A218553 -id:A218553 - cp's OEIS frontend

A061547 Number of 132 and 213-avoiding derangements of {1,2,...,n}.

1, 0, 1, 2, 6, 10, 26, 42, 106, 170, 426, 682, 1706, 2730, 6826, 10922, 27306, 43690, 109226, 174762, 436906, 699050, 1747626, 2796202, 6990506, 11184810, 27962026, 44739242, 111848106, 178956970, 447392426, 715827882, 1789569706, 2863311530, 7158278826

Offset: 0

Emeric Deutsch, May 16 2001

Or, number of permutations with no fixed points avoiding 213 and 132.
Number of derangements of {1,2,...,n} having ascending runs consisting of consecutive integers. Example: a(4)=6 because we have 234/1, 34/12, 34/2/1, 4/123, 4/3/12, 4/3/2/1, the ascending runs being as indicated. - Emeric Deutsch, Dec 08 2004
Let c be twice the sequence A002450 interlaced with itself (from the second term), i.e., c = 2*(0, 1, 1, 5, 5, 21, 21, 85, 85, 341, 341, ...). Let d be powers of 4 interlaced with the zero sequence: d = (1, 0, 4, 0, 16, 0, 64, 0, 256, 0, ...). Then a(n+1) = c(n) + d(n). - Creighton Dement, May 09 2005
Inverse binomial transform of A094705 (0, 1, 4, 15). - Paul Curtz, Jun 15 2008
Equals row sums of triangle A177993. - Gary W. Adamson, May 16 2010
a(n-1) is also the number of order preserving partial isometries (of an n-chain) of fix 1 (fix of alpha equals the number of fixed points of alpha). - Abdullahi Umar, Dec 28 2010
a(n+1) <= A218553(n) is also the Moore lower bound on the order of a (5,n)-cage. - Jason Kimberley, Oct 31 2011
For n > 0, a(n) is the location of the n-th new number to make a first appearance in A087230. E.g., the 17th number to make its first appearance in A087230 is 18 and this occurs at A087230(43690) and a(17)=43690. - K D Pegrume, Jan 26 2022
Position in A002487 of 2 adjacent terms of A000045. E.g., 3/5 at 10, 5/8 at 26, 8/13 at 42, ... - Ed Pegg Jr, Dec 27 2022

			a(4)=6 because the only 132 and 213-avoiding permutations of {1,2,3,4} without fixed points are: 2341, 3412, 3421, 4123, 4312 and 4321.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
F. Al-Kharousi, R. Kehinde and A. Umar, Combinatorial results for certain semigroups of partial isometries of a finite chain, The Australasian Journal of Combinatorics, Volume 58 (3) (2014), 363-375.
J. Brillhart and P. Morton, A case study in mathematical research: the Golay-Rudin-Shapiro sequence, Amer. Math. Monthly, 103 (1996) 854-869 (contains the sequence of the odd-subscripted terms and that of the even-subscripted terms).
Emeric Deutsch, Derangements That Don't Rise Too Fast: 10902, Amer. Math. Monthly, Vol. 110, No. 7 (2003), pp. 639-640.
K. Dilcher and K. B. Stolarsky, Stern polynomials and double-limit continued fractions, Acta Arithmetica 140 (2009), 119-134
R. Kehinde and A. Umar, On the semigroup of partial isometries of a finite chain, arXiv:1101.0049 [math.GR], 2010.
T. Mansour and A. Robertson, Refined restricted permutations avoiding subsets of patterns of length three, arXiv:math/0204005 [math.CO], 2002
Index entries for linear recurrences with constant coefficients, signature (1,4,-4).

Cf. A000166, A020988, A020989, A068018.
Cf. A177993. - Gary W. Adamson, May 16 2010
Cf. A183158, A183159. - Abdullahi Umar, Dec 28 2010
Moore lower bound on the order of a (k,g) cage: A198300 (square); rows: A000027 (k=2), A027383 (k=3), A062318 (k=4), this sequence (k=5), A198306 (k=6), A198307 (k=7), A198308 (k=8), A198309 (k=9), A198310 (k=10), A094626 (k=11); columns: A020725 (g=3), A005843 (g=4), A002522 (g=5), A051890 (g=6), A188377 (g=7). - Jason Kimberley, Oct 31 2011

[(3/8)*2^n +(1/24)*(-2)^n - 2/3: n in [1..35]]; // Vincenzo Librandi, Aug 13 2011

A061547:=n->ceil(abs((3/8)*2^n +(1/24)*(-2)^n - 2/3)); seq(A061547(n), n=1..30); # Wesley Ivan Hurt, Apr 03 2014

f[n_] := (9*2^(n-3) - (-2)^(n-3) - 2)/3; Array[f, 32] (* Robert G. Wilson v, Aug 13 2011 *)

a(n)=(3/8)*2^n+(1/24)*(-2)^n-2/3 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = (3/8)*2^n + (1/24)*(-2)^n - 2/3 for n>=1.
a(n) = 4*a(n-2) + 2, a(0)=1, a(1)=0, a(2)=1.
G.f: (5*z^3-3*z^2-z+1)/((z-1)*(4*z^2-1)).
a(n) = A020989((n-2)/2) for n=2, 4, 6, ... and A020988((n-3)/2) for n=3, 5, 7, ... .
a(n+1)-2*a(n) = A078008 signed. Differences: doubled A000302. - Paul Curtz, Jun 15 2008
a(2i+1) = 2*Sum_{j=0..i-1} 4^j = string "2"^i read in base 4.
a(2i+2) = 4^i + 2*Sum_{j=0..i-1} 4^j = string "1"*"2"^i read in base 4.
a(n+2) = Sum_{k=0..n} A144464(n,k)^2 = Sum_{k=0..n} A152716(n,k). - Philippe Deléham and Michel Marcus, Feb 26 2014
a(2*n-1) = A176965(2*n), a(2*n) = A176965(2*n-1) for n>0. - Yosu Yurramendi, Dec 23 2016
a(2*n-1) = A020988(k-1), a(2*n)= A020989(n-1) for n>0. - Yosu Yurramendi, Jan 03 2017
a(n+2) = 2*A086893(n), n > 0. - Yosu Yurramendi, Mar 07 2017
E.g.f.: (15 - 8*cosh(x) + 5*cosh(2*x) - 8*sinh(x) + 4*sinh(2*x))/12. - Stefano Spezia, Apr 07 2022

a(0)=1 prepended by Alois P. Heinz, Jan 27 2022

A054760 Table T(n,k) = order of (n,k)-cage (smallest n-regular graph of girth k), n >= 2, k >= 3, read by antidiagonals.

Original entry on oeis.org

3, 4, 4, 5, 6, 5, 6, 8, 10, 6, 7, 10, 19, 14, 7, 8, 12, 30, 26, 24, 8, 9, 14, 40, 42, 67, 30, 9, 10, 16, 50, 62

Offset: 0

N. J. A. Sloane, Apr 26 2000

			First eight antidiagonals are:
   3  4  5  6  7  8  9 10
   4  6 10 14 24 30 58
   5  8 19 26 67 80
   6 10 30 42  ?
   7 12 40 62
   8 14 50
   9 16
  10

P. R. Christopher, Degree monotonicity of cages, Graph Theory Notes of New York, 38 (2000), 29-32.

Andries E. Brouwer, Cages
M. Daven and C. A. Rodger, (k,g)-cages are 3-connected, Discr. Math., 199 (1999), 207-215.
Geoff Exoo, Regular graphs of given degree and girth
G. Exoo and R. Jajcay, Dynamic cage survey, Electr. J. Combin. (2008, 2011).
Gordon Royle, Cubic Cages
Gordon Royle, Cages of higher valency
Pak Ken Wong, Cages-a survey, J. Graph Theory 6 (1982), no. 1, 1-22.

Moore lower bound: A198300.
Orders of cages: this sequence (n,k), A000066 (3,n), A037233 (4,n), A218553 (5,n), A218554 (6,n), A218555 (7,n),  A191595 (n,5).
Graphs not required to be regular: A006787, A006856.

T(k,g) >= A198300(k,g) with equality if and only if: k = 2 and g >= 3; g = 3 and k >= 2; g = 4 and k >= 2; g = 5 and k = 2, 3, 7 or possibly 57; or g = 6, 8, or 12, and there exists a symmetric generalized g/2-gon of order k - 1. - Jason Kimberley, Jan 01 2013

Edited by Jason Kimberley, Apr 25 2010, Oct 26 2011, Dec 21 2012, Jan 01 2013

A000066 Smallest number of vertices in trivalent graph with girth (shortest cycle) = n.

Original entry on oeis.org

4, 6, 10, 14, 24, 30, 58, 70, 112, 126

Offset: 3

N. J. A. Sloane

Also called the order of the (3,n) cage graph.
Recently (unpublished) McKay and Myrvold proved that the minimal graph on 112 vertices is unique. - May 20 2003
If there are n vertices and e edges, then 3n=2e, so n is always even.
Current lower bounds for a(13)..a(32) are 202, 258, 384, 512, 768, 1024, 1536, 2048, 3072, 4096, 6144, 8192, 12288, 16384, 24576, 32768, 49152, 65536, 98304, 131072. - from Table 3 of the Dynamic cage survey via Jason Kimberley, Dec 31 2012
Current upper bounds for a(13)..a(32) are 272, 384, 620, 960, 2176, 2560, 4324, 5376, 16028, 16206, 49326, 49608, 108906, 109200, 285852, 415104, 1141484, 1143408, 3649794, 3650304. - from Table 3 of the Dynamic cage survey via Jason Kimberley, Dec 31 2012

A. T. Balaban, Trivalent graphs of girth nine and eleven and relationships among cages, Rev. Roum. Math. Pures et Appl. 18 (1973) 1033-1043.
Brendan McKay, personal communication.
H. Sachs, On regular graphs with given girth, pp. 91-97 of M. Fiedler, editor, Theory of Graphs and Its Applications: Proceedings of the Symposium, Smolenice, Czechoslovakia, 1963. Academic Press, NY, 1964.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Gabriela Araujo-Pardo, Geoffrey Exoo, and Robert Jajcay, Small bi-regular graphs of even girth Discrete Mathematics 339.2 (2016): 658-667.
Andries E. Brouwer, Cages
Geoff Exoo, Regular graphs of given degree and girth
G. Exoo and R. Jajcay, Dynamic cage survey, Electr. J. Combin. (2008, 2011).
Brendan McKay, W. Myrvold and J. Nadon, Fast backtracking principles applied to find new cages, 9th Annual ACM-SIAM Symposium on Discrete Algorithms, 1998, 188-191.
M. O'Keefe and P. K. Wong, A smallest graph of girth 10 and valency 3, J. Combin. Theory, B 29 (1980), 91-105.
Gordon Royle, Cubic Cages
Eric Weisstein's World of Mathematics, Cage Graph
Pak Ken Wong, Cages-a survey, J. Graph Theory 6 (1982), no. 1, 1-22.

Cf. A006787, A052453 (number of such graphs).

Orders of cages: A054760 (n,k), this sequence (3,n), A037233 (4,n), A218553 (5,n), A218554 (6,n), A218555 (7,n), A191595 (n,5).

For all g > 2, a(g) >= A027383(g-1), with equality if and only if g = 3, 4, 5, 6, 8, or 12. - Jason Kimberley, Dec 21 2012 and Jan 01 2013

Additional comments from Matthew Cook, May 15 2003

Balaban proved 112 as an upper bound for a(11). The proof that it is also a lower bound is in the paper by Brendan McKay, W. Myrvold and J. Nadon.

A037233 Order of (4,n) cage, i.e., minimal order of 4-regular graph of girth n.

Original entry on oeis.org

5, 8, 19, 26, 67, 80

Offset: 3

Erich Friedman

a(9) <= 275, a(10) <= 384, a(12) = 728. - From Royle's page via Jason Kimberley, Dec 26 2012

Andries E. Brouwer, Cages
Geoff Exoo, Regular graphs of given degree and girth
G. Exoo and R. Jajcay, Dynamic cage survey, Electr. J. Combin. (2008, 2011).
Gordon Royle, Cages of higher valency
Eric Weisstein's World of Mathematics, Cage Graph (claims too much)

Orders of cages: A054760 (n,k), A000066 (3,n), this sequence (4,n), A218553 (5,n), A218554 (6,n), A218555 (7,n), A191595 (n,5).

a(n) >= A062318(n+1). - Jason Kimberley, Dec 21 2012

Extended by Jason Kimberley, Apr 25 2010

A191595 Order of smallest n-regular graph of girth 5.

Original entry on oeis.org

5, 10, 19, 30, 40, 50

Offset: 2

N. J. A. Sloane, Jun 07 2011

Current upper bounds for a(8)..a(20) are 80, 96, 124, 154, 203, 230, 288, 312, 336, 448, 480, 512, 576. - Corrected from "Lower" to "Upper" and updated, from Table 4 of the Dynamic cage survey, by Jason Kimberley, Dec 29 2012

Current lower bounds for a(8)..a(20) are 67, 86, 103, 124, 147, 174, 199, 230, 259, 294, 327, 364, 403. - from Table 4 of the Dynamic cage survey via Jason Kimberley, Dec 31 2012

M. Abreu et al., A family of regular graphs of girth 5, Discrete Math., 308 (2008), 1810-1815.
Andries E. Brouwer, Cages
Geoff Exoo, Regular graphs of given degree and girth
G. Exoo and R. Jajcay, Dynamic cage survey, Electr. J. Combin. (2008, 2011).
G. Royle, Cages of higher valency

Orders of cages: A054760 (n,k), A000066 (3,n), A037233 (4,n), A218553 (5,n), A218554 (6,n), A218555 (7,n), this sequence (n,5).

Moore lower bound on the orders of (k,g) cages: A198300 (square); rows: A000027 (k=2), A027383 (k=3), A062318 (k=4), A061547 (k=5), A198306(k=6), A198307 (k=7), A198308 (k=8), A198309 (k=9), A198310 (k=10),A094626 (k=11); columns: A020725 (g=3), A005843 (g=4), A002522 (g=5), A051890 (g=6), A188377 (g=7). - Jason Kimberley, Nov 02 2011

a(n) >= A002522(n) with equality if and only if n = 2, 3, 7 or possibly 57. - Jason Kimberley, Nov 02 2011

a(2) = 5 prepended by Jason Kimberley, Jan 02 2013

cp's OEIS Frontend

A061547 Number of 132 and 213-avoiding derangements of {1,2,...,n}.

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A054760 Table T(n,k) = order of (n,k)-cage (smallest n-regular graph of girth k), n >= 2, k >= 3, read by antidiagonals.

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A000066 Smallest number of vertices in trivalent graph with girth (shortest cycle) = n.

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A037233 Order of (4,n) cage, i.e., minimal order of 4-regular graph of girth n.

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A191595 Order of smallest n-regular graph of girth 5.

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A218554 Order of (6,n) cage, i.e., minimal order of 6-regular graph of girth n.

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A218555 Order of (7,n) cage, i.e., minimal order of 7-regular graph of girth n.

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