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A080036
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a(n) = n + round(sqrt(2*n)) + 1.
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24
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1, 3, 5, 6, 8, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86
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OFFSET
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0,2
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COMMENTS
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a(0)=1, a(1)=3; for n>1, a(n)=a(n-1)+1 if n is already in the sequence, a(n)=a(n-1)+2 otherwise.
Sequence (without first term) is the complement of A000124 (central polygonal numbers). - Jaroslav Krizek, Jun 16 2009
a(n) is the Ramsey core number rc(2,n). The Ramsey core number rc(s,t) is the smallest n such that for all edge 2-colorings of K_n, either the factor induced by the first color contains an s-core or the second factor contains a t-core. (A k-core is a subgraph with minimum degree at least k.) - Allan Bickle, Mar 29 2023
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REFERENCES
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R. Klein and J. Schönheim, Decomposition of K_{n} into degenerate graphs, In Combinatorics and Graph Theory Hefei 6-27, April 1992. World Scientific. Singapore, New Jersey, London, Hong Kong, 141-1
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LINKS
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FORMULA
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a(n) = ceiling(n + 1/2 + sqrt(2*(n-1)+9/4)). - Allan Bickle, Mar 29 2023
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EXAMPLE
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For order 5, one of the two factors has at least 5 edges, and so contains a cycle. For order 4, K_4 decomposes into two paths. Thus rc(2,2)=5.
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MATHEMATICA
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PROG
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(Python)
from math import isqrt
def A080036(n): return (k:=isqrt(m:=n<<1))+int((m<<2)>(k<<2)*(k+1)+1)+n+1 # Chai Wah Wu, Jul 26 2022
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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