A054760 -id:A054760 - cp's OEIS frontend

A051890 a(n) = 2*(n^2 - n + 1).

2, 2, 6, 14, 26, 42, 62, 86, 114, 146, 182, 222, 266, 314, 366, 422, 482, 546, 614, 686, 762, 842, 926, 1014, 1106, 1202, 1302, 1406, 1514, 1626, 1742, 1862, 1986, 2114, 2246, 2382, 2522, 2666, 2814, 2966, 3122, 3282, 3446, 3614, 3786, 3962

Offset: 0

Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), Apr 30 2000

Draw n ellipses in the plane (n > 0); sequence gives maximum number of regions into which the plane is divided (cf. A014206, A386480).
Least k such that Z(k,2) <= Z(n,3) where Z(m,s) = Sum_{i>=m} 1/i^s = zeta(s) - Sum_{i=1..m-1} 1/i^s. - Benoit Cloitre, Nov 29 2002
For n > 2, third diagonal of A154685. - Vincenzo Librandi, Aug 06 2010
a(k) is also the Moore lower bound A198300(k,6) on the order A054760(k,6) of a (k,6)-cage. Equality is achieved if and only if there exists a finite projective plane of order k - 1. A sufficient condition for this is that k - 1 be a prime power. - Jason Kimberley, Oct 17 2011 and Jan 01 2013
From Jess Tauber, May 20 2013: (Start)
For neutron shell filling in spherical atomic nuclei, this sequence shows numerical differences between filled spin-split suborbitals sharing all quantum numbers except the principal quantum number n, and here all n's must differ by 1. Only a small handful of exceptions exist.
This sequence consists of summed pairs of every other doubled triangular number. It also can be created by taking differences between nuclear magic numbers from the harmonic oscillator (HO)(doubled tetrahedral) set and the spin-orbit (SO) set (2,6,14,28,50,82,126,184,...), with either set being larger. So SO-HO: 2-0=2, 6-0=6, 14-0=14, 28-2=26, 50-8=42, 82-20=62, 126-40=86, 184-70=114, and HO-SO: 2-0=2, 8-2=6, 20-6=14, 40-14=26, 70-28=42, 112-50=62, 168-82=86, 240-126=114. From the perspective of idealized HO periodic structure, with suborbitals in order from largest to smallest spin, alternating by parity, the HO-SO set is spaced two period analogs PLUS one suborbital, while the SO-HO set is spaced two period analogs MINUS one suborbital. (End)
The known values of f(k,6) and F(k,6) in Brown (1967), Table 1, closely match this sequence. - N. J. A. Sloane, Jul 09 2015
Numbers k such that 2*k - 3 is a square. - Bruno Berselli, Nov 08 2017
Numbers written 222 in number base B, including binary with 'digit' 2: 222(2)=14, 222(3)=26, ... - Ron Knott, Nov 14 2017

G. C. Greubel, Table of n, a(n) for n = 0..5000
William G. Brown, On Hamiltonian regular graphs of girth six, J. London Math. Soc., 42 (1967): 514-520.
Steven Edwards and William Griffiths, On Generalized Delannoy Numbers, J. Int. Seq., Vol. 23 (2020), Article 20.3.6.
William Q. Erickson and Jan Kretschmann, The structure and normalized volume of Monge polytopes, arXiv:2311.07522 [math.CO], 2023. See p. 10.
Parabola, Problem #Q607, vol. 20, no. 2, 1984, p. 27.
Eric Weisstein's World of Mathematics, Plane Division by Ellipses
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001844, A002061, A014206, A154685, A195600, A386480.

Moore lower bound on the order of a (k,g) cage: 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), this sequence (g=6), A188377 (g=7).

List([0..50], n-> 2*(n^2-n+1)); # G. C. Greubel, Feb 21 2019

[2*(n^2-n+1): n in [0..50]]; // G. C. Greubel, Feb 21 2019

A051890 := n->2*(n^2-n+1); seq(A051890(n) = n=0..50);

Table[2*(n^2-n+1), {n, 0, 50}] (* G. C. Greubel, Jul 14 2017 *)
LinearRecurrence[{3,-3,1},{2,2,6},50] (* Harvey P. Dale, Jul 14 2025 *)

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

[2*(n^2-n+1) for n in (0..50)] # G. C. Greubel, Feb 21 2019

a(n) = 4*binomial(n, 2) + 2. - Francois Jooste (phukraut(AT)hotmail.com), Mar 05 2003
For n > 2, nearest integer to (Sum_{k>=n} 1/k^3)/(Sum_{k>=n} 1/k^5). - Benoit Cloitre, Jun 12 2003
a(n) = 2*A002061(n). - Jonathan Vos Post, Jun 19 2005
a(n) = 4*n + a(n-1) - 4 for n > 0, a(0)=2. - Vincenzo Librandi, Aug 06 2010
a(n) = 2*(n^2 - n +1) = 2*(n-1)^2 + 2(n-1) + 2 = 222 read in base n-1 (for n > 3). - Jason Kimberley, Oct 20 2011
G.f.: 2*(1 - 2*x + 3*x^2)/(1 - x)^3. - Colin Barker, Jan 10 2012
a(n) = A001844(n-1) + 1 = A046092(n-1) + 2. - Jaroslav Krizek, Dec 27 2013
E.g.f.: 2*(x^2 + 1)*exp(x). - G. C. Greubel, Jul 14 2017

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.

A198300 Square array M(k,g), read by antidiagonals, of the Moore lower bound on the order of a (k,g)-cage.

Original entry on oeis.org

3, 4, 4, 5, 6, 5, 6, 8, 10, 6, 7, 10, 17, 14, 7, 8, 12, 26, 26, 22, 8, 9, 14, 37, 42, 53, 30, 9, 10, 16, 50, 62, 106, 80, 46, 10, 11, 18, 65, 86, 187, 170, 161, 62, 11, 12, 20, 82, 114, 302, 312, 426, 242, 94, 12, 13, 22, 101, 146, 457, 518, 937, 682, 485, 126, 13

Offset: 1

Jason Kimberley, Oct 27 2011

k >= 2; g >= 3.
The base k-1 reading of the base 10 string of A094626(g).
Exoo and Jajcay Theorem 1: M(k,g) <= A054760(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 n-gon of order k - 1.

			This is the table formed from the antidiagonals for k+g = 5..20:
3   4   5   6    7    8    9     10    11    12    13    14    15   16  17 18
4   6  10  14   22   30    46    62    94   126   190   254   382  510 766
5   8  17  26   53   80   161   242   485   728  1457  2186  4373 6560
6  10  26  42  106  170   426   682  1706  2730  6826 10922 27306
7  12  37  62  187  312   937  1562  4687  7812 23437 39062
8  14  50  86  302  518  1814  3110 10886 18662 65318
9  16  65 114  457  800  3201  5602 22409 39216
10 18  82 146  658 1170  5266  9362 42130
11 20 101 182  911 1640  8201 14762
12 22 122 222 1222 2222 12222
13 24 145 266 1597 2928
14 26 170 314 2042
15 28 197 366
16 30 226
17 32
18

E. Bannai and T. Ito, On finite Moore graphs, J. Fac. Sci. Tokyo, Sect. 1A, 20 (1973) 191-208.
R. M. Damerell, On Moore graphs, Proc. Cambridge Phil. Soc. 74 (1973) 227-236.

Jason Kimberley, Table of n, a(n) for n = 1..20100 (k+g = 5..204)
Jason Kimberley, Table of n, k+g, k, g, M(k,g)=a(n) for k+g = 5..204 (n = 1..20100)
G. Exoo and R. Jajcay, Dynamic cage survey, Electr. J. Combin. (2008, 2011).
Gordon Royle, Cages of higher valency

Moore lower bound on the order of a (k,g) cage: this sequence (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), 2*A053698 (g=8), 2*A053699 (g=10), 2*A053700 (g=12), 2*A053716 (g=14), 2*A053716 (g=16), 2*A102909 (g=18), 2*A103623 (g=20), 2*A060885 (g=22), 2*A105067 (g=24), 2*A060887 (g=26), 2*A104376 (g=28), 2*A104682 (g=30), 2*A105312 (g=32).

Cf. A054760 (the actual order of a (k,g)-cage).

ExtendedStringToInt:=func;
M:=func;
k_:=2;g_:=3;
anti:=func;
[anti(kg):kg in[5..15]];

Table[Function[g, FromDigits[#, k - 1] &@ IntegerDigits@ SeriesCoefficient[x (1 + x)/((1 - x) (1 - 10 x^2)), {x, 0, g}]][n - k + 3], {n, 2, 12}, {k, n, 2, -1}] // Flatten (* Michael De Vlieger, May 15 2017 *)

M(k,2i) = 2 sum_{j=0}^{i-1}(k-1)^j =  string "2"^i read in base k-1.
M(k,2i+1) = (k-1)^i +  2 sum_{j=0}^{i-1}(k-1)^j = string "1"*"2"^i read in base k-1.
Recurrence:
M(k,3) = k + 1,
M(k,2i) = M(k,2i-1) + (k-1)^(i-1),
M(k,2i+1) = M(k,2i) + (k-1)^i.

A058276 Number of connected 6-regular simple graphs on n vertices with girth at least 4.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 9, 6, 267, 3727, 483012, 69823723, 14836130862

Offset: 0

N. J. A. Sloane, Dec 17 2000

The null graph on 0 vertices is vacuously connected and 6-regular; since it is acyclic, it has infinite girth. - Jason Kimberley, Jan 30 2011

Other than at n=0, this sequence first differs from A184964 at n = A054760(6,5) = 40.

Jason Kimberley, Connected regular graphs with girth at least 4/
Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth at least g/
Markus Meringer, Fast Generation of Regular Graphs and Construction of Cages, Journal of Graph Theory, Vol. 30, No. 2 (1999), 137-146.
Markus Meringer, Tables of Regular Graphs/

6-regular simple graphs with girth at least 4: this sequence (connected), A185264 (disconnected), A185364 (not necessarily connected).
Connected k-regular simple graphs with girth at least 4: A186724 (any k), A186714 (triangle); specified degree k: A185114 (k=2), A014371 (k=3), A033886 (k=4), A058275 (k=5), this sequence (k=6), A181153 (k=7), A181154 (k=8), A181170 (k=9).
Connected 6-regular simple graphs with girth at least g: A006822 (g=3), this sequence (g=4).
Connected 6-regular simple graphs with girth exactly g: A184963 (g=3), A184964 (g=4).

a(n) = A014377(n) - A184963(n).

Terms a(19), a(20), and a(21), were appended, from running Meringer's GENREG at U. Ncle. for 51 processor days, by Jason Kimberley on Dec 11 2009

a(22) was appended, from running Meringer's GENREG at U. Ncle. for 1620 processor days, by Jason Kimberley on Dec 10 2011

A184964 Number of connected 6-regular simple graphs on n vertices with girth exactly 4.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 9, 6, 267, 3727, 483012, 69823723, 14836130862

Offset: 0

Jason Kimberley, Feb 28 2011

Other than at n=0, this sequence first differs from A058276 at n = A054760(6,5) = 40.

			a(0)=0 because even though the null graph (on zero vertices) is vacuously 6-regular and connected, since it is acyclic, it has infinite girth.
The a(12)=1 graph is the complete bipartite graph K_{6,6}.

Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth exactly g

Connected k-regular simple graphs with girth exactly 4: A006924 (k=3), A184944 (k=4), A184954 (k=5), this sequence (k=6), A184974 (k=7).
Connected 6-regular simple graphs with girth at least g: A006822 (g=3), A058276 (g=4).
Connected 6-regular simple graphs with girth exactly g: A184963 (g=3), this sequence (g=4).

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

A184981 Irregular triangle C(n,g) counting the connected 8-regular simple graphs on n vertices with girth at least g.

Original entry on oeis.org

1, 1, 6, 94, 10786, 3459386, 1470293676, 733351105935, 1

Offset: 9

Jason Kimberley, Jan 19 2012

The first column is for girth at least 3. The row length is incremented to g-2 when n reaches A054760(8,g).

			1;
1;
6;
94;
10786;
3459386;
1470293676;
733351105935, 1;
?, 0;
?, 1;
?, 0;
?, 13;
?, 1;

Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth at least g

Connected 8-regular simple graphs with girth at least g: this sequence (triangle); chosen g: A014378 (g=3), A181154 (g=4).
Connected 8-regular simple graphs with girth exactly g: A184980 (triangle); chosen g: A184983 (g=3).
Triangular arrays C(n,g) counting connected simple k-regular graphs on n vertices with girth at least g: A185131 (k=3), A184941 (k=4), A184951 (k=5), A184961 (k=6), A184971 (k=7), this sequence (k=8), A184991 (k=9).

A184980 Irregular triangle C(n,g) counting the connected 8-regular simple graphs on n vertices with girth exactly g.

Original entry on oeis.org

1, 1, 6, 94, 10786, 3459386, 1470293676, 733351105934, 1

Offset: 9

Jason Kimberley, Jan 19 2012

The first column is for girth at least 3. The row length is incremented to g-2 when 2n reaches A054760(8,g).

			1;
1;
6;
94;
10786;
3459386;
1470293676;
733351105934, 1;
?, 0;
?, 1;
?, 0;
?, 13;
?, 1;

Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth exactly g

Connected 8-regular simple graphs with girth at least g: A184981 (triangle); chosen g: A014378 (g=3), A181154 (g=4).
Connected 8-regular simple graphs with girth exactly g: this sequence (triangle); chosen g: A184983 (g=3).
Triangular arrays C(n,g) counting connected simple k-regular graphs on n vertices with girth exactly g: A198303 (k=3), A184940 (k=4), A184950 (k=5), A184960 (k=6), A184970 (k=7), this sequence (k=8).

A184950 Irregular triangle C(n,g) counting the connected 5-regular simple graphs on 2n vertices with girth exactly g.

Original entry on oeis.org

1, 3, 59, 1, 7847, 1, 3459376, 7, 2585136287, 388, 2807104844073, 406824

Offset: 3

Jason Kimberley, Feb 24 2011

The first column is for girth exactly 3. The row length sequence starts: 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4. The row length is incremented to g-2 when 2n reaches A054760(5,g).

			1;
3;
59, 1;
7847, 1;
3459376, 7;
2585136287, 388;
2807104844073, 406824;
?, 1125022325;
?, 3813549359274;

Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth exactly g
Jason Kimberley, Incomplete table of i, n, g, C(n,g)=a(i) for row n = 3..22

Connected 5-regular simple graphs with girth at least g: A184951 (triangle); chosen g: A006821 (g=3), A058275 (g=4).
Connected 5-regular simple graphs with girth exactly g: this sequence (triangle); chosen g: A184953 (g=3), A184954 (g=4), A184955 (g=5).
Triangular arrays C(n,g) counting connected simple k-regular graphs on n vertices with girth exactly g: A198303 (k=3), A184940 (k=4), this sequence (k=5), A184960 (k=6), A184970 (k=7), A184980 (k=8).

cp's OEIS Frontend

A051890 a(n) = 2*(n^2 - n + 1).

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

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A198300 Square array M(k,g), read by antidiagonals, of the Moore lower bound on the order of a (k,g)-cage.

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A058276 Number of connected 6-regular simple graphs on n vertices with girth at least 4.

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A184964 Number of connected 6-regular simple graphs on n vertices with girth exactly 4.

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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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A184981 Irregular triangle C(n,g) counting the connected 8-regular simple graphs on n vertices with girth at least g.

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