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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. 17
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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.

REFERENCES

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.

LINKS

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

FORMULA

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.

EXAMPLE

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

MATHEMATICA

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 *)

PROG

(MAGMA)

ExtendedStringToInt:=func<seq, base|&+[Integers()|seq[i]*base^(#seq-i):i in[1..#seq]]>;

M:=func<k, g|ExtendedStringToInt((IsOdd(g)select[1]else[])cat[2^^(g div 2)], k-1)>;

k_:=2; g_:=3;

anti:=func<kg|[M(kg-g, g):g in[g_..kg-k_]]>;

[anti(kg):kg in[5..15]];

CROSSREFS

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).

Sequence in context: A196379 A316353 A204002 * A054760 A079107 A205837

Adjacent sequences:  A198297 A198298 A198299 * A198301 A198302 A198303

KEYWORD

nonn,tabl,easy,base

AUTHOR

Jason Kimberley, Oct 27 2011

STATUS

approved

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Last modified August 18 13:28 EDT 2018. Contains 313832 sequences. (Running on oeis4.)