A094555 -id:A094555 - cp's OEIS frontend

A140431 2*A094555(n).

0, 2, 2, 12, 22, 92, 222, 772, 2102, 6732, 19342, 59732, 175782, 534172, 1588862, 4793892, 14327062, 43090412, 129052782, 387595252, 1161911942, 3487483452, 10458955102, 31383855812, 94137586422, 282440721292, 847266239822

Offset: 0

Paul Curtz, Jun 19 2008

Index entries for linear recurrences with constant coefficients, signature (2,5,-6).

b(n) = A091002(n-1); b(n+1)-3b(n)= A077925(n-2), where b(n)=floor(a(n)/10).

a(n) = (1-(-2)^n+3^n)/3 for n>0. a(n) = 2*a(n-1)+5*a(n-2)-6*a(n-3) for n>3. G.f.: 2*x*(1-x-x^2)/((1-x)*(1+2*x)*(1-3*x)). [Colin Barker, Sep 21 2012]

Edited and extended by R. J. Mathar, Aug 02 2008

A140966 a(n) = (5 + (-2)^n)/3.

Original entry on oeis.org

2, 1, 3, -1, 7, -9, 23, -41, 87, -169, 343, -681, 1367, -2729, 5463, -10921, 21847, -43689, 87383, -174761, 349527, -699049, 1398103, -2796201, 5592407, -11184809, 22369623, -44739241, 89478487, -178956969, 357913943, -715827881, 1431655767, -2863311529, 5726623063

Offset: 0

Paul Curtz, Jul 27 2008

Inverse binomial transform of A048573.
This is an example of the case k=-1 of sequences with recurrences a(n) = k*a(n-1) + (k+3)*a(n-2) - (2*k+2)*a(n-3).
The case k=1 is covered, for example, by A097163, A135520, A136326, A136336, or A137208.
Sequences with k=2 are A094554 and A094555.
Sequences with k=3 are A084175, A108924, and A139818.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-1,2).

Cf. A048573.
Cf. A097163, A135520, A136326, A136336, A137208.
Cf. A094554, A094555.
Cf. A084175, A108924, A139818.
Cf. A000079, A001045, A078008, A083582, A083584, A163834.

[( 5+(-2)^n)/3: n in [0..35]]; // Vincenzo Librandi, Jul 05 2011

(5+(-2)^Range[0,30])/3 (* or *) LinearRecurrence[{-1,2},{2,1},40] (* Harvey P. Dale, Apr 23 2019 *)

a(n)=(5+(-2)^n)/3 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = -a(n-1) + 2*a(n-2).
G.f.: (2+3*x)/((1-x)*(1+2*x)).
a(n+1) - a(n) = (-1)^(n+1)*A000079(n).
a(n+3) = (-1)^n*A083582(n).
a(n+1) - 2*a(n) = -a(n+2).
a(n+1) - 3*a(n) = 5*(-1)^(n+1)*A078008(n) = (-1)^(n+1)*A001045(n-1).
a(2n+3) = -A083584(n), a(2n) = A163834(n). - Philippe Deléham, Feb 24 2014
E.g.f.: (5*exp(x) + exp(-2*x))/3. - Stefano Spezia, Jul 27 2024

Definition simplified by R. J. Mathar, Sep 11 2009

A140429 a(n) = floor(3^(n-1)).

Original entry on oeis.org

0, 1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907, 43046721, 129140163, 387420489, 1162261467, 3486784401, 10460353203, 31381059609, 94143178827, 282429536481, 847288609443

Offset: 0

Paul Curtz, Jun 19 2008

Binomial transform of Jacobsthal numbers A001045.

Implicit use in A094555 (Barry).

Index entries for linear recurrences with constant coefficients, signature (3).

Cf. A000079, A131577, A133494.

a(n)=if(n<2,n,3^(n-1)) \\ Charles R Greathouse IV, Oct 03 2016

a(n)=floor(3^(n-1)) \\ M. F. Hasler, Apr 13 2018

a(n) = floor(3^(n-1)) = A000244(n-1) = A133494(n), n >= 1.

O.g.f.: x/(1-3x). - R. J. Mathar, Aug 27 2008

Extended by R. J. Mathar, Aug 28 2008

New name by M. F. Hasler, Apr 13 2018

A094554 Number of closed walks of length n at a base vertex of a truncated tetrahedron (triangular prism).

Original entry on oeis.org

1, 0, 3, 2, 19, 30, 143, 322, 1179, 3110, 10183, 28842, 89939, 262990, 802623, 2380562, 7196299, 21479670, 64657463, 193535482, 581480259, 1742693150, 5231574703, 15687733602, 47077181819, 141203583430, 423666674343

Offset: 0

Paul Barry, May 11 2004

For n > 0, 6*a(n) is the number of 3-colorings of the prism of size 2 X n (i.e., C_2 X C_n).More generally, the number of k-colorings of the prism of size 2 X n is given by (k^2 - 3*k + 3)^n + (k - 1) * ((3 - k)^n + (1 - k)^n) + k^2 - 3*k + 1 (chromatic polynomial). - Sela Fried, Oct 07 2023

Andrew Howroyd, Table of n, a(n) for n = 0..1000
N. L. Biggs, R. M. Damerell, and D. A. Sands, Recursive families of graphs, Journal of Combinatorial Theory Series B Volume 12 (1972), 123-131.
R. J. Mathar, Counting Walks on Finite Graphs, Nov 2020, Section 3.
Index entries for linear recurrences with constant coefficients, signature (2,5,-6).

Cf. A002001, A025192, A094555, A094556, A328778.

LinearRecurrence[{2, 5, -6}, {1, 0, 3, 2}, 30] (* Greg Dresden, Jun 19 2021 *)

a(n) = if(n==0, 1, (1 + 3^n + 2*(-2)^n)/6) \\ Andrew Howroyd, Jun 14 2021

G.f.: (1 - 2*x - 2*x^2 + 2*x^3)/((1 - x)*(1 + 2*x)*(1 - 3*x)).
a(n) = 1/6 + 3^n/6 + (-2)^n/3 for n > 0.
a(n) = 2*a(n-1) + 5*a(n-2) - 6*a(n-3) for n >= 4.
E.g.f.: exp(-2*x)*(1 + exp(2*x))*(2 + exp(3*x))/6. - Stefano Spezia, Sep 26 2023

A094556 Number of walks of length n between opposite vertices on a triangular prism.

Original entry on oeis.org

0, 1, 0, 7, 8, 51, 100, 407, 1008, 3451, 9500, 30207, 87208, 268451, 791700, 2402407, 7152608, 21567051, 64482700, 193885007, 580781208, 1744091251, 5228778500, 15693326007, 47065997008, 141225953051, 423621935100, 1270977653407

Offset: 0

Paul Barry, May 11 2004

Andrew Howroyd, Table of n, a(n) for n = 0..1000
R. J. Mathar, Counting Walks on Finite Graphs, Nov 2020, Section 3.
Index entries for linear recurrences with constant coefficients, signature (2,5,-6).

Cf. A094554, A094555.

LinearRecurrence[{2,5,-6},{0,1,0,7},30] (* or *) CoefficientList[ Series[ x (1-2x+2x^2)/((1-x)(1+2x)(1-3x)),{x,0,30}],x] (* Harvey P. Dale, Jul 13 2011 *)

a(n) = if(n==0, 0, (3^n - 2*(-2)^n - 1)/6) \\ Andrew Howroyd, Jun 15 2021

G.f.: x*(1 - 2*x + 2*x^2)/((1 - x)*(1 + 2*x)*(1 - 3*x)).
a(n) = 3^n/6 - (-2)^n/3 - 1/6 + 0^n/3.
a(n) = 2*a(n-1) + 5*a(n-2) - 6*a(n-3) for n >= 4.
E.g.f.: exp(-x)*(2+exp(3*x))*sinh(x)/3. - Stefano Spezia, Sep 26 2023

A344747 a(n) = (1/6)*(3^n + (-2)^n - 1).

Original entry on oeis.org

0, 2, 3, 16, 35, 132, 343, 1136, 3195, 10012, 29183, 89256, 264355, 799892, 2386023, 7185376, 21501515, 64613772, 193622863, 581305496, 1743042675, 5230875652, 15689131703, 47074385616, 141209175835, 423655489532, 1270910544543, 3812843481736, 11438306748995

Offset: 1

Ryan Brooks, Jun 14 2021

			a(4) = (1/6)*(3^4 + (-2)^4 - 1) = (1/6)*(81+16-1) = 16.

Index entries for linear recurrences with constant coefficients, signature (2,5,-6).

Potentially related to A094554, A094555, and A094556 (which have the same recurrence).

LinearRecurrence[{2, 5, -6}, {0, 2, 3}, 30] (* Greg Dresden, Jun 19 2021 *)

G.f.: x*(2 - x)/((1 - x)*(1 + 2*x)*(1 - 3*x)). - Andrew Howroyd, Jun 15 2021

a(n) = A094554(n-1) + 2*A094556(n-1). - Greg Dresden, Jun 19 2021

cp's OEIS Frontend

A140431 2*A094555(n).

Views

Author

Keywords

Links

Formula

Extensions

A140966 a(n) = (5 + (-2)^n)/3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A140429 a(n) = floor(3^(n-1)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

Formula

Extensions

A094554 Number of closed walks of length n at a base vertex of a truncated tetrahedron (triangular prism).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A094556 Number of walks of length n between opposite vertices on a triangular prism.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A344747 a(n) = (1/6)*(3^n + (-2)^n - 1).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula