A099787 -id:A099787 - cp's OEIS frontend

A099783 a(n) = Sum_{k=0..floor(n/3)} C(n-k,2k)3^(n-2*k).

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

Paul Barry, Oct 26 2004

In general a(n) = Sum_{k=0..floor(n/3)} C(n-k,2*k) * u^k * v^(n-3*k) has g.f. (1-v*x)/((1-v*x)^2 - u*x^2) and satisfies the recurrence a(n) = 2*u*v*a(n-1) - v^2*a(n-2) + u*a(n-3).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-9,3).

Cf. A099780, A099781, A099782, A099784, A099785, A099786, A099787.

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

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

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

LinearRecurrence[{6,-9,3}, {1,3,9}, 30] (* G. C. Greubel, Sep 04 2019 *)

my(x='x+O('x^30)); Vec((1-3*x)/((1-3*x)^2 - 3*x^3)) \\ G. C. Greubel, Sep 04 2019

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

G.f.: (1-3*x)/((1-3*x)^2 - 3*x^3).

a(n) = 6*a(n-1) - 9*a(n-2) + 3*a(n-3).

A099784 a(n) = Sum_{k=0..floor(n/3)} C(n-k,2k) 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

Paul Barry, Oct 26 2004

In general a(n) = Sum_{k=0..floor(n/3)} C(n-k,2*k) * u^k * v^(n-3*k) has g.f. (1-v*x)/((1-v*x)^2 - u*x^2) and satisfies the recurrence a(n) = 2*u*v*a(n-1) - v^2*a(n-2) + u*a(n-3).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-4,-4,2).

Cf. A099780, A099781, A099782, A099783, A099785, A099786, A099787.

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

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

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

LinearRecurrence[{-4,-4,2}, {1,-2,4}, 30] (* G. C. Greubel, Sep 04 2019 *)

my(x='x+O('x^30)); Vec((1-2*x)/((1-2*x)^2 - 2*x^3)) \\ G. C. Greubel, Sep 04 2019

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

G.f.: (1+2*x)/((1+2*x)^2 - 2*x^3).
a(n) = Sum_{k=0..floor(n/3)} C(n-k, 2*k)*2^(n-2*k)*(-1)^(n-3*k).
a(n) = -4*a(n-1) - 4*a(n-2) + 2*a(n-3).

A099785 a(n) = Sum_{k=0..floor(n/4)} C(n-k,3k) 2^(n-3*k).

Original entry on oeis.org

1, 2, 4, 8, 18, 48, 144, 448, 1380, 4152, 12224, 35456, 102024, 292768, 840416, 2416384, 6959504, 20069280, 57913536, 167158656, 482462752, 1392319488, 4017460224, 11590946816, 33439639616, 96470796672, 278311599616

Offset: 0

Paul Barry, Oct 26 2004

In general a(n) = Sum_{k=0..floor(n/4)} C(n-k,3*k) * u^k * v^(n-4*k) has g.f. (1-v*x)^2/((1-v*x)^3 - u*x^4) and satisfies the recurrence a(n) = 3*v*a(n-1) - 3*v^2*a(n-2) + v^3*a(n-3) + u*a(n-4).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-12,8,2).

Cf. A003522, A097119.

Cf. A099780, A099781, A099783, A099784, A099785, A099786, A099787.

a:=[1,2,4,8];; for n in [5..30] do a[n]:=6*a[n-1] -12*a[n-2] + 8*a[n-3] +2*a[n-4]; od; a; # G. C. Greubel, Sep 04 2019

I:=[1,2,4,8]; [n le 4 select I[n] else 6*Self(n-1) - 12*Self(n-2) + 8*Self(n-3) + 2*Self(n-4): n in [1..30]]; // G. C. Greubel, Sep 04 2019

seq(coeff(series((1-2*x)^2/((1-2*x)^3 - 2*x^4), x, n+1), x, n), n = 0 .. 40); # G. C. Greubel, Sep 04 2019

Table[Sum[Binomial[n-k,3k]2^(n-3k),{k,0,Floor[n/4]}],{n,0,30}] (* or *) LinearRecurrence[{6,-12,8,2},{1,2,4,8},30] (* Harvey P. Dale, Apr 01 2012 *)

my(x='x+O('x^30)); Vec((1-2*x)^2/((1-2*x)^3 - 2*x^4)) \\ G. C. Greubel, Sep 04 2019

def A099785_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-2*x)^2/((1-2*x)^3 - 2*x^4)).list()
A099785_list(30) # G. C. Greubel, Sep 04 2019

G.f.: (1-2*x)^2/((1-2*x)^3 - 2*x^4).

a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3) + 2*a(n-4).

A099786 a(n) = Sum_{k=0..floor(n/4)} C(n-k,3k)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

Paul Barry, Oct 26 2004

In general a(n) = Sum_{k=0..floor(n/4)} C(n-k,3*k) * u^k * v^(n-4*k) has g.f. (1-v*x)^2/((1-v*x)^3 - u*x^4) and satisfies the recurrence a(n) = 3*v*a(n-1) - 3*v^2*a(n-2) + v^3*a(n-3) + u*a(n-4).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-27,27,1).

Cf. A003522, A097119.

Cf. A099780, A099781, A099782, A099783, A099784, A099785, A099787.

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

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

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

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 *)

my(x='x+O('x^30)); Vec((1-3*x)^2/((1-3*x)^3 - x^4)) \\ G. C. Greubel, Sep 04 2019

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

G.f.: (1-3*x)^2/((1-3*x)^3 - x^4).

a(n) = 9*a(n-1) - 27*a(n-2) + 27*a(n-3) + a(n-4).

A099782 a(n) = Sum_{k=0..floor(n/3)} C(n-k,2k) 2^k * 4^(n-3*k).

Original entry on oeis.org

1, 4, 16, 66, 280, 1216, 5380, 24144, 109504, 500488, 2300128, 10612224, 49096720, 227578432, 1056304384, 4907373600, 22813275520, 106100835328, 493609021504, 2296885357824, 10689540189184, 49753373831296, 231588118339072

Offset: 0

Paul Barry, Oct 26 2004

In general a(n) = Sum_{k=0..floor(n/3)} C(n-k,2*k) * u^k * v^(n-3*k) has g.f. (1-v*x)/((1-v*x)^2 - u*x^2) and satisfies the recurrence a(n) = 2*u*v*a(n-1) - v^2*a(n-2) + u*a(n-3).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-16,2).

Cf. A099780, A099781, A099783, A099784, A099785, A099786, A099787.

a:=[1,4,16];; for n in [4..30] do a[n]:=8*a[n-1]-16*a[n-2] + 2*a[n-3]; od; a; # G. C. Greubel, Sep 04 2019

I:=[1,4,16]; [n le 3 select I[n] else 8*Self(n-1) - 16*Self(n-2) + 2*Self(n-3): n in [1..30]]; // G. C. Greubel, Sep 04 2019

seq(coeff(series((1-4*x)/((1-4*x)^2 - 2*x^3), x, n+1), x, n), n = 0 .. 40); # G. C. Greubel, Sep 04 2019

LinearRecurrence[{8,-16,2}, {1,4,16}, 30] (* G. C. Greubel, Sep 04 2019 *)

my(x='x+O('x^30)); Vec((1-4*x)/((1-4*x)^2 - 2*x^3)) \\ G. C. Greubel, Sep 04 2019

def A099782_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-4*x)/((1-4*x)^2 - 2*x^3)).list()
A099782_list(30) # G. C. Greubel, Sep 04 2019

G.f.: (1-4*x)/((1-4*x)^2 - 2*x^3).

a(n) = 8*a(n-1) - 16*a(n-2) + 2*a(n-3).

cp's OEIS Frontend

A099783 a(n) = Sum_{k=0..floor(n/3)} C(n-k,2*k)*3^(n-2*k).

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A099784 a(n) = Sum_{k=0..floor(n/3)} C(n-k,2*k) * 2^k * (-2)^(n-3*k).

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A099785 a(n) = Sum_{k=0..floor(n/4)} C(n-k,3*k) * 2^(n-3*k).

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Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A099786 a(n) = Sum_{k=0..floor(n/4)} C(n-k,3*k)*3^(n-4*k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A099782 a(n) = Sum_{k=0..floor(n/3)} C(n-k,2*k) * 2^k * 4^(n-3*k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A099783 a(n) = Sum_{k=0..floor(n/3)} C(n-k,2k)3^(n-2*k).

A099784 a(n) = Sum_{k=0..floor(n/3)} C(n-k,2k) 2^k * (-2)^(n-3*k).

A099785 a(n) = Sum_{k=0..floor(n/4)} C(n-k,3k) 2^(n-3*k).

A099786 a(n) = Sum_{k=0..floor(n/4)} C(n-k,3k)3^(n-4*k).

A099782 a(n) = Sum_{k=0..floor(n/3)} C(n-k,2k) 2^k * 4^(n-3*k).