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A257637
Maximal number of edges in an n-vertex triangle-free graph with maximal degree at most 4.
1
0, 1, 2, 4, 6, 9, 12, 16, 17, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120
OFFSET
1,3
COMMENTS
For n <= 8, a(n) is the Turan number T(3,n), realized by a complete bipartite graph K(m, m) if n = 2m or K(m, m+1) if n = 2m+1, since then that graph has maximal degree <= 4.
For n >= 10, any 4-regular triangle-free graph (i.e., graph with no three-cliques) will do.
For n = 9, there is no 4-regular triangle-free graph, as can be seen from inspection.
FORMULA
a(n) = floor(n^2/4) = A002620(n), if 1 <= n <= 8.
a(9) = 17.
a(n) = 2*n = A005843(n), if n >= 10.
From Chai Wah Wu, Feb 03 2021: (Start)
a(n) = 2*a(n-1) - a(n-2) for n > 11.
G.f.: x*(-x^10 + 2*x^9 - 3*x^8 + x^7 + x^5 + x^3 + x)/(x - 1)^2. (End)
PROG
(PARI) a(n)=if(n>9, 2*n, if(n<9, n^2\4, 17)) \\ Charles R Greathouse IV, Jan 06 2016
CROSSREFS
Sequence in context: A375983 A135146 A256956 * A258027 A053096 A155752
KEYWORD
easy,nonn
AUTHOR
Jörgen Backelin, Nov 04 2015
STATUS
approved