 21 2019
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LinearRecurrence[{3,0,-1,3}, {2,8,28,82}, 30] (* G. C. Greubel, Oct 07 2016 *)
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my(x='x+O('x^30)); Vec(2*(1+x+2*x^2)/((1+x)*(1-3*x)*(1-x+x^2))) \\ G. C. Greubel, Nov 21 2019
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def A135263_list(prec):
P. = PowerSeriesRing(ZZ, prec)
return P(2*(1+x+2*x^2)/((1+x)*(1-3*x)*(1-x+x^2))).list()
A135263_list(30) # G. C. Greubel, Nov 21 2019
Original entry on oeis.org
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
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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
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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
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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
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LinearRecurrence[{3,0,-1,3}, {3,12,42,123}, 25] (* G. C. Greubel, Oct 07 2016 *)
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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
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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
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