A084567 -id:A084567 - cp's OEIS frontend

A015518 a(n) = 2a(n-1) + 3a(n-2), with a(0)=0, a(1)=1.

0, 1, 2, 7, 20, 61, 182, 547, 1640, 4921, 14762, 44287, 132860, 398581, 1195742, 3587227, 10761680, 32285041, 96855122, 290565367, 871696100, 2615088301, 7845264902, 23535794707, 70607384120, 211822152361, 635466457082

Offset: 0

Olivier Gérard

Number of walks of length n between any two distinct vertices of the complete graph K_4. - Paul Barry and Emeric Deutsch, Apr 01 2004
For n >= 1, a(n) is the number of integers k, 1 <= k <= 3^(n-1), whose ternary representation ends in an even number of zeros (see A007417). - Philippe Deléham, Mar 31 2004
Form the digraph with matrix A=[0,1,1,1;1,0,1,1;1,1,0,1;1,0,1,1]. A015518(n) corresponds to the (1,3) term of A^n. - Paul Barry, Oct 02 2004
The same sequence may be obtained by the following process. Starting a priori with the fraction 1/1, the denominators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 4 times the bottom to get the new top. The limit of the sequence of fractions is 2. - Cino Hilliard, Sep 25 2005
(A046717(n))^2 + (2*a(n))^2 = A046717(2n). E.g., A046717(3) = 13, 2*a(3) = 14, A046717(6) = 365. 13^2 + 14^2 = 365. - Gary W. Adamson, Jun 17 2006
For n >= 2, number of ordered partitions of n-1 into parts of sizes 1 and 2 where there are two types of 1 (singletons) and three types of 2 (twins). For example, the number of possible configurations of families of n-1 male (M) and female (F) offspring considering only single births and twins, where the birth order of M/F/pair-of-twins is considered and there are three types of twins; namely, both F, both M, or one F and one M - where birth order within a pair of twins itself is disregarded. In particular, for a(3)=7, two children could be either: (1) F, then M; (2) M, then F; (3) F,F; (4) M,M; (5) F,F twins; (6) M,M twins; or (7) M,F twins (emphasizing that birth order is irrelevant here when both/all children are the same gender and when two children are within the same pair of twins). - Rick L. Shepherd, Sep 18 2004
a(n) is prime for n = {2, 3, 5, 7, 13, 23, 43, 281, 359, ...}, where only a(2) = 2 corresponds to a prime of the form (3^k - 1)/4. All prime terms, except a(2) = 2, are the primes of the form (3^k + 1)/4. Numbers k such that (3^k + 1)/4 is prime are listed in A007658. Note that all prime terms have prime indices. Prime terms are listed in A111010. - Alexander Adamchuk, Nov 19 2006
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-2, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=charpoly(A,1). - Milan Janjic, Jan 26 2010
Select an odd size subset S from {1,2,...,n}, then select an even size subset from S. - Geoffrey Critzer, Mar 02 2010
a(n) is the number of ternary sequences of length n where the numbers of (0's, 1's) are (even, odd) respectively, and, by symmetry, the number of such sequences where those numbers are (odd, even) respectively. A122983 covers (even, even), and A081251 covers (odd, odd). - Toby Gottfried, Apr 18 2010
An elephant sequence, see A175654. For the corner squares just one A[5] vector, with decimal value 341, leads to this sequence (without the leading 0). For the central square this vector leads to the companion sequence A046717 (without the first leading 1). - Johannes W. Meijer, Aug 15 2010
Let R be the commutative algebra resulting from adjoining the elements of the Klein four-group to the integers (equivalently, K = Z[x,y,z]/{x*y - z, y*z - x, x*z - y, x^2 - 1, y^2 - 1, z^2 - 1}). Then a(n) is equal to the coefficients of x, y, and z in the expansion of (x + y + z)^n. - Joseph E. Cooper III (easonrevant(AT)gmail.com), Nov 06 2010
Pisano period lengths:  1, 2, 2, 4, 4, 2, 6, 8, 2, 4, 10, 4, 6, 6, 4, 16, 16, 2, 18, 4, ... - R. J. Mathar, Aug 10 2012
The ratio a(n+1)/a(n) converges to 3 as n approaches infinity. - Felix P. Muga II, Mar 09 2014
This is a divisibility sequence, also the values of Chebyshev polynomials, and also the number of ways of packing a 2 X n-1 rectangle with dominoes and unit squares. - R. K. Guy, Dec 16 2016
For n>0, gcd(a(n),a(n+1))=1. - Kengbo Lu, Jul 02 2020

John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see p. 16.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
A. Abdurrahman, CM Method and Expansion of Numbers, arXiv:1909.10889 [math.NT], 2019.
Jean-Paul Allouche, Jeffrey Shallit, Zhixiong Wen, Wen Wu, and Jiemeng Zhang, Sum-free sets generated by the period-k-folding sequences and some Sturmian sequences, arXiv:1911.01687 [math.CO], 2019.
Katerina Böhmová, Cristina Dalfó, and Clemens Huemer, On cyclic Kautz digraphs, Research Report, 2015.
Garry Bowlin and Matthew G. Brin, Coloring Planar Graphs via Colored Paths in the Associahedra, arXiv preprint arXiv:1301.3984 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 12 2013
AJ Bu and Doron Zeilberger, Using Symbolic Computation to Explore Generalized Dyck Paths and Their Areas, arXiv:2305.09030 [math.CO], 2023.
Ji Young Choi, A Generalization of Collatz Functions and Jacobsthal Numbers, J. Int. Seq., Vol. 21 (2018), Article 18.5.4.
Sergio Falcón, Binomial Transform of the Generalized k-Fibonacci Numbers, Communications in Mathematics and Applications (2019) Vol. 10, No. 3, 643-651.
Dale Gerdemann, Fractal generated from (2,3) recursion, YouTube Video, Dec 05 2014.
Toufik Mansour and Mark Shattuck, Pattern avoidance in flattened derangements, Disc. Math. Lett. (2025) Vol. 15, 67-74. See p. 74.
R. J. Mathar, Counting Walks on Finite Graphs, Nov 2020, Section 2.
Felix Pozon Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, March 2014.
Index entries for linear recurrences with constant coefficients, signature (2,3).
Index entries for sequences related to Chebyshev polynomials.

a(n) = A080926(n-1) + 1 = (1/3)*A054878(n+1) = (1/3)*abs(A084567(n+1)).
First differences of A033113 and A039300.
Partial sums of A046717.
The following sequences (and others) belong to the same family: A000129, A001333, A002532, A002533, A002605, A015518, A015519, A026150, A046717, A063727, A083098, A083099, A083100, A084057.
Cf. A046717.
Cf. A007658, A111010.
Cf. A001045, A078008, A097073, A115341. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008
Cf. A007482, A059260.

[Round(3^n/4): n in [0..30]]; // Vincenzo Librandi, Jun 24 2011

Table[(3^n-(-1)^n)/4,{n,0,30}] (* Alexander Adamchuk, Nov 19 2006 *)

a(n):= round(3^n/4)$ /* Dimitri Papadopoulos, Nov 28 2023 */

PARI
```
a(n)=round(3^n/4)
```

for n in range(0, 20): print(int((3**n-(-1)**n)/4), end=', ') # Stefano Spezia, Nov 30 2018

Sage
```
[round(3^n/4) for n in range(0,27)]
```

G.f.: x/((1+x)*(1-3*x)).
a(n) = (3^n - (-1)^n)/4 = floor(3^n/4 + 1/2).
a(n) = 3^(n-1) - a(n-1). - Emeric Deutsch, Apr 01 2004
E.g.f.: (exp(3*x) - exp(-x))/4. Second inverse binomial transform of (5^n-1)/4, A003463. Inverse binomial transform for powers of 4, A000302 (when preceded by 0). - Paul Barry, Mar 28 2003
a(n) = Sum_{k=0..floor(n/2)} C(n, 2k+1)*2^(2k). - Paul Barry, May 14 2003
a(n) = Sum_{k=1..n} binomial(n, k)*(-1)^(n+k)*4^(k-1). - Paul Barry, Apr 02 2003
a(n+1) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*2^(n-2*k)*3^k. - Paul Barry, Jul 13 2004
a(n) = U(n-1, i/sqrt(3))(-i*sqrt(3))^(n-1), i^2=-1. - Paul Barry, Nov 17 2003
G.f.: x*(1+x)^2/(1 - 6*x^2 - 8*x^3 - 3*x^4) = x(1+x)^2/characteristic polynomial(x^4*adj(K_4)(1/x)). - Paul Barry, Feb 03 2004
a(n) = sum_{k=0..3^(n-1)} A014578(k) = -(-1)^n*A014983(n) = A051068(3^(n-1)), for n > 0. - Philippe Deléham, Mar 31 2004
E.g.f.: exp(x)*sinh(2*x)/2. - Paul Barry, Oct 02 2004
a(2*n+1) = A054880(n) + 1. - M. F. Hasler, Mar 20 2008
2*a(n) + (-1)^n = A046717(n). - M. F. Hasler, Mar 20 2008
a(n) = ((1+sqrt(4))^n - (1-sqrt(4))^n)/4. - Al Hakanson (hawkuu(AT)gmail.com), Dec 31 2008
a(n) = abs(A014983(n)). - Zerinvary Lajos, May 28 2009
a(n) = round(3^n/4). - Mircea Merca, Dec 28 2010
a(n) = Sum_{k=1,3,5,...} binomial(n,k)*2^(k-1). - Geoffrey Critzer, Mar 02 2010
From Sergei N. Gladkovskii, Jul 19 2012: (Start)
G.f.: G(0)/4 where G(k)= 1 - 1/(9^k - 3*x*81^k/(3*x*9^k - 1/(1 + 1/(3*9^k - 27*x*81^k/(9*x*9^k + 1/G(k+1)))))); (continued fraction).
E.g.f.: G(0)/4 where G(k)= 1 - 1/(9^k - 3*x*81^k/(3*x*9^k - (2*k+1)/(1 + 1/(3*9^k - 27*x*81^k/(9*x*9^k + (2*k+2)/G(k+1)))))); (continued fraction).  (End)
G.f.: G(0)*x/(2*(1-x)), where G(k) = 1 + 1/(1 - x*(4*k-1)/(x*(4*k+3) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013
a(n+1) = Sum_{k = 0..n} A238801(n,k)*2^k. - Philippe Deléham, Mar 07 2014
a(n) = (-1)^(n-1)*Sum_{k=0..n-1} A135278(n-1,k)*(-4)^k = (-1)^(n-1)*Sum_{k=0..n-1} (-3)^k. Equals (-1)^(n-1)*Phi(n,-3), where Phi is the cyclotomic polynomial when n is an odd prime. (For n > 0.) - Tom Copeland, Apr 14 2014
a(n) = 2*A006342(n-1) - n mod 2 if n > 0, a(0)=0. - Yuchun Ji, Nov 30 2018
a(n) = 2*A033113(n-2) + n mod 2 if n > 0, a(0)=0. - Yuchun Ji, Aug 16 2019
a(2*k) = 2*A002452(k), a(2*k+1) = A066443(k). - Yuchun Ji, Aug 14 2019
a(n+1) = 2*Sum_{k=0..n} a(k) if n odd, and 1 + 2*Sum_{k=0..n} a(k) if n even. - Kengbo Lu, May 30 2020
a(n) = F(n) + Sum_{k=1..(n-1)} a(k)*L(n-k), for F(n) and L(n) the Fibonacci and Lucas numbers. - Kengbo Lu and Greg Dresden, Jun 05 2020
From Kengbo Lu, Jun 11 2020: (Start)
a(n) = A002605(n) + Sum_{k = 1..n-2} a(k)*A002605(n-k-1).
a(n) = A006130(n-1) + Sum_{k = 1..n-1} a(k)*A006130(n-k-1). (End)
a(2n) = Sum_{i>=0, j>=0} binomial(n-j-1,i)*binomial(n-i-1,j)* 2^(2n-2i-2j-1)* 3^(i+j). - Kengbo Lu, Jul 02 2020
a(n) = 3*a(n-1) - (-1)^n. - Dimitri Papadopoulos, Nov 28 2023
G.f.: x/((1 + x)*(1 - 3*x)) = Sum_{n >= 0} x^(n+1) * Product_{k = 1..n} (k + 3*x + 1)(1 + k*x) (a telescoping series). Cf. A007482. - Peter Bala, May 08 2024
From Peter Bala, Jun 29 2025: (Start)
For n >= 1, a(n+1) = 2^n * hypergeom([1/2 - (1/2)*n, -(1/2)*n], [-n], -3).
G.f. A(x) = x*exp(Sum_{n >= 1} a(2*n)/a(n)*x^n/n) =  x + 2*x^2 + 7*x^3 + 20*x^4 + ....
sqrt(A(x)/x) is the g.f. of A002426.
The following series telescope:
Sum_{n >= 1} (-3)^n/(a(n)*a(n+1)) = -1; Sum_{n >= 1} (-3)^n/(a(n)*a(n+1)*a(n+2)*a(n+3)) = -1/98.
In general, for k >= 0, Sum_{n >= 1} (-3)^n/(a(n)*a(n+1)*...*a(n+2*k+1)) = -1/((a(1)*a(2)*...*a(2*k+1))*a(2*k+1)).
Sum_{n >= 1} 3^n/(a(n)*a(n+1)*a(n+2)) = 1/4; Sum_{n >= 1} 3^n/(a(n)*a(n+1)*a(n+2)* a(n+3)*a(n+4)) = 1/5600.
In general, for k >= 1, Sum_{n >= 1} 3^n/(a(n)*a(n+1)*...*a(n+2*k)) = 1/((a(1)*a(2)*...*a(2*k))*a(2*k)). (End)

More terms from Emeric Deutsch, Apr 01 2004

Edited by Ralf Stephan, Aug 30 2004

A054878 Number of closed walks of length n along the edges of a tetrahedron based at a vertex.

Original entry on oeis.org

1, 0, 3, 6, 21, 60, 183, 546, 1641, 4920, 14763, 44286, 132861, 398580, 1195743, 3587226, 10761681, 32285040, 96855123, 290565366, 871696101, 2615088300, 7845264903, 23535794706, 70607384121, 211822152360, 635466457083

Offset: 0

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

Number of closed walks of length n at a vertex of C_4, the cyclic graph on 4 nodes. 3*A015518(n) + a(n) = 3^n. - Paul Barry, Feb 03 2004
Form the digraph with matrix A = [0,1,1,1; 1,0,1,1; 1,1,0,1; 1,0,1,1]; a(n) corresponds to the (1,1) term of A^n. - Paul Barry, Oct 02 2004
Absolute values of A084567 (compare generating functions).
For n > 1, 4*a(n)=A218034(n)= the trace of the n-th power of the adjacency matrix for a complete 4-graph, a 4 X 4 matrix with a null diagonal and all ones for off-diagonal elements. The diagonal elements for the n-th power are a(n) and the off-diagonal are a(n)+1 for an odd power and a(n)-1 for an even (cf. A001045). - Tom Copeland, Nov 06 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Ji Young Choi, A Generalization of Collatz Functions and Jacobsthal Numbers, J. Int. Seq., Vol. 21 (2018), Article 18.5.4.
M. Dukes and C. D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016.
R. J. Mathar, Counting Walks on Finite Graphs, (Nov 2020), Section 2.
Index entries for linear recurrences with constant coefficients, signature (2,3).

Row n=4 of A109502. A084567 (signed version).
{a(n)/3} for n>0 is A015518, non-closed walks.
Cf. A001045, A078008, A097073, A115341, A015518 (sequences where a(n)=3^n-a(n-1)). - Vladimir Joseph Stephan Orlovsky, Dec 11 2008

[(3^n+(-1)^n*3)/4: n in [0..35]]; // Vincenzo Librandi, Jun 30 2011

A054878:=n->(3^n + (-1)^n*3)/4: seq(A054878(n), n=0..50); # Wesley Ivan Hurt, Sep 16 2017

Table[(3^n + (-1)^n*3)/4, {n, 0, 26}] (* or *)
CoefficientList[Series[1/4*(3/(1 + x) + 1/(1 - 3 x)), {x, 0, 26}], x] (* Michael De Vlieger, Sep 15 2017 *)

a(n) = (3^n + 3*(-1)^n)/4; \\ Altug Alkan, Sep 17 2017

a(n) = (3^n + (-1)^n*3)/4.
G.f.: 1/4*(3/(1+x) + 1/(1-3*x)).
E.g.f.: (exp(3*x) + 3*exp(-x))/4. - Paul Barry, Apr 20 2003
a(n) = 3^n - a(n-1) with a(0)=0. - Labos Elemer, Apr 26 2003
G.f.: (1 - 3*x^2 - 2*x^3)/(1 - 6*x^2 - 8*x^3 - 3*x^4) = (1 - 3*x^2 - 2*x^3)/charpoly(adj(C_4)). - Paul Barry, Feb 03 2004
From Paul Barry, Oct 02 2004: (Start)
G.f.: (1-2*x)/(1 - 2*x - 3*x^2).
a(n) = 2*a(n-1) + 3*a(n-2). (End)
G.f.: 1 - x + x/Q(0), where Q(k) = 1 + 3*x^2 - (3*k+4)*x + x*(3*k+1 - 3*x)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 07 2013
a(n+m) = a(n)*a(m) + a(n+1)*a(m+1)/3. - Yuchun Ji, Sep 12 2017
a(n) = 3*a(n-1) + 3*(-1)^n. - Yuchun Ji, Sep 13 2017
From Peter Bala, May 28 2024: (Start)
a(n) = (-1)^n + Sum_{k = 1..n} (-1)^(n-k)*binomial(n, k)*4^(k-1).
G.f.: A(x) = 1/(1 - x^2) o 1/(1 - x^2), where o denotes the black diamond product of power series as defined by Dukes and White. Cf. A015575.
The black diamond product A(x) o A(x) is the g.f. for the number of closed walks of length n at a vertex along the edges of the 15-simplex. (End)

A213173 a(n) = 4^floor(n/2), Powers of 4 repeated.

Original entry on oeis.org

1, 1, 4, 4, 16, 16, 64, 64, 256, 256, 1024, 1024, 4096, 4096, 16384, 16384, 65536, 65536, 262144, 262144, 1048576, 1048576, 4194304, 4194304, 16777216, 16777216, 67108864, 67108864, 268435456, 268435456, 1073741824, 1073741824, 4294967296, 4294967296

Offset: 0

Philippe Deléham, Apr 14 2013

1, followed by A056450. - Joerg Arndt, Sep 17 2013
Binomial transform of A084567.
Unsigned version of A164111.
Also, number of walks of length n on the star S_4 = K_1,4 starting at a specific vertex of degree 1. - Sean A. Irvine, Jun 03 2025

G. C. Greubel, Table of n, a(n) for n = 0..1000
Sean A. Irvine, Walks on Graphs.
Wikipedia, Star (graph theory)
Index entries for linear recurrences with constant coefficients, signature (0,4).

Cf. A000302, A056450, A164111.

LinearRecurrence[{0, 4}, {1, 1}, 40] (* T. D. Noe, Apr 17 2013 *)
CoefficientList[Series[(1 + x)/(1 - 4*x^2), {x,0,50}], x] (* G. C. Greubel, Apr 30 2017 *)
With[{p4=4^Range[0,30]},Riffle[p4,p4]] (* Harvey P. Dale, Mar 17 2022 *)

a(n)=4^(n\2) \\ Charles R Greathouse IV, Oct 03 2016

O.g.f.: (1+x)/(1-4*x^2).
a(n) = A016116(n)^2.
a(n) = 4*a(n-2) with a(0) = a(1) = 1.
a(n) = a(n-1)*a(n-2)/a(n-3) = 4^A004526(n).
a(n) = sum(A152815(n,k)*3^k, 0<=k<=n). - Philippe Deléham, Apr 22 2013

A208532 Mirror image of triangle in A125185; unsigned version of A120058.

Original entry on oeis.org

1, 2, 1, 3, 4, 2, 4, 9, 10, 4, 5, 16, 28, 24, 8, 6, 25, 60, 80, 56, 16, 7, 36, 110, 200, 216, 128, 32, 8, 49, 182, 420, 616, 560, 288, 64, 9, 64, 280, 784, 1456, 1792, 1408, 640, 128, 10, 81, 408, 1344, 3024, 4704, 4992, 3456, 1408, 256

Offset: 0

Philippe Deléham, Feb 27 2012

Subtriangle of the triangle given by (1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Equals A007318*A134309*A097806 as infinite lower triangular matrix.
Row sums are powers of 3 (A000244).
Diagonal sums are powers of 2 (A000079).

			Triangle begins :
1
2, 1
3, 4, 2
4, 9, 10, 4
5, 16, 28, 24, 8
6, 25, 60, 80, 56, 16
7, 36, 110, 200, 216, 128, 32
8, 49, 182, 420, 616, 560, 288, 64
9, 64, 280, 784, 1456, 1792, 1408, 640, 128
10, 81, 408, 1344, 3024, 4704, 4992, 3456, 1408, 256
Triangle (1, 1, -1, 1, 0, 0, 0, ...) DELTA (0, 1, 1, 0, 0, 0, ...) begins :
1
1, 0
2, 1, 0
3, 4, 2, 0
4, 9, 10, 4, 0
5, 16, 28, 24, 8, 0
6, 25, 60, 80, 56, 16, 0

Cf. Columns: A000027, A000290, A006331, A112742.

Cf. Diagonals: A011782, 2*A045623,

T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) - 2*T(n-1,k-1), T(0,0) = T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n.
G.f.: (1-y*x)/((1-x)*(1-(1+2*y)*x)).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A083085(n), A084567(n), A000012(n), A000027(n+1), A000244(n), A083065(n), A083076(n) for x = -3, -2, -1, 0, 1, 2, 3 respectively.

A251634 Numerators of inverse Riordan triangle of Riordan triangle A029635. Riordan (1/(1-x), x/(1+2*x)). Triangle read by rows for 0 <= m <= n.

Original entry on oeis.org

1, 1, 1, 1, -1, 1, 1, 3, -3, 1, 1, -5, 9, -5, 1, 1, 11, -23, 19, -7, 1, 1, -21, 57, -61, 33, -9, 1, 1, 43, -135, 179, -127, 51, -11, 1, 1, -85, 313, -493, 433, -229, 73, -13, 1, 1, 171, -711, 1299, -1359, 891, -375, 99, -15, 1, 1, -341, 1593, -3309, 4017, -3141, 1641, -573, 129, -17, 1

Offset: 0

Wolfdieter Lang, Jan 09 2015

The denominators are given by 2*A130321(n,m).
The rational lower triangular matrix with entries R(n,m) = T(n,m)/(2*A130321(n,m)) = T(n,m)/2^(n-m+1) for n >= m >= 0 and 0 otherwise is the inverse of the Riordan matrix A029635.
R is the rational Riordan triangle (1/(2-x), x/(1+x)).
The numerator triangle T is the Riordan array (1/(1-x), x/(1+2*x)). From the o.g.f. of the column sequences of R and T(n,m) = 2^(n-m+1)*R(n,m).
Row sums of the rational triangle R are [1/2, seq(3/2^(n+1), for n >= 1)].
Row sums of the present triangle T give [repeat(1,2,)].
Alternating row sums of the rational triangle R give (-1)^n*A102900(n)/2^(n+1), n >= 0: 1/2, -1/4, 7/8, -25/16, 103/32, -409/64, 1639/128, -6553/256, 26215/512, ... .
Alternating row sums of the present triangle T give A084567.
The inverse of the T Riordan matrix is ((1-3*x)/(1-2*x), x/(1-2*x)) = A251636.
Equals A248810 when the first column (m = 0) of ones is removed. - Georg Fischer, Jul 26 2023

			The triangle T(n,m) begins:
  n\m  0    1    2     3     4     5    6    7    8   9 ...
  0:   1
  1:   1    1
  2:   1   -1    1
  3:   1    3   -3     1
  4:   1   -5    9    -5     1
  5:   1   11  -23    19    -7     1
  6:   1  -21   57   -61    33    -9    1
  7:   1   43 -135   179  -127    51  -11    1
  8:   1  -85  313  -493   433  -229   73  -13    1
  9:   1  171 -711  1299 -1359   891 -375   99  -15   1
  ...
The rational Riordan triangle R(n,m) begins:
  n\m  0      1      2      3     4    5  ...
  0:  1/2
  1:  1/4    1/2
  2:  1/8   -1/4    1/2
  3:  1/16   3/8   -3/4    1/2
  4:  1/32  -5/16   9/8   -5/4   1/2
  5:  1/64  11/3  -23/1   19/8  -7/4  1/2
  ...
For more rows see the link.

Wolfdieter Lang, Eleven rows of the triangle, and rational triangle.

Cf. A029635, A084567, A102900, A130321, A251636, A248810, A331333.

A251634 := proc(n, k) local S; S := proc(n, k) option remember; `if`(k = 0, 1,
`if`(k > n, 0, S(n-1, k-1)/k - 2*S(n-1, k))) end: k!*S(n, k) end:
seq(seq(A251634(n, k), k=0..n)), n=0..9); # Peter Luschny, Jan 19 2020

O.g.f. of the row polynomials P(n,x) = Sum_{m=0..n} R(n,m)*x^m of the rational triangle R: G(z,x) = Sum_{n>=0} P(n,x)*z^n = (1+z)/((2-z)*(1+(1-x)*z)).
O.g.f. column m of the rational triangle R: (1/(2-x))*(x/(1+x))^m, m >= 0 (Riordan property of R).
O.g.f. column m of the numerator triangle T: (1/(1-x))*(x/(1+2*x))^m, m >= 0. (Riordan property of T).
T(n, k) = k!*S(n, k) where S(n, k) is recursively defined by:
if k = 0 then 1 else if k > n then 0 else S(n-1, k-1)/k - 2*S(n-1, k). - Peter Luschny, Jan 19 2020

cp's OEIS Frontend

A015518 a(n) = 2*a(n-1) + 3*a(n-2), with a(0)=0, a(1)=1.

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A054878 Number of closed walks of length n along the edges of a tetrahedron based at a vertex.

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A213173 a(n) = 4^floor(n/2), Powers of 4 repeated.

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A208532 Mirror image of triangle in A125185; unsigned version of A120058.

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A251634 Numerators of inverse Riordan triangle of Riordan triangle A029635. Riordan (1/(1-x), x/(1+2*x)). Triangle read by rows for 0 <= m <= n.

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A015518 a(n) = 2a(n-1) + 3a(n-2), with a(0)=0, a(1)=1.