A099783
a(n) = Sum_{k=0..floor(n/3)} C(n-k,2*k)*3^(n-2*k).
Original entry on oeis.org
1, 3, 9, 30, 108, 405, 1548, 5967, 23085, 89451, 346842, 1345248, 5218263, 20242872, 78528609, 304640595, 1181814705, 4584708702, 17785841652, 68998115709, 267670245492, 1038395956527, 4028337876861, 15627474388899, 60624993311226
Offset: 0
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a:=[1,3,9];; for n in [4..30] do a[n]:=6*a[n-1]-9*a[n-2]+3*a[n-3]; od; a; # G. C. Greubel, Sep 04 2019
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I:=[1,3,9]; [n le 3 select I[n] else 6*Self(n-1) - 9*Self(n-2) + 3*Self(n-3): n in [1..30]]; // G. C. Greubel, Sep 04 2019
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seq(coeff(series((1-3*x)/((1-3*x)^2 - 3*x^3), x, n+1), x, n), n = 0..30); # G. C. Greubel, Sep 04 2019
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LinearRecurrence[{6,-9,3}, {1,3,9}, 30] (* G. C. Greubel, Sep 04 2019 *)
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my(x='x+O('x^30)); Vec((1-3*x)/((1-3*x)^2 - 3*x^3)) \\ G. C. Greubel, Sep 04 2019
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def A099783_list(prec):
P. = PowerSeriesRing(ZZ, prec)
return P((1-3*x)/((1-3*x)^2 - 3*x^3)).list()
A099783_list(30) # G. C. Greubel, Sep 04 2019
A099784
a(n) = Sum_{k=0..floor(n/3)} C(n-k,2*k) * 2^k * (-2)^(n-3*k).
Original entry on oeis.org
1, -2, 4, -6, 4, 16, -92, 312, -848, 1960, -3824, 5760, -3824, -15392, 88384, -299616, 814144, -1881344, 3669568, -5524608, 3657472, 14807680, -84909824, 287723520, -781639424, 1805843968, -3521371136, 5298829824
Offset: 0
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a:=[1,-2,4];; for n in [4..30] do a[n]:=-4*a[n-1]-4*a[n-2] + 2*a[n-3]; od; a; # G. C. Greubel, Sep 04 2019
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I:=[1,-2,4]; [n le 3 select I[n] else -4*Self(n-1) - 4*Self(n-2) + 2*Self(n-3): n in [1..30]]; // G. C. Greubel, Sep 04 2019
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seq(coeff(series((1-2*x)/((1-2*x)^2 - 2*x^3), x, n+1), x, n), n = 0 .. 30); # G. C. Greubel, Sep 04 2019
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LinearRecurrence[{-4,-4,2}, {1,-2,4}, 30] (* G. C. Greubel, Sep 04 2019 *)
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my(x='x+O('x^30)); Vec((1-2*x)/((1-2*x)^2 - 2*x^3)) \\ G. C. Greubel, Sep 04 2019
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def A099784_list(prec):
P. = PowerSeriesRing(ZZ, prec)
return P((1-2*x)/((1-2*x)^2 - 2*x^3)).list()
A099784_list(30) # G. C. Greubel, Sep 04 2019
A099786
a(n) = Sum_{k=0..floor(n/4)} C(n-k,3*k)*3^(n-4*k).
Original entry on oeis.org
1, 3, 9, 27, 82, 255, 819, 2727, 9397, 33312, 120537, 441855, 1631017, 6036879, 22345074, 82589247, 304612975, 1120960983, 4116353265, 15088372416, 55224373105, 201895801851, 737506551321, 2692518758163, 9826402960882
Offset: 0
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a:=[1,3,9,27];; for n in [5..30] do a[n]:=9*a[n-1]-27*a[n-2] + 27*a[n-3] +a[n-4]; od; a; # G. C. Greubel, Sep 04 2019
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I:=[1,3,9,27]; [n le 4 select I[n] else 9*Self(n-1) - 27*Self(n-2) + 27*Self(n-3) +Self(n-4): n in [1..30]]; // G. C. Greubel, Sep 04 2019
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seq(coeff(series((1-3*x)^2/((1-3*x)^3 - x^4), x, n+1), x, n), n = 0..30); # G. C. Greubel, Sep 04 2019
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LinearRecurrence[{9,-27,27,1},{1,3,9,27},40] (* or *) CoefficientList[ Series[-((1-3 x)^2/(x (x (x (x+27)-27)+9)-1)),{x,0,40}],x] (* Harvey P. Dale, Jun 06 2011 *)
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my(x='x+O('x^30)); Vec((1-3*x)^2/((1-3*x)^3 - x^4)) \\ G. C. Greubel, Sep 04 2019
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def A099786_list(prec):
P. = PowerSeriesRing(ZZ, prec)
return P((1-3*x)^2/((1-3*x)^3 - x^4)).list()
A099786_list(30) # G. C. Greubel, Sep 04 2019
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