Miquel A. Fiol - cp's OEIS frontend

A376313 Independence number of the 2-supertoken graph FF_2(C_n) of the cycle C_n on n vertices.

2, 3, 6, 7, 12, 14, 20, 22, 30, 33, 42, 45, 56, 60, 72, 76, 90, 95, 110, 115, 132, 138, 156, 162, 182, 189, 210, 217, 240, 248, 272, 280, 306, 315, 342, 351, 380, 390, 420, 430, 462, 473, 506, 517, 552, 564, 600, 612, 650, 663, 702, 715, 756, 770, 812, 826, 870, 885, 930, 945, 992, 1008

Offset: 2

Miquel A. Fiol, Sep 26 2024

Given a graph G on n vertices and an integer k>=1, the k-supertoken (or reduced k-th power) FF_k(G) of G has vertices representing configurations of k indistinguishable tokens in the (not necessarily different) vertices of G, with two configurations being adjacent if one can be obtained from the other by moving one token along an edge of G.

R. H. Hammack and G. D. Smith, Cycle bases of reduced powers of graphs, Ars Math. Contemp. 12 (2017) 183-203.

a(n) = k*(n+2) if n=4*k or n=4*k+1, and a(n)=(k+1)*n if n=4*k+2 or n=4*k+3.

A375747 Table read by rows: T(n,k) for n >= 3 and k=1,2,...,2*n-1 are the distinct eigenvalues of the twisted odd graph O^(sigma)_n.

Original entry on oeis.org

3, 2, 1, -1, -2, 4, 3, 2, 1, -1, -2, -3, 5, 4, 3, 2, 1, -1, -2, -3, -4, 6, 5, 4, 3, 2, 1, -1, -2, -3, -4, -5, 7, 6, 5, 4, 3, 2, 1, -1, -2, -3, -4, -5, -6, 8, 7, 6, 5, 4, 3, 2, 1, -1, -2, -3, -4, -5, -6, -7, 9, 8, 7, 6, 5, 4, 3, 2, 1, -1, -2, -3, -4, -5, -6, -7, -8

Offset: 3

Miquel A. Fiol, Aug 26 2024

			The table begins:
  3  2  1 -1 -2
  4  3  2  1 -1 -2 -3
  5  4  3  2  1 -1 -2 -3 -4
  6  5  4  3  2  1 -1 -2 -3 -4 -5
  7  6  5  4  3  2  1 -1 -2 -3 -4 -5 -6
  ...

Miquel A. Fiol, E. Garriga, and J. L. A. Yebra, On twisted odd graphs, Combin. Probab. Comput. 9 (2000), 227-240.

Cf. A001700.

T[n_,k_]:=If[k==1, n, If[1Stefano Spezia, Aug 27 2024 *)

T(n,1) = n; T(n,k) = n+1-k for k=2,...,n; T(n,k) = n-k for k=n+1,...,2n-1.

A375309 Number of walks of length n along the edges of a dodecahedron graph between two vertices at distance two.

Original entry on oeis.org

0, 0, 1, 1, 7, 11, 51, 105, 399, 967, 3299, 8789, 28271, 79443, 247507, 716353, 2193583, 6452639, 19575075, 58095597, 175350735, 522947755, 1574075603, 4706879321, 14146450127, 42363311991, 127217598691, 381275400325, 1144458922159

Offset: 0

Miquel A. Fiol, Aug 11 2024

Index entries for linear recurrences with constant coefficients, signature (2,10,-16,-25,30).

Cf. A054883.

LinearRecurrence[{2, 10, -16, -25, 30}, {0, 0, 1, 1, 7, 11}, 30] (* Amiram Eldar, Aug 13 2024 *)

For n>=6, a(n) = 2*a(n-1) + 10*a(n-2) - 16*a(n-3) - 25*a(n-4) + 30*a(n-5).
From Stefano Spezia, Aug 13 2024: (Start)
G.f.: x^2*(1 - x - 5*x^2 + 3*x^3)/((1 - x)*(1 + 2*x)*(1 - 3*x)*(1 - 5*x^2)).
a(n) = (3*5^(n/2)*(1 + (-1)^n) + 3^(1+n) + (-1)^n*2^(1+n) - 5)/60 for n > 0. (End)

A374795 The minimum diameter of a chordal ring mixed graph with 2*n vertices.

Original entry on oeis.org

1, 3, 3, 3, 4, 5, 5, 6, 5, 7, 7, 7, 7, 7, 9, 7, 8, 9, 9, 9, 9, 10, 9, 11, 9, 11, 11, 11, 11, 11, 11, 11, 11, 11, 13, 11, 12, 13, 13, 13, 13, 13, 13, 14, 13, 15, 13, 15, 13, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 17, 15, 16

Offset: 2

Miquel A. Fiol, Jul 20 2024

nonn

Let N >= 2 and c(< N ) be, respectively, even and odd numbers. The chordal ring mixed graph CRM(N, c) is a mixed graph with vertex set V = Z_N (all arithmetic is modulo N ), with arcs i -> i + 1 (forming a directed cycle) and edges i ∼ i + c if i is even (and, hence, i ∼ i - c if i is odd, forming the ‘chords’).

Cf. A373503.

A374622 Maximum number of vertices of a chordal ring mixed graph CRM(N,c) with diameter n.

Original entry on oeis.org

8, 10, 18, 16, 32, 34, 50, 44, 72, 74, 98, 88, 128, 130, 162, 148, 200, 202, 242, 224, 288, 290, 338, 316, 392, 394, 450, 424, 512, 514, 578, 548, 648, 650, 722, 688, 800, 802, 882, 844, 968, 970, 1058, 1016, 1152, 1154

Offset: 3

Miquel A. Fiol, Jul 14 2024

nonn

			For n = 9, the maximum number of vertices a(9) = 50 is attained by the chordal ring mixed graph CRM(50,9).

C. Dalfó, G. Erskine, G. Exoo, M. A. Fiol, and J. Tuite, On bipartite (1, 1, k)-mixed graphs, (2024).

Cf. A371396.

If n is odd, a(n) = (n+1)^2/2.
Conjecture: If n is even, n=0 mod 4, a(n) = n^2/2+2;
            If n (> 2) is even, n=2 mod 4, a(n) = n*(n/2 - 1) + 4.
Conjectured g.f.: 2*(1 + x + 2*x^2 + x^3 + 2*x^4 - 3*x^5 + 4*x^6 - x^7 + x^8)/((1 - x)^3*(1 + x + x^2 + x^3)^2). - Stefano Spezia, Jul 14 2024

A373503 The minimum diameter of a chordal ring graph with 2*n vertices.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 5, 6, 6, 6, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 7, 8, 8, 8, 8, 9, 9, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 9, 9, 10, 10, 10, 9, 10, 11, 11, 10, 10, 11, 11, 11, 10, 11, 11, 11, 11, 11, 11, 12, 11, 11, 11, 11, 11, 12, 12, 11, 11, 12, 12

Offset: 1

Miquel A. Fiol, Jul 10 2024

nonn

P. Morillo, F. Comellas, and M. A. Fiol, The optimization of Chordal Ring Networks, Commun. Technol., Eds. Q. Yasheng and W. Xiuying. World Scientific, (1987), 295-299.

Cf. A369859, A371396.

A373554 Number of ternary length-2 squarefree words of length n not containing the subwords 021 or 10.

Original entry on oeis.org

1, 3, 5, 7, 11, 16, 24, 36, 53, 80, 118, 177, 263, 392, 585, 870, 1299, 1933, 2883, 4295, 6400, 9540, 14212, 21185, 31564, 47042, 70101, 104463, 155680, 231985, 345722, 515187, 767749, 1144111, 1704963, 2540784, 3786288, 5642420, 8408397

Offset: 0

Miquel A. Fiol, Jun 09 2024

			For n=3 the a(3)=7 solutions are  012, 020, 120, 121, 201, 202, 212.

Index entries for linear recurrences with constant coefficients, signature (0,2,1,-1).

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

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

A373080 a(n) is the number of binary strings of length n not containing the substrings 0000, 0001, 0011, 0111, 1111.

Original entry on oeis.org

1, 2, 4, 8, 11, 18, 28, 40, 64, 96, 144, 224, 336, 512, 784, 1184, 1808, 2752, 4176, 6368, 9680, 14720, 22416, 34080, 51856, 78912, 120016, 182624, 277840, 422656, 643088, 978336, 1488400, 2264512, 3445072, 5241312, 7974096, 12131456, 18456720, 28079648

Offset: 0

Miquel A. Fiol, Jun 03 2024

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

Cf. A052947, A164395, A164408, A368430.

LinearRecurrence[{0, 1, 2}, {1, 2, 4, 8, 11, 18, 28}, 50] (* Paolo Xausa, Jun 24 2024 *)

a(n) = a(n-2) + 2*a(n-3) for n >= 7.

G.f.: -(x+1)^2*(x^2+1)^2/(2*x^3+x^2-1). - Alois P. Heinz, Jun 03 2024

A373447 Number of ternary strings of length n avoiding the substrings 00, 11, 22, 121, 212, 202.

Original entry on oeis.org

1, 3, 6, 9, 14, 23, 36, 57, 90, 143, 226, 357, 566, 895, 1416, 2241, 3546, 5611, 8878, 14049, 22230, 35175, 55660, 88073, 139362, 220519, 348938, 552141, 873678, 1382463, 2187536, 3461441, 5477202, 8666835, 13713942, 21700217, 34337278, 54333495, 85974452

Offset: 0

Miquel A. Fiol, Jun 05 2024

			For n=3 the a(3)=9 solutions are 010, 012, 020, 021, 101, 102, 120, 201, 210.

Index entries for linear recurrences with constant coefficients, signature (0,1,2,1,0,-1).

Cf. A003945.

LinearRecurrence[{0,1,2,1,0,-1},{1,3,6,9,14,23,36},40] (* Harvey P. Dale, Aug 04 2025 *)

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

G.f.: (x+1)*(x^2+x+1)*(x^3-x^2-x-1)/(-x^6+x^4+2*x^3+x^2-1). - Alois P. Heinz, Jun 05 2024

A369859 Minimum chord of a chordal ring graph with 2*n vertices and minimal diameter.

Original entry on oeis.org

1, 1, 3, 3, 3, 5, 5, 5, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 15, 7, 7, 9, 7, 19, 7, 9, 9, 9, 9, 9, 9, 11, 19, 19, 9, 9, 21, 11, 29, 23, 23, 11, 11, 25, 11, 11, 11, 11, 11, 13, 13, 23, 23, 41, 11, 25, 25, 13, 15, 27, 27, 27, 13, 13, 29, 29, 13, 13, 13, 31, 13, 15, 15, 15, 17, 27, 13, 43, 29, 29, 29, 45, 17, 17, 31, 31, 31, 39

Offset: 1

Miquel A. Fiol, Apr 30 2024

nonn

Given integers n and c (odd), the chordal ring graph CR(2*n,c) is a bipartite graph with vertex set Z_{2*n}, and edges {i,i+1}, {i,i-1}, {i,i+c} if i is odd, and {i,i-c} if i is even.

Cf. A371396.

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User: Miquel A. Fiol

A376313 Independence number of the 2-supertoken graph FF_2(C_n) of the cycle C_n on n vertices.

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A375747 Table read by rows: T(n,k) for n >= 3 and k=1,2,...,2*n-1 are the distinct eigenvalues of the twisted odd graph O^(sigma)_n.

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A375309 Number of walks of length n along the edges of a dodecahedron graph between two vertices at distance two.

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A374795 The minimum diameter of a chordal ring mixed graph with 2*n vertices.

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A374622 Maximum number of vertices of a chordal ring mixed graph CRM(N,c) with diameter n.

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A373503 The minimum diameter of a chordal ring graph with 2*n vertices.

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A373554 Number of ternary length-2 squarefree words of length n not containing the subwords 021 or 10.

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A373080 a(n) is the number of binary strings of length n not containing the substrings 0000, 0001, 0011, 0111, 1111.

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A373447 Number of ternary strings of length n avoiding the substrings 00, 11, 22, 121, 212, 202.

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A369859 Minimum chord of a chordal ring graph with 2*n vertices and minimal diameter.

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