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%I #14 Apr 01 2021 14:55:56
%S 1,1,1,3,4,4,5,4,4,5,6,7,8,8,7,8,9,9,10,11,10,12,13,12,13,13,13,15,16,
%T 16,17,16,16,17,18,19,20,20,19,20,21,21,22,23,22,24,25,24,25,25,25,27,
%U 28,28,29,28,28,29,30,31,32,32,31,32,33,33,34,35,34,36
%N a(n) is the Skolem number of the cycle graph C_n.
%C For the meaning of Skolem number of a graph, see Definitions 1.4 and 1.5 in Carrigan and Green.
%H Braxton Carrigan and Garrett Green, <a href="https://research.library.kutztown.edu/contact/vol2/iss1/2/">Skolem Number of Subgraphs on the Triangular Lattice</a>, Communications on Number Theory and Combinatorial Theory 2 (2021), Article 2.
%H <a href="/index/Rec#order_18">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,2,-2,2,-1,0,0,0,0,0,-1,2,-2,2,-2,2,-1).
%F G.f.: x*(1 - x + x^2 + x^3 - x^7 + x^8 - 2*x^9 + 2*x^10 - x^11 + 2*x^12 - x^13 + 2*x^15 - x^16)/(1 - 2*x + 2*x^2 - 2*x^3 + 2*x^4 - 2*x^5 + x^6 + x^12 - 2*x^13 + 2*x^14 - 2*x^15 + 2*x^16 - 2*x^17 + x^18).
%F a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - 2*a(n-4) + 2*a(n-5) - a(n-6) - a(n-12) + 2*a(n-13) - 2*a(n-14) + 2*a(n-15) - 2*a(n-16) + 2*a(n-17) - a(n-18) for n > 18.
%F a(n) = (n - 3)/2 + 1 when n is congruent to 3, 9, 15, 21 mod 24; ceiling(n/2) when n is congruent to 0, 1, 2, 8, 10, 11, 16, 17, 18, 19 mod 24; ceiling(n/2) + 1 when n is congruent to 4, 5, 6, 7, 12, 13, 14, 20, 22, 23 mod 24 (see Theorem 1.6 in Carrigan and Green).
%t a[n_]:=If[MemberQ[{3,9,15,21},Mod[n,24]],(n-3)/2+1,If[MemberQ[{0,1,2,8,10,11,16,17,18,19},Mod[n,24]],Ceiling[n/2],Ceiling[n/2]+1]]; Array[a,70]
%Y Cf. A342939, A342940.
%Y For n > 1, 3*A002061(n) gives the Skolem number of the hexagonal grid graph H_n.
%K nonn,easy
%O 1,4
%A _Stefano Spezia_, Mar 30 2021