A102518 -id:A102518 - cp's OEIS frontend

A094555 Number of walks of length n between two vertices on the same triangular face of a truncated tetrahedron (triangular prism).

0, 1, 1, 6, 11, 46, 111, 386, 1051, 3366, 9671, 29866, 87891, 267086, 794431, 2396946, 7163531, 21545206, 64526391, 193797626, 580955971, 1743741726, 5229477551, 15691927906, 47068793211, 141220360646, 423633119911, 1270955283786

Offset: 0

Paul Barry, May 11 2004

Average of binomial and inverse binomial transforms of the Jacobsthal numbers A001045. - Paul Barry, Jan 04 2005

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. A001045, A094554, A094556.

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

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

G.f.: x*(1 - x - x^2)/((1 - x)*(1 + 2*x)*(1 - 3*x)).
a(n) = 3^n/6 - (-2)^n/6 + 1/6 - 0^n/6.
a(n) = 2*a(n-1) + 5*a(n-2) - 6*a(n-3) for n >= 4.
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2k)*A001045(n-2k). - Paul Barry, Jan 04 2005
E.g.f.: exp(-2*x)*(exp(5*x) + exp(3*x) - exp(2*x) - 1)/6. - Stefano Spezia, Dec 26 2021

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

A054880 a(n) = 3*(9^n - 1)/4.

Original entry on oeis.org

0, 6, 60, 546, 4920, 44286, 398580, 3587226, 32285040, 290565366, 2615088300, 23535794706, 211822152360, 1906399371246, 17157594341220, 154418349070986, 1389765141638880, 12507886274749926, 112570976472749340, 1013138788254744066, 9118249094292696600, 82064241848634269406, 738578176637708424660

Offset: 0

Paolo Dominici (pl.dm(AT)libero.it), May 23 2000

Number of walks of length 2n+1 along the edges of a (3 dimensional) cube between two opposite vertices.

Urn A initially contains 3 labeled balls while urn B is empty. A ball is randomly selected and switched from one urn to the other. a(n)/3^(2n+1) is the probability that urn A is empty after 2n+1 switches. - Geoffrey Critzer, May 23 2013

G. C. Greubel, Table of n, a(n) for n = 0..1000
G. Benkart and D. Moon, A Schur-Weyl Duality Approach to Walking on Cubes, arXiv preprint arXiv:1409.8154 [math.RT], 2014 and Ann. Combin. 20 (3) (2016) 397-417
R. J. Mathar, Counting Walks on Finite Graphs, Nov 2020, Section 5.
Index entries for linear recurrences with constant coefficients, signature (10,-9).

Cf. A002452, A015518, A046717, A066443.

List([0..30], n-> 3*(9^n -1)/4); # G. C. Greubel, Jul 14 2019

[3*(9^n -1)/4: n in [0..30]]; // G. C. Greubel, Jul 14 2019

Table[(2 n + 1)! Coefficient[Series[Sinh[x]^3, {x, 0, 2 n + 1}],
x^(2 n + 1)], {n, 0, 30}]  (* Geoffrey Critzer, May 23 2013 *)
LinearRecurrence[{10,-9},{0,6},30] (* Harvey P. Dale, Sep 17 2024 *)

vector(30, n, n--; 3*(9^n -1)/4) \\ G. C. Greubel, Jul 14 2019

[3*(9^n -1)/4 for n in (0..30)] # G. C. Greubel, Jul 14 2019

G.f.: (3/4)/(1 - 9*x) - (3/4)/(1 - x).
a(n) = 6*A002452(n).
sin(x)^3 = Sum_{k>=0} (-1)^(k+1)*a(k)*x^(2k+1)/(2k+1)!. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 08 2001
a(n) = A015518(2n+1) - 1 = (A046717(2n+1) - 1)/2. - M. F. Hasler, Mar 20 2008
a(n) = 9*a(n-1) + 6 with n > 0, a(0) = 0. - Vincenzo Librandi, Aug 07 2010
a(n) = A066443(n) - 1. - Georg Fischer, Nov 25 2018
E.g.f.: 3*(exp(9*x) - exp(x))/4. - G. C. Greubel, Jul 14 2019
a(n) = 10*a(n-1) - 9*a(n-2) with a(0) = 0 and a(1) = 6. - Miquel A. Fiol, Mar 09 2024

A125857 Numbers whose base-9 representation is 22222222.......2.

Original entry on oeis.org

0, 2, 20, 182, 1640, 14762, 132860, 1195742, 10761680, 96855122, 871696100, 7845264902, 70607384120, 635466457082, 5719198113740, 51472783023662, 463255047212960, 4169295424916642, 37523658824249780, 337712929418248022

Offset: 1

Zerinvary Lajos, Feb 03 2007

If f(1) := 1/x and f(n+1) = (f(n) + 2/f(n))/3, then f(n) = 3^(1-n) * (1/x + a(n)*x + O(x^3)). - Michael Somos, Jul 28 2020

			G.f. = 2*x^2 + 20*x^3 + 182*x^4 + 1640*x^5 + 14762*x^6 + 132860*x^7 + ... - _Michael Somos_, Jul 28 2020

G. Benkart, D. Moon, A Schur-Weyl Duality Approach to Walking on Cubes, arXiv preprint arXiv:1409.8154 [math.RT], 2014 and Ann. Combin. 20 (3) (2016) 397-417
E. Estrada and J. A. de la Pena, From Integer Sequences to Block Designs via Counting Walks in Graphs, arXiv preprint arXiv:1302.1176 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 28 2013
E. Estrada and J. A. de la Pena, Integer sequences from walks in graphs, Notes on Number Theory and Discrete Mathematics, Vol. 19, 2013, No. 3, 78-84.
R. J. Mathar, Counting Walks on Finite Graphs, Nov 2020, Section 5.
Vladimir Pletser, Congruence conditions on the number of terms in sums of consecutive squared integers equal to squared integers, arXiv:1409.7969 [math.NT], 2014.
Index entries for linear recurrences with constant coefficients, signature (10,-9).

Cf. A002452.

Maple
```
seq((9^n-1)*2/8, n=0..19);
```

FromDigits[#, 9]&/@Table[PadRight[{2}, n, 2], {n, 0, 20}] (* Harvey P. Dale, Feb 02 2011 *)
Table[(9^(n - 1) - 1)*2/8, {n, 20}] (* Wesley Ivan Hurt, Mar 29 2014 *)

Vec(2*x^2/((x-1)*(9*x-1)) + O(x^100)) \\ Colin Barker, Sep 30 2014

{a(n) = (9^(n-1) - 1)/4}; /* Michael Somos, Jul 02 2017 */

a(n) = (9^(n-1) - 1)*2/8.
a(n) = 9*a(n-1) + 2 (with a(1)=0). - Vincenzo Librandi, Sep 30 2010
a(n) = 2 * A002452(n). - Vladimir Pletser, Mar 29 2014
From Colin Barker, Sep 30 2014: (Start)
a(n) = 10*a(n-1) - 9*a(n-2).
G.f.: 2*x^2 / ((x-1)*(9*x-1)). (End)
a(n) = -a(2-n) * 9^(n-1) for all n in Z. - Michael Somos, Jul 02 2017
a(n) = A191681(n-1)/2. - Klaus Purath, Jul 03 2020

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

A270369 Expansion of g.f. (1-7x)/(1-9x).

Original entry on oeis.org

1, 2, 18, 162, 1458, 13122, 118098, 1062882, 9565938, 86093442, 774840978, 6973568802, 62762119218, 564859072962, 5083731656658, 45753584909922, 411782264189298, 3706040377703682, 33354363399333138, 300189270593998242, 2701703435345984178, 24315330918113857602, 218837978263024718418

Offset: 0

Colin Barker, Mar 18 2016

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9).

Cf. A001019 (powers of 9), A054879 (partial sums), A132025.

Cf. similar sequences with g.f. (1-k*x)/(1-9*x) and k=0..8: A001019 (k=0; k=8 gives two initial 1's ), A055275 (k=1), A270472 (k=2), A092810 (k=3), A067403 (k=4), A270473 (k=5), A102518 (k=6), this sequence (k=7).

CoefficientList[Series[(1-7x)/(1-9x),{x,0,20}],x] (* or *) Join[ {1}, NestList[9#&,2,20]] (* Harvey P. Dale, Oct 15 2017 *)

PARI
```
Vec((1-7*x)/(1-9*x) + O(x^30))
```

G.f.: (1-7*x)/(1-9*x).
a(n) = 9*a(n-1) for n>1.
a(n) = 2*9^(n-1) for n>0.
From Amiram Eldar, May 08 2023: (Start)
Sum_{n>=0} 1/a(n) = 25/16.
Sum_{n>=0} (-1)^n/a(n) = 11/20.
Product_{n>=1} (1 - 1/a(n)) = A132025. (End)
E.g.f.: (2*exp(9*x) + 7)/9. - Elmo R. Oliveira, Mar 25 2025

A270473 Expansion of g.f. (1-5x)/(1-9x).

Original entry on oeis.org

1, 4, 36, 324, 2916, 26244, 236196, 2125764, 19131876, 172186884, 1549681956, 13947137604, 125524238436, 1129718145924, 10167463313316, 91507169819844, 823564528378596, 7412080755407364, 66708726798666276, 600378541187996484, 5403406870691968356, 48630661836227715204

Offset: 0

Colin Barker, Mar 17 2016

Also squares that can be expressed as the sum of two powers of three (3^x + 3^y), except a(0). - Karl-Heinz Hofmann, Sep 03 2022

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9).

Cf. A001019 (powers of 9), A083884 (partial sums).
Cf. A067403: (1-4*x)/(1-9*x); A102518: (1-6*x)/(1-9*x).
Cf. A356879, A356880.

Join[{1},NestList[9#&,4,20]] (* Harvey P. Dale, Oct 23 2022 *)

PARI
```
Vec((1-5*x)/(1-9*x) + O(x^30))
```

G.f.: (1-5*x)/(1-9*x).
a(n) = 9*a(n-1) for n>1.
a(n) = 4*9^(n-1) for n>0.
E.g.f.: (4*exp(9*x) + 5)/9. - Stefano Spezia, Jul 09 2024

A199573 Number of round trips of length n from any of the four vertices of the cycle graph C_4.

Original entry on oeis.org

1, 0, 2, 0, 8, 0, 32, 0, 128, 0, 512, 0, 2048, 0, 8192, 0, 32768, 0, 131072, 0, 524288, 0, 2097152, 0, 8388608, 0, 33554432, 0, 134217728, 0, 536870912, 0, 2147483648, 0, 8589934592, 0, 34359738368, 0, 137438953472, 0

Offset: 0

Wolfdieter Lang, Nov 08 2011

See the array w(N,L) and the triangle a(K,N) given in A199571.
Essentially the same as A103424.
This is A081294 and A000004 interleaved. - Omar E. Pol, Nov 09 2011

			a(4)=8 from the eight round trips of length 4 (starting from, say, vertex no. 1): 12121, 14141, 12141, 14121, 12321, 14341, 12341 and 14321.

R. J. Mathar, Counting Walks on Finite Graphs, Section 1.
Index entries for linear recurrences with constant coefficients, signature (0,4).

Cf. A078008 (N=3), A054877 (N=5), A199571.

CoefficientList[Series[(1 - 2 x^2)/(1 - (2 x)^2), {x, 0, 40}], x] (* or *) Riffle[Join[{1},NestList[4#&,2,20]],0] (* or *) LinearRecurrence[ {0,4},{1,0,2},80] (* Harvey P. Dale, Dec 04 2015 *)

a(n) =  2^(n-2)*(1+(-1)^n), n>=2, a(0)=1.
O.g.f.: (1-2*x^2)/(1-(2*x)^2).
E.g.f.: 1+(1 + 2*x^2/(U(0) - 2*x^2 + 1))*x^2 where U(k)=  4*k+5 + 2*x^2/(1 + (2*k+3)*(k+2)/U(k+1)) ; (continued fraction, 3rd kind, 2-step). - Sergei N. Gladkovskii, Oct 28 2012

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A094555 Number of walks of length n between two vertices on the same triangular face of a truncated tetrahedron (triangular prism).

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

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A054880 a(n) = 3*(9^n - 1)/4.

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A125857 Numbers whose base-9 representation is 22222222.......2.

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

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A270369 Expansion of g.f. (1-7*x)/(1-9*x).

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A270473 Expansion of g.f. (1-5*x)/(1-9*x).

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A270369 Expansion of g.f. (1-7x)/(1-9x).

A270473 Expansion of g.f. (1-5x)/(1-9x).