A000066 -id:A000066 - cp's OEIS frontend

A266730 Appears to be an erroneous version of A000066.

4, 6, 10, 14, 24, 30, 58, 70, 116

Offset: 3

N. J. A. Sloane, Jan 04 2016

nonn

Listed in accordance with OEIS policy of including published but erroneous sequences, to serve as pointers to the correct versions.

Araujo-Pardo, Gabriela, Geoffrey Exoo, and Robert Jajcay. "Small bi-regular graphs of even girth." Discrete Mathematics 339.2 (2016): 658-667.

A027383 a(2n) = 32^n - 2; a(2*n+1) = 2^(n+2) - 2.

Original entry on oeis.org

1, 2, 4, 6, 10, 14, 22, 30, 46, 62, 94, 126, 190, 254, 382, 510, 766, 1022, 1534, 2046, 3070, 4094, 6142, 8190, 12286, 16382, 24574, 32766, 49150, 65534, 98302, 131070, 196606, 262142, 393214, 524286, 786430, 1048574, 1572862, 2097150, 3145726, 4194302, 6291454

Offset: 0

R. K. Guy

Number of balanced strings of length n: let d(S) = #(1's) - #(0's), # == count in S, then S is balanced if every substring T of S has -2 <= d(T) <= 2.
Number of "fold lines" seen when a rectangular piece of paper is folded n+1 times along alternate orthogonal directions and then unfolded. - Quim Castellsaguer (qcastell(AT)pie.xtec.es), Dec 30 1999
Also the number of binary strings with the property that, when scanning from left to right, once the first 1 is seen in position j, there must be a 1 in positions j+2, j+4, ... until the end of the string. (Positions j+1, j+3, ... can be occupied by 0 or 1.) - Jeffrey Shallit, Sep 02 2002
a(n-1) is also the Moore lower bound on the order of a (3,n)-cage. - Eric W. Weisstein, May 20 2003 and Jason Kimberley, Oct 30 2011
Partial sums of A016116. - Hieronymus Fischer, Sep 15 2007
Equals row sums of triangle A152201. - Gary W. Adamson, Nov 29 2008
From John P. McSorley, Sep 28 2010: (Start)
a(n) = DPE(n+1) is the total number of k-double-palindromes of n up to cyclic equivalence. See sequence A180918 for the definitions of a k-double-palindrome of n and of cyclic equivalence. Sequence A180918 is the 'DPE(n,k)' triangle read by rows where DPE(n,k) is the number of k-double-palindromes of n up to cyclic equivalence. For example, we have a(4) = DPE(5) = DPE(5,1) + DPE(5,2) + DPE(5,3) + DPE(5,4) + DPE(5,5) = 0 + 2 + 2 + 1 + 1 = 6.
The 6 double-palindromes of 5 up to cyclic equivalence are 14, 23, 113, 122, 1112, 11111. They come from cyclic equivalence classes {14,41}, {23,32}, {113,311,131}, {122,212,221}, {1112,2111,1211,1121}, and {11111}. Hence a(n)=DPE(n+1) is the total number of cyclic equivalence classes of n containing at least one double-palindrome.
(End)
From Herbert Eberle, Oct 02 2015: (Start)
For n > 0, there is a red-black tree of height n with a(n-1) internal nodes and none with less.
In order a red-black tree of given height has minimal number of nodes, it has exactly 1 path with strictly alternating red and black nodes. All nodes outside this height defining path are black.
Consider:
mrbt5                R
                    / \
                   /   \
                  /     \
                 /       B
                /       / \
mrbt4          B       /   B
              / \     B   E E
             /   B   E E
mrbt3       R   E E
           / \
          /   B
mrbt2    B   E E
        / E
mrbt1  R
      E E
(Red nodes shown as R, blacks as B, externals as E.)
Red-black trees mrbt1, mrbt2, mrbt3, mrbt4, mrbt5 of respective heights h = 1, 2, 3, 4, 5; all minimal in the number of internal nodes, namely 1, 2, 4, 6, 10.
Recursion (let n = h-1): a(-1) = 0, a(n) = a(n-1) + 2^floor(n/2), n >= 0.
(End)
Also the number of strings of length n with the digits 1 and 2 with the property that the sum of the digits of all substrings of uneven length is not divisible by 3. An example with length 8 is 21221121. - Herbert Kociemba, Apr 29 2017
a(n-2) is the number of achiral n-bead necklaces or bracelets using exactly two colors.  For n=4, the four arrangements are AAAB, AABB, ABAB, and ABBB. - Robert A. Russell, Sep 26 2018
Partial sums of powers of 2 repeated 2 times, like A200672 where is 3 times. - Yuchun Ji, Nov 16 2018
Also the number of binary words of length n with cuts-resistance <= 2, where, for the operation of shortening all runs by one, cuts-resistance is the number of applications required to reach an empty word. Explicitly, these are words whose sequence of run-lengths, all of which are 1 or 2, has no odd-length run of 1's sandwiched between two 2's. - Gus Wiseman, Nov 28 2019
Also the number of up-down paths with n steps such that the height difference between the highest and lowest points is at most 2. - Jeremy Dover, Jun 17 2020
Also the number of non-singleton integer compositions of n + 2 with no odd part other than the first or last. Including singletons gives A052955. This is an unsorted (or ordered) version of A351003. The version without even (instead of odd) interior parts is A001911, complement A232580. Note that A000045(n-1) counts compositions without odd parts, with non-singleton case A077896, and A052952/A074331 count non-singleton compositions without even parts. Also the number of compositions y of n + 1 such that y_i = y_{i+1} for all even i. - Gus Wiseman, Feb 19 2022

			After 3 folds one sees 4 fold lines.
Example: a(3) = 6 because the strings 001, 010, 100, 011, 101, 110 have the property.
Binary: 1, 10, 100, 110, 1010, 1110, 10110, 11110, 101110, 111110, 1011110, 1111110, 10111110, 11111110, 101111110, 111111110, 1011111110, 1111111110, 10111111110, ... - _Jason Kimberley_, Nov 02 2011
Example: Partial sums of powers of 2 repeated 2 times:
a(3) = 1+1+2 = 4;
a(4) = 1+1+2+2 = 6;
a(5) = 1+1+2+2+4 = 10.
_Yuchun Ji_, Nov 16 2018

John P. McSorley: Counting k-compositions of n with palindromic and related structures. Preprint, 2010. [John P. McSorley, Sep 28 2010]

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
J. Jordan and R. Southwell, Further Properties of Reproducing Graphs, Applied Mathematics, Vol. 1 No. 5, 2010, pp. 344-350. - From _N. J. A. Sloane_, Feb 03 2013
Leonard F. Klosinski, Gerald L. Alexanderson and Loren C. Larson, Under misprinted head B3, Amer Math. Monthly, 104(1997) 753-754.
Laurent Noé, Spaced seed design on profile HMMs for precise HTS read-mapping efficient sliding window product on the matrix semi-group, in Rapide Bilan 2012-2013, LIFL, Université Lille 1 - INRIA Journées au vert 11 et 12 juin 2013.
Eric Weisstein's World of Mathematics, Cage Graph
Index entries for sequences obtained by enumerating foldings
Index entries for linear recurrences with constant coefficients, signature (1,2,-2).

Cf. A132666, A152201.
Moore lower bound on the order of a (k,g) cage: A198300 (square); rows: A000027 (k=2), this sequence (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, Oct 30 2011
Cf. A000066 (actual order of a (3,g)-cage).
Bisections are A033484 (even) and A000918 (odd).
a(n) = A305540(n+2,2), the second column of the triangle.
Numbers whose binary expansion is a balanced word are A330029.
Binary words counted by cuts-resistance are A319421 or A329860.
Cf. A000975, A062011, A329862, A329863.
The complementary compositions are counted by A274230(n-1) + 1, with bisections A060867 (even) and A134057 (odd).
Cf. A000346, A000984, A001405, A001700, A011782 (compositions).
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A029744 = {s(n), n>=1}, the numbers 2^k and 3*2^k, as the parent: A029744 (s(n)); A052955 (s(n)-1), A027383 (s(n)-2), A354788 (s(n)-3), A347789 (s(n)-4), A209721 (s(n)+1), A209722 (s(n)+2), A343177 (s(n)+3), A209723 (s(n)+4); A060482, A136252 (minor differences from A354788 at the start); A354785 (3*s(n)), A354789 (3*s(n)-7). The first differences of A029744 are 1,1,1,2,2,4,4,8,8,... which essentially matches eight sequences: A016116, A060546, A117575, A131572, A152166, A158780, A163403, A320770. The bisections of A029744 are A000079 and A007283. - N. J. A. Sloane, Jul 14 2022

import Data.List (transpose)
a027383 n = a027383_list !! n
a027383_list = concat $ transpose [a033484_list, drop 2 a000918_list]
-- Reinhard Zumkeller, Jun 17 2015

[2^Floor((n+2)/2)+2^Floor((n+1)/2)-2: n in [0..50]]; // Vincenzo Librandi, Aug 16 2011

a[0]:=0:a[1]:=1:for n from 2 to 100 do a[n]:=2*a[n-2]+2 od: seq(a[n], n=1..41); # Zerinvary Lajos, Mar 16 2008

a[n_?EvenQ] := 3*2^(n/2)-2; a[n_?OddQ] := 2^(2+(n-1)/2)-2; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Oct 21 2011, after Quim Castellsaguer *)
LinearRecurrence[{1, 2, -2}, {1, 2, 4}, 41] (* Robert G. Wilson v, Oct 06 2014 *)
Table[Length[Select[Tuples[{0,1},n],And[Max@@Length/@Split[#]<=2,!MatchQ[Length/@Split[#],{_,2,ins:1..,2,_}/;OddQ[Plus[ins]]]]&]],{n,0,15}] (* Gus Wiseman, Nov 28 2019 *)

a(n)=2^(n\2+1)+2^((n+1)\2)-2 \\ Charles R Greathouse IV, Oct 21 2011

def a(n): return 2**((n+2)//2) + 2**((n+1)//2) - 2
print([a(n) for n in range(43)]) # Michael S. Branicky, Feb 19 2022

a(0)=1, a(1)=2; thereafter a(n+2) = 2*a(n) + 2.
a(2n) = 3*2^n - 2 = A033484(n);
a(2n-1) = 2^(n+1) - 2 = A000918(n+1).
G.f.: (1 + x)/((1 - x)*(1 - 2*x^2)). - David Callan, Jul 22 2008
a(n) = Sum_{k=0..n} 2^min(k, n-k).
a(n) = 2^floor((n+2)/2) + 2^floor((n+1)/2) - 2. - Quim Castellsaguer (qcastell(AT)pie.xtec.es)
a(n) = 2^(n/2)*(3 + 2*sqrt(2) + (3-2*sqrt(2))*(-1)^n)/2 - 2. - Paul Barry, Apr 23 2004
a(n) = A132340(A052955(n)). - Reinhard Zumkeller, Aug 20 2007
a(n) = A052955(n+1) - 1. - Hieronymus Fischer, Sep 15 2007
a(n) = A132666(a(n+1)) - 1. - Hieronymus Fischer, Sep 15 2007
a(n) = A132666(a(n-1)+1) for n > 0. - Hieronymus Fischer, Sep 15 2007
A132666(a(n)) = a(n-1) + 1 for n > 0. - Hieronymus Fischer, Sep 15 2007
G.f.: (1 + x)/((1 - x)*(1 - 2*x^2)). - David Callan, Jul 22 2008
a(n) = 2*( (a(n-2)+1) mod (a(n-1)+1) ), n > 1. - Pierre Charland, Dec 12 2010
a(n) = A136252(n-1) + 1, for n > 0. - Jason Kimberley, Nov 01 2011
G.f.: (1+x*R(0))/(1-x), where R(k) = 1 + 2*x/( 1 - x/(x + 1/R(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 16 2013
a(n) = 2^((2*n + 3*(1-(-1)^n))/4)*3^((1+(-1)^n)/2) - 2. - Luce ETIENNE, Sep 01 2014
a(n) = a(n-1) + 2^floor((n-1)/2) for n>0, a(0)=1. - Yuchun Ji, Nov 23 2018
E.g.f.: 3*cosh(sqrt(2)*x) - 2*cosh(x) + 2*sqrt(2)*sinh(sqrt(2)*x) - 2*sinh(x). - Stefano Spezia, Apr 06 2022

More terms from Larry Reeves (larryr(AT)acm.org), Mar 24 2000

Replaced definition with a simpler one. - N. J. A. Sloane, Jul 09 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

A185131 Irregular triangle C(n,g) counting connected trivalent simple graphs on 2n vertices with girth at least g.

Original entry on oeis.org

1, 2, 1, 5, 2, 19, 6, 1, 85, 22, 2, 509, 110, 9, 1, 4060, 792, 49, 1, 41301, 7805, 455, 5, 510489, 97546, 5783, 32, 7319447, 1435720, 90938, 385, 117940535, 23780814, 1620479, 7574, 1, 2094480864, 432757568, 31478584, 181227, 3, 40497138011, 8542471494

Offset: 2

Jason Kimberley, Jan 09 2012

The first column is for girth at least 3. The row length is incremented to g-2 when 2n reaches A000066(g).

			                  1;
                  2,             1;
                  5,             2;
                 19,             6,            1;
                 85,            22,            2;
                509,           110,            9,          1;
               4060,           792,           49,          1;
              41301,          7805,          455,          5;
             510489,         97546,         5783,         32;
            7319447,       1435720,        90938,        385;
          117940535,      23780814,      1620479,       7574,         1;
         2094480864,     432757568,     31478584,     181227,         3;
        40497138011,    8542471494,    656783890,    4624501,        21;
       845480228069,  181492137812,  14621871204,  122090544,       546,    1;
     18941522184590, 4127077143862, 345975648562, 3328929954,     30368,    0;
    453090162062723,        ?,            ?,     93990692595,   1782840,    1;
  11523392072541432,        ?,            ?,   2754222605376,  95079083,    3;
 310467244165539782,        ?,            ?,          ?,     4686063120,   13;
8832736318937756165,        ?,            ?,          ?,   220323447962,  155;
          ?,                ?,            ?,          ?, 10090653722861, 4337;

Jason Kimberley, Table of i, a(i) for i = 2..59 (n = 2..16)
B. Brinkmann, J. Goedgebeur, and B. D. McKay, Generation of cubic graphs, Discr. Math. Theor. Comp. Sci. 13 (2) (2011) 69-80.
House of Graphs, Cubic graphs
Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth at least g
M. Meringer, Tables of Regular Graphs

Connected 3-regular simple graphs with girth at least g: this sequence (triangle); chosen g: A002851 (g=3), A014371 (g=4), A014372 (g=5), A014374 (g=6), A014375 (g=7), A014376 (g=8).
Connected 3-regular simple graphs with girth exactly g: A198303 (triangle); chosen g: A006923 (g=3), A006924 (g=4), A006925 (g=5), A006926 (g=6), A006927 (g=7).
Triangular arrays C(n,g) counting connected simple k-regular graphs on n vertices with girth at least g: this sequence (k=3), A184941 (k=4), A184951 (k=5), A184961 (k=6), A184971 (k=7), A184981 (k=8).

Terms C(18,6), C(20,7) and C(21,7) from House of Graphs via Jason Kimberley, May 21 2017

A198303 Irregular triangle C(n,g) counting connected trivalent simple graphs on 2n vertices with girth exactly g.

Original entry on oeis.org

1, 1, 1, 3, 2, 13, 5, 1, 63, 20, 2, 399, 101, 8, 1, 3268, 743, 48, 1, 33496, 7350, 450, 5, 412943, 91763, 5751, 32, 5883727, 1344782, 90553, 385, 94159721, 22160335, 1612905, 7573, 1, 1661723296, 401278984, 31297357, 181224, 3, 31954666517

Offset: 2

Jason Kimberley, Nov 16 2011

The first column is for girth exactly 3. The row length is incremented to g-2 when 2n reaches A000066(g).

			1;
1, 1;
3, 2;
13, 5, 1;
63, 20, 2;
399, 101, 8, 1;
3268, 743, 48, 1;
33496, 7350, 450, 5;
412943, 91763, 5751, 32;
5883727, 1344782, 90553, 385;
94159721, 22160335, 1612905, 7573, 1;
1661723296, 401278984, 31297357, 181224, 3;
31954666517, 7885687604, 652159389, 4624480, 21;
663988090257, 166870266608, 14499780660, 122089998, 545;
14814445040728, 3781101495300, 342646718608, 3328899586, 30368;

F. C. Bussemaker, S. Cobeljic, L. M. Cvetkovic and J. J. Seidel, Computer investigations of cubic graphs, T.H.-Report 76-WSK-01, Technological University Eindhoven, Dept. Mathematics, 1976.
Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth exactly g

The sum of the n-th row of this sequence is A002851(n).
Connected 3-regular simple graphs with girth exactly g: this sequence (triangle); chosen g: A006923 (g=3), A006924 (g=4), A006925 (g=5), A006926 (g=6), A006927 (g=7).
Connected 3-regular simple graphs with girth at least g: A185131 (triangle); chosen g: A002851 (g=3), A014371 (g=4), A014372 (g=5), A014374 (g=6), A014375 (g=7), A014376 (g=8).
Triangular arrays C(n,g) counting connected simple k-regular graphs on n vertices with girth exactly g: this sequence (k=3), A184940 (k=4), A184950 (k=5), A184960 (k=6), A184970 (k=7), A184980 (k=8).

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

A006787 Number of n-node graphs with no cycles of length less than 5.

Original entry on oeis.org

1, 2, 3, 6, 11, 23, 48, 114, 293, 869, 2963, 12066, 58933, 347498, 2455693, 20592932, 202724920, 2322206466, 30743624324, 468026657815, 8161170076257

Offset: 1

N. J. A. Sloane

Includes graphs with no cycles at all as well as graphs with girth greater than 5.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

CombOS - Combinatorial Object Server, Generate graphs
Brendan McKay, Emails to N. J. A. Sloane, 1991
Brendan D. McKay, Isomorph-free exhaustive generation, Table 2.
Brendan D. McKay, Isomorph-Free Exhaustive Generation, J. Algorithms, vol. 26 iss. 2 (1998), 306-324.

Cf. A000066, A000088, A054760, A159847, A126757 (connected, inv. Eul. Transf.), A128236, A128237, A300705.

a(n) = A000088(n) - A128236(n) - A128237(n). - Andrew Howroyd, May 06 2021

Definition corrected by Brendan McKay, Apr 27 2007
a(18)-a(19) (from the McKay reference) added by R. J. Mathar, Jun 17 2008
a(20)-a(21) from Brendan McKay, Mar 11 2018

A052453 Number of nonisomorphic (3,n) cage graphs.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 18, 3, 1, 1

Offset: 3

Eric W. Weisstein

A (3,n) cage graph is a 3-regular (or cubic, or trivalent) graph which has girth n, and has the fewest possible number of vertices. - Harry Richman, Jan 14 2025

			a(3) = 1 since the complete graph K_4 is the unique smallest cubic graph with girth 3.
a(5) = 1 since the Petersen graph is the unique smallest cubic graph with girth 5.
a(12) = 1 from the unique generalized hexagon of order 2.

Andries E. Brouwer, Cages
Geoffrey Exoo, Regular graphs of given degree and girth.
Geoffrey Exoo and Robert 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)
Wikipedia, Cage (graph theory)

Cf. A000066 (size of these graphs).

a(11) from Brendan McKay and W. Myrvold

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

A218553 Order of (5,n) cage, i.e., minimal order of 5-regular graph of girth n.

Original entry on oeis.org

6, 10, 30, 42

Offset: 3

Arkadiusz Wesolowski, Nov 02 2012

a(7) <= 152, a(8) = 170, a(12) = 2730. - From Royle's page via Jason Kimberley, Dec 21 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), A037233 (4,n), this sequence (5,n), A218554 (6,n), A218555 (7,n), A191595 (n,5).

a(n) >= A061547(n+1).

a(7) deleted by Jason Kimberley, Dec 21 2012

cp's OEIS Frontend

A266730 Appears to be an erroneous version of A000066.

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A027383 a(2*n) = 3*2^n - 2; a(2*n+1) = 2^(n+2) - 2.

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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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A185131 Irregular triangle C(n,g) counting connected trivalent simple graphs on 2n vertices with girth at least g.

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A198303 Irregular triangle C(n,g) counting connected trivalent simple graphs on 2n vertices with girth exactly g.

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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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A006787 Number of n-node graphs with no cycles of length less than 5.

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A052453 Number of nonisomorphic (3,n) cage graphs.

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A027383 a(2n) = 32^n - 2; a(2*n+1) = 2^(n+2) - 2.