A198633 -id:A198633 - cp's OEIS frontend

A198632 Triangle version of the array of the number of closed paths of even length on the graph P_n (n vertices, n-1 edges).

1, 0, 2, 0, 2, 3, 0, 2, 4, 4, 0, 2, 8, 6, 5, 0, 2, 16, 14, 8, 6, 0, 2, 32, 36, 20, 10, 7, 0, 2, 64, 94, 56, 26, 12, 8, 0, 2, 128, 246, 164, 76, 32, 14, 9, 0, 2, 256, 644, 488, 234, 96, 38, 16, 10, 0, 2, 512, 1686, 1460, 740, 304, 116, 44, 18, 11, 0, 2, 1024, 4414, 4376, 2372, 992, 374, 136, 50, 20, 12, 0, 2, 2048, 11556, 13124, 7654, 3296, 1244, 444, 156, 56, 22, 13

Offset: 0

Wolfdieter Lang, Nov 02 2011

This array is an example of counting walks on a graph whose adjacency matrix is given by a special symmetric tridiagonal matrix with nonnegative integer entries, appropriate for orthogonal polynomials. These are quadratic Jacobi matrices J_n with nonnegative entries. The corresponding graphs could be called Jacobi graphs. Here Chebyshev S-polynomials (coefficients A049310) are considered, which belong to the Jacobi class of the classical orthogonal polynomial systems. The corresponding graph has adjacency matrix [[0,1,0,...],[1,0,1,...],[0,1,0,1,...]...[0,...0,1,0]] (n rows and n columns), with characteristic polynomial S(n,x) (see also a comment by Michael Somos on A049310).
w(n,l;p_k->p_m) = ((J_n)^l)(k,m) is the number of walks of length l from vertex p_k to vertex p_m on such a Jacobi graph. w(n,0; p_k->p_m) = delta(k,m), with the Kronecker symbol delta. The total number of closed walks of length l is w(n,l):=Sum_{i=1..n} w(n,l; p_i->p_i) = trace(J_n^l), which is the l-th power sum of the eigenvalues of J_n, i.e., the zeros of the characteristic polynomial for J_n. There are theorems for the o.g.f. of the normalized power sums of these zeros. See, e.g., the given W. Lang reference, p. 244. This results for the S-polynomial in the o.g.f. G(n,x) = Sum_{l=0..infinity} w(n,l)*x^l = y*(d/dy)S(n,y)/S(y) with y=1/x. This can be rewritten in the form given in the formula section (this results from eq. (3.8b) of the W. Lang reference, and in eq. (3.8d) it should be coth, not tanh).
From Wolfdieter Lang, Oct 10 2012: (Start)
For an accompanying paper on path counting on Jacobi graphs see the W. Lang link under A201198.
The total number of round trips of length L on the graph P_n, taken per site, becomes for n -> infinity A126869(L). See the just mentioned link, p. 8. This limit is derived from the limit of G(n,x)/n with G(n,x) given in the formula section.
Thanks go to Clyde P. Kruskal for asking a question which led to this comment.
(End)

			The array w(n,2*k) is
n\k  0  1   2   3   4    5    6     7     8      9 ...
1:   1  0   0   0   0    0    0     0     0      0
2:   2  2   2   2   2    2    2     2     2      2
3:   3  4   8  16  32   64  128   256   512   1024
4:   4  6  14  36  94  246  644  1686  4414  11556
5:   5  8  20  56 164  488 1460  4376 13124  39368
6:   6 10  26  76 234  740 2372  7654 24778  80338
7:   7 12  32  96 304  992 3296 11072 37440 127104
8:   8 14  38 116 374 1244 4220 14504 50294 175454
9:   9 16  44 136 444 1496 5144 17936 63164 224056
...
The triangle is
k\n 1  2    3    4    5    6   7    8   9 10 11 12 ...
0:  1
1:  0  2
2:  0  2    3
3:  0  2    4    4
4:  0  2    8    6    5
5:  0  2   16   14    8    6
6:  0  2   32   36   20   10   7
7:  0  2   64   94   56   26  12    8
8:  0  2  128  246  164   76  32   14   9
9:  0  2  256  644  488  234  96   38  16 10
10: 0  2  512 1686 1460  740 304  116  44 18 11
11: 0  2 1024 4414 4376 2372 992  374 136 50 20 12
...
n=3, l=2*k = 4: graph P_3 as 1-2-3, with eight walks of length 4, namely 12121, 12321, 21212, 23232, 21232, 23212, 32323 and 32123.

S. Barbero, Dickson Polynomials, Chebyshev Polynomials, and Some Conjectures of Jeffery, Journal of Integer Sequences, 17 (2014), #14.3.8.
S. Barbero, U. Cerruti, N. Murru, Identities Involving Zeros of Ramanujan and Shanks Cubic Polynomials, J. Integer Seq., Vol. 16 (2013), Article 13.8.1.
Carlos M. da Fonseca, M. Lawrence Glasser, Victor Kowalenko, Basic trigonometric power sums with applications, arXiv:1601.07839 [math.NT], 2016. See Theorem 5.1.
W. Lang, On sums of powers of zeros of polynomials, J. Comp. Appl. Math., 89 (1998) 237-256.

Column sequences: A000007, 2*A000012, A198633, 2*A005248, A198635, ...

a(k,n)=w(n,2*(k-n+2)), the total number of closed walks (paths) of length 2*(k-n+2) on the graph P_n, which looks like o-o-o...-o, with n vertices (nodes) and n-1 edges (lines), k+1>=n>=1.
O.g.f. G(n,x) for w(n,l), which vanishes for odd l, is
((n+1)*coth((n+1)*log((2*x)/(1-sqrt(1-(2*x)^2)))) - 1/sqrt(1-(2*x)^2))/sqrt(1-(2*x)^2). See the comment above for a version with Chebyshev S-polynomials.
Conjecture: For the array w(n,2*k) in the example below, w(2*q,2*k)/2 = A185095(q,k), q >= 1, k >= 0. - L. Edson Jeffery, Nov 23 2013

A198636 One half of total number of round trips, each of length 2n, on the graph P_6 (o-o-o-o-o-o).

Original entry on oeis.org

3, 5, 13, 38, 117, 370, 1186, 3827, 12389, 40169, 130338, 423065, 1373466, 4459278, 14478659, 47011093, 152642789, 495626046, 1609284589, 5225309458, 16966465802, 55089756851, 178875298901, 580804419201, 1885860059450, 6123349080945

Offset: 0

Wolfdieter Lang, Nov 03 2011

See the array and triangle A198632 for the general graph P_N case (there N is n and the length is l=2*k).

			With the graph P_6 as 1-2-3-4-5-6:
n=0: a(0)=3 because w(6,0)=6, the number of vertices.
n=2: a(2)=5 because the 10 round trips of length 2 are 121, 212, 232, 323, 343, 434, 454, 545, 565 and 656.

M. F. Hasler, Table of n, a(n) for n = 0..499
S. Barbero, Dickson Polynomials, Chebyshev Polynomials, and Some Conjectures of Jeffery, Journal of Integer Sequences, 17 (2014), #14.3.8.
S. Barbero, U. Cerruti, N. Murru, Identities Involving Zeros of Ramanujan and Shanks Cubic Polynomials, J. Integer Seq., Vol. 16 (2013), Article 13.8.1, pp. 10-12.
Index entries for linear recurrences with constant coefficients, signature (5,-6,1).

Cf. A198632, A198633, A005248, A198635.

Table[7 (Binomial[2 n - 1, n - 1] + Sum[Binomial[2 n, n - 7 k], {k, Floor[n/7]}]) - 2^(2 n - 1) - (7/2) Boole[n == 0], {n, 0, 25}] (* Michael De Vlieger, Jul 17 2017 *)

vec_A198636(Nmax)=Vec((3-10*x+6*x^2)/(1-5*x+6*x^2-x^3)+O(x^Nmax)) \\ Indices will start at 1 in this vector. - M. F. Hasler, Nov 03 2013

{a(n) = if( n<0, n=-n; polcoeff( (3 - 12*x + 5*x^2) / (1 - 6*x + 5*x^2 - x^3) + x * O(x^n), n), polcoeff( (3 - 10*x + 6*x^2) / (1 - 5*x + 6*x^2 -x^3) + x * O(x^n), n))}; /* Michael Somos, Jul 17 2017 */

a(n) = w(6,2*n)/2, n>=0, with w(6,l) the total number of closed walks on the graph P_6 (the simple path with 6 points (vertices) and 5 lines (or edges)).
O.g.f. for w(6,l) (with zeros for odd l): y*(d/dy)S(6,y)/S(6,y) with y=1/x and Chebyshev S-polynomials (coefficients A049310). See also A198632 for a rewritten form.
O.g.f.: (3-10*x+6*x^2)/(1-5*x+6*x^2-x^3). - Colin Barker, Jan 02 2012
Conjecture: a(n) = 2^(2*n)*(sum_{k=1,2,3} (cos(k*Pi/7))^(2*n)). - L. Edson Jeffery, Jan 21 2012 (in fact this conjecture was recently proved in [Barbero, et al.])
a(n) = 7*(binomial(2n-1,n-1) + sum_{k = 1..floor(n/7)} binomial(2n,n-7k)) - 2^(2n-1). - M. Lawrence Glasser, Feb 20 2013
Let r,s,t be the roots of x^3 + x^2 - 2x - 1; then apparently a(n) = r^(2n) + s^(2n) + t^(2n). - James R. Buddenhagen, Nov 03 2013 [This is equivalent to the conjecture by L. Edson Jeffery.]
a(n) = 5*a(n-1) - 6*a(n-2) + a(n-3). - M. F. Hasler, Nov 05 2013
G.f.: F(x) = (sum_{r=0..2} ((3-r)*(-1)^r*binomial(6-r,r))*x^r)/(sum_{s=0..3} ((-1)^s*binomial(6-s,s))*x^s). - L. Edson Jeffery, Nov 23 2013

A198635 Total number of round trips, each of length 2*n on the graph P_5 (o-o-o-o-o).

Original entry on oeis.org

5, 8, 20, 56, 164, 488, 1460, 4376, 13124, 39368, 118100, 354296, 1062884, 3188648, 9565940, 28697816, 86093444, 258280328, 774840980, 2324522936, 6973568804, 20920706408, 62762119220, 188286357656, 564859072964, 1694577218888, 5083731656660, 15251194969976

Offset: 0

Wolfdieter Lang, Nov 02 2011

See the array and triangle A198632 for the general case for the graph P_N (there N is n and the length is l = 2*k).

			With the graph P_5 as 1-2-3-4-5:
n=0: 5, from the length 0 walks starting at 1,2,3,4 and 5.
n=1: 8, from the walks of length 2, namely 121, 212, 232, 323, 343, 434, 454 and 545.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A005248, A198632, A198633.

Essentially the same as A115099.

a[0] = 5; a[n_] := 2*3^n + 2; Array[a, 28, 0] (* Jean-François Alcover, Nov 01 2017, after Andrew Howroyd *)
CoefficientList[Series[(5 - 12 x + 3 x^2)/(1 - 4 x + 3 x^2), {x, 0, 27}], x] (* Michael De Vlieger, Dec 18 2017 *)
LinearRecurrence[{4,-3},{5,8,20},30] (* Harvey P. Dale, Nov 27 2024 *)

a(n) = w(5,2*n), n >= 0, with w(5,l) the total number of closed walks on the graph P_5 (the simple path with 5 points (vertices) and 4 lines (or edges)).
O.g.f. for w(5,l) (with zeros for odd l): y*(d/dy)S(5,y)/S(5,y) with y = 1/x and Chebyshev S-polynomials (coefficients A049310). See also A198632 for a rewritten form.
G.f.: (5-12*x+3*x^2)/(1-4*x+3*x^2). - Colin Barker, Jan 02 2012
a(n) = 3*a(n-1) - 4, n > 1. - Vincenzo Librandi, Jan 02 2012
a(n) = 2*3^n + 2 for n > 0. - Andrew Howroyd, Mar 18 2017
a(n) = 2*A034472(n) for n > 0. - Andrew Howroyd, Mar 18 2017

A199571 Table version of the array of number of round trips of length L from any of the N vertices of the cycle graph C_N.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 4, 0, 1, 0, 0, 2, 0, 1, 0, 16, 2, 2, 0, 1, 0, 0, 6, 0, 2, 0, 1, 0, 64, 10, 8, 0, 2, 0, 1, 0, 0, 22, 0, 6, 0, 2, 0, 1, 0, 256, 42, 32, 2, 6, 0, 2, 0, 1, 0, 0, 86, 0, 20, 0, 6, 0, 2, 0, 1, 0, 1024, 170, 128, 14, 22, 0, 6, 0, 2, 0, 1, 0, 0

Offset: 0

Wolfdieter Lang, Nov 08 2011

Let w(N,L) be the number of return paths (round trip walks) of length L >= 0 from any vertex of the cycle graph C_N, N >= 1. (Due to cyclic symmetry, this array w(N,L) is independent of the start vertex.) w(N,L) = trace(AC(N)^L)/N = Sum_{k=0..N-1} x^{(N)}_k, with the N X N adjacency matrix AC(N) of the cycle graph C_N, and x^{(N)}_k are the zeros of the characteristic polynomial C(N,x) of AC(N). See A198637 for the coefficient triangle for C(N,x). C(N,x) = 2*(T(N,x/2)-1) for N >= 2. These zeros are x^{(N)}_k = 2*cos(2*Pi*k/N), N >= 2 (from T(N,x/2)=1). For N=1 one has C(1,x)=x with x^{(1)}_0 = 0. This sum formula for w(n,L) has been given in a comment to A054877 (N=5 case) by H. Kociemba. For N=1 one uses 0^0 := 1 to obtain w(1,L) = delta(L,0) (Kronecker's delta-symbol).
The o.g.f. G(N,x) := Sum_{L>=0} w(N,L)*x^L is, by a general result on moments of zeros of polynomials (see the W. Lang reference, theorem 5, p. 244),
  y*(d/dx)C(N,x)/(N*C(N,x)), with y=1/x. This becomes for N >= 2: G(N,x) = y*S(N-1,y)/(2*T(N,y/2)-1) with y=1/x. For N=1 one has G(1,x)=1 (not 1/(1-2*x)). In the formula section this N >= 2 result is given explicitly, using the Binet-de Moivre form of the S- and T-polynomials.

			The triangle a(K,N) = w(N,K-N+1) begins
K\N  1    2     3    4    5    6    7   8   9  10 ...
0:   1
1:   0    1
2:   0    0     1
3:   0    4     0    1
4:   0    0     2    0    1
5:   0   16     2    2    0    1
6:   0    0     6    0    2    0    1
7:   0   64    10    8    0    2    0   1
8:   0    0    22    0    6    0    2   0   1
9:   0  256    42   32    2    6    0   2   0   1
...
The array w(N,L) begins
N\L   0   1   2   3   4   5   6   7    8    9    10 ...
1:    1   0   0   0   0   0   0   0    0    0     0
2:    1   0   4   0  16   0  64   0  256    0  1024
3:    1   0   2   2   6  10  22  42   86  170   342
4:    1   0   2   0   8   0  32   0  128    0   512
5:    1   0   2   0   6   2  20  14   70   72   254
6:    1   0   2   0   6   0  22   0   86    0   342
7:    1   0   2   0   6   0  20   2   70   18   252
8:    1   0   2   0   6   0  20   0   72    0   272
9:    1   0   2   0   6   0  20   0   70    2   252
10:   1   0   2   0   6   0  20   0   70    0   254
...
w(1,0)=1, one vertex considered.
For N >= 2 the vertices (nodes) of C_N are numbered consecutively in the positive sense by 1,2,...,N. W.l.o.g. one can take the vertex number 1 as start of the return trip.
w(3,4)=6 from the six return paths 12121, 13131, 12131, 13121, 12321 and 13231.
w(5,5)=2 from the two return paths 123451 and 154321.

Wolfdieter Lang, On sums of powers of zeros of polynomials, J. Comp. Appl. Math. 89 (1998) 237-256.

Cf. A198633 (walks on the P_N graph).

The N=1,...,10 sequences are A000007, A199572, A078008, A199573, A054877, A047849, A094659, A063376, A094233, A095929.

a(K,L) = w(N,K-N+1), K >= 0, n=1,...,K+1, with w(N,L) defined as return walk numbers of length L of the cycle graph C_N in the comment section above.
w(N,L) = Sum_{k=0..N-1} (2*cos(2*Pi*k)/N)^L, N >= 2. For N=1 one has w(1,0)=1 and w(1,L)=0 if L >= 1.
O.g.f. G(N,x) for w(N,L): for N >= 2:
  y*S(N-1,y)/(2*(T(N,y/2)-1)) with y=1/x, and for N=1 one has G(1,x)=1. This can, for N >= 2, be written as
G(N,x) = sinh(N*log(2*x/(1-sqrt(1-(2*x)^2))))/(sqrt(1-(2*x)^2)*(cosh(N*log(2*x/(1-sqrt(1-(2*x)^2))))-1)).

A256096 Expansion of (4+3x)/(1+3x).

Original entry on oeis.org

4, -9, 27, -81, 243, -729, 2187, -6561, 19683, -59049, 177147, -531441, 1594323, -4782969, 14348907, -43046721, 129140163, -387420489, 1162261467, -3486784401, 10460353203, -31381059609, 94143178827, -282429536481, 847288609443, -2541865828329, 7625597484987

Offset: 0

Wolfdieter Lang, Mar 23 2015

This is the Z-sequence of the Riordan triangle ((1+2*x)/(1-x)^2, x/(1-x)). See A135857.

The expansion can be seen as a special case of the family of generating functions 1+1/(x+1/k) for k>=1 which relates this sequence to A054977 and A198633 (neglecting questions of sign). - Peter Luschny, Mar 24 2015

Winston de Greef, Table of n, a(n) for n = 0..2080
Index entries for linear recurrences with constant coefficients, signature (-3).

Cf. A135857, A054977, A198633.

Join[{4},NestList[-3#&,-9,30]] (* Harvey P. Dale, Jan 27 2023 *)

a(n)=if(!n, 4, (-1)^n*3^(n+1)) \\ Winston de Greef, Mar 19 2023

O.g.f.: (4+3*x)/(1+3*x).
a(0) = 4; for n >= 1, a(n) = (-1)^n*3^(n+1).
a(0) = 4, a(1) = -9; for n >= 2, a(n) = -3*a(n-1).
E.g.f.: 1 + 3*exp(-3*x). - Alejandro J. Becerra Jr., Jan 28 2021

A258935 Independence number of Keller graphs.

Original entry on oeis.org

4, 5, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592

Offset: 1

Stan Wagon, Nov 06 2015

			For G(2), a maximum independent set is {03,10,12,13,23}.

W. Jarnicki, W. Myrvold, P. Saltzman, S. Wagon, Properties, proved and conjectured, of Keller, queen, and Mycielski graphs, Ars Mathematica Contemporanea 13:2 (2017) 427-460.

Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Eric Weisstein's World of Mathematics, Independence Number
Eric Weisstein's World of Mathematics, Keller Graph

Cf. A006946, A135831, A135907.

Essentially the same as A143858, A240951, A198633, A171497, A151821, A146541 and A077552.

Mathematica
```
Join[{4, 5}, 2^Range[3, 10]]
```

a(n)=if(n>2,2^n,n+3) \\ Charles R Greathouse IV, Nov 07 2015

a(n) = 2^n except a(1) = 4 and a(2) = 5.

G.f.: x*(x*(3+2*x)-4)/(2*x-1), e.g.f.: exp(2*x)+x^2/2+2*x-1. - Benedict W. J. Irwin, Jul 15 2016

A306548 Triangle T(n,k) read by rows, where the k-th column is the shifted self-convolution of the power function n^k, n >= 0, 0 <= k <= n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 2, 1, 0, 0, 3, 4, 1, 0, 0, 4, 10, 8, 1, 0, 0, 5, 20, 34, 16, 1, 0, 0, 6, 35, 104, 118, 32, 1, 0, 0, 7, 56, 259, 560, 418, 64, 1, 0, 0, 8, 84, 560, 2003, 3104, 1510, 128, 1, 0, 0, 9, 120, 1092, 5888, 16003, 17600, 5554, 256, 1, 0, 0, 10, 165, 1968, 14988, 64064, 130835, 101504, 20758, 512, 1, 0, 0

Offset: 0

Kolosov Petro, Feb 23 2019

For n > 0 an odd-power identity n^(2m+1)+1, m >= 0 can be found using the current sequence. The sum of the n-th diagonal of T(n,k) over 0 <= k <= m multiplied by A(m,k) gives n^(2m+1)-1, where A(m,k) = A302971(m,k)/A304042(m,k). For example, consider the case n=4, m=2: the n-th diagonal of T(n, 0 <= k <= m) is {5, 10, 34}, and the m-th row of triangle A(m, 0 <= k <= m) is {1, 0, 30}, thus (3+1)^5 + 1 = 5*1 + 10*0 + 34*30 = 1025.

			==================================================================
k=    0     1     2     3      4      5     6    7    8    9    10
==================================================================
n=0:  2;
n=1:  2,    0;
n=2:  3,    0,    0;
n=3:  4,    1,    0,    0;
n=4:  5,    4,    1,    0,     0;
n=5:  6,   10,    8,    1,     0,     0;
n=6:  7,   20,   34,   16,     1,     0,    0;
n=7:  8,   35,  104,  118,    32,     1,    0,   0;
n=8:  9,   56,  259,  560,   418,    64,    1,   0,   0;
n=9:  10,  84,  560, 2003,  3104,  1510,  128,   1,   0,   0;
n=10: 11, 120, 1092, 5888, 16003, 17600, 5554, 256,   1,   0;   0;
...

D. V. Widder et al., The Convolution Transform, Bull. Amer. Math. Soc. 60 (1954), 444-456.
Wikipedia, Convolution.
Wikipedia, Convolution power.

Nonzero terms of columns k=0..5 give: A000027, A000292, A033455, A145216, A145217, A145218.
Partial sums of columns k=1..2 give: A000332, A259181.
Cf. A302971, A304042, A198633, A000079.

f[m_, s_] := Piecewise[{{s^m, s >= 0}, {0, True}}];
F[n_, m_] := Sum[f[m, n - k]*f[m, k], {k, -Infinity, +Infinity}];
T[n_, k_] := F[n - k, k];
Column[Table[T[n, k], {n, 0, 12}, {k, 0, n}], Left]

f(m, s) = s^m, if s >= 0;
f(m, s) = 0, otherwise.
F(n,m) = Sum_{k} f(m, n-k) * f(m, k), -oo < k < +oo;
T(n,k) = F(n-k, k).

Edited by Kolosov Petro, Mar 13 2019

A228305 a(1) = 3; for n >= 1, a(2n) = 2^(n+1), a(2n+1) = 5*2^(n-1).

Original entry on oeis.org

3, 4, 5, 8, 10, 16, 20, 32, 40, 64, 80, 128, 160, 256, 320, 512, 640, 1024, 1280, 2048, 2560, 4096, 5120, 8192, 10240, 16384, 20480, 32768, 40960, 65536, 81920, 131072, 163840, 262144, 327680, 524288, 655360, 1048576, 1310720, 2097152, 2621440, 4194304, 5242880

Offset: 1

Arkadiusz Wesolowski, Aug 20 2013

Union of A020714 and A198633.
Essentially the same as A094958.
For every n, a(1)^3 + a(2)^3 + a(3)^3 + ... + a(2*n-1)^3 is a cube.

			a(9) = 40 because it is equal to 5*2^(4-1).

Wikipedia, Cube (algebra)
Index entries for linear recurrences with constant coefficients, signature (0,2).

Cf. A020714, A094958, A121451, A164682, A198633.

[n le 3 select n+2 else 2*Self(n-2) : n in [1..43]];

CoefficientList[Series[(3 + 4 x - x^2)/(1 - 2 x^2), {x, 0, 50}], x] (* Bruno Berselli, Aug 20 2013 *)

r=43; print1(3); print1(", "); for(n=2, r, if(bitand(n, 1), print1(5*2^((n-3)/2)), print1(2^(n/2+1))); print1(", "));

a(n) = ceiling((9 - (- 1)^n)*2^(floor(n/2) - 2)).
a(n) = n + 2 for n <= 3; a(n) = 2*a(n-2) for n > 3.
From Bruno Berselli, Aug 20 2013: (Start)
G.f.: x*(3+4*x-x^2)/(1-2*x^2).
a(n) = (16-(8-5*r)*(1-(-1)^n))*r^(n-6) for n>1, r=sqrt(2). (End)
E.g.f.: (8*cosh(sqrt(2)*x) + 5*sqrt(2)*sinh(sqrt(2)*x) + 2*x - 8)/4. - Stefano Spezia, Apr 09 2025

cp's OEIS Frontend

A198632 Triangle version of the array of the number of closed paths of even length on the graph P_n (n vertices, n-1 edges).

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A198636 One half of total number of round trips, each of length 2n, on the graph P_6 (o-o-o-o-o-o).

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A198635 Total number of round trips, each of length 2*n on the graph P_5 (o-o-o-o-o).

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A199571 Table version of the array of number of round trips of length L from any of the N vertices of the cycle graph C_N.

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A256096 Expansion of (4+3*x)/(1+3*x).

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A258935 Independence number of Keller graphs.

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A306548 Triangle T(n,k) read by rows, where the k-th column is the shifted self-convolution of the power function n^k, n >= 0, 0 <= k <= n.

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A228305 a(1) = 3; for n >= 1, a(2*n) = 2^(n+1), a(2*n+1) = 5*2^(n-1).

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A256096 Expansion of (4+3x)/(1+3x).

A228305 a(1) = 3; for n >= 1, a(2n) = 2^(n+1), a(2n+1) = 5*2^(n-1).