A342940 -id:A342940

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A342938

a(n) is the Skolem number of the cycle graph C_n.

+10
2

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, 16, 17, 16, 16, 17, 18, 19, 20, 20, 19, 20, 21, 21, 22, 23, 22, 24, 25, 24, 25, 25, 25, 27, 28, 28, 29, 28, 28, 29, 30, 31, 32, 32, 31, 32, 33, 33, 34, 35, 34, 36

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

For the meaning of Skolem number of a graph, see Definitions 1.4 and 1.5 in Carrigan and Green.

LINKS

Table of n, a(n) for n=1..70.

Braxton Carrigan and Garrett Green, Skolem Number of Subgraphs on the Triangular Lattice, Communications on Number Theory and Combinatorial Theory 2 (2021), Article 2.

Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-2,2,-1,0,0,0,0,0,-1,2,-2,2,-2,2,-1).

FORMULA

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).

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.

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).

MATHEMATICA

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]

CROSSREFS

Cf. A342939, A342940.

For n > 1, 3*A002061(n) gives the Skolem number of the hexagonal grid graph H_n.

KEYWORD

nonn,easy

AUTHOR

Stefano Spezia, Mar 30 2021

STATUS

approved

A342939

a(n) is the Skolem number of the triangular grid graph T_n.

+10
2

1, 2, 5, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154, 172, 191, 211, 232, 254, 277, 301, 326, 352, 379, 407, 436, 466, 497, 529, 562, 596, 631, 667, 704, 742, 781, 821, 862, 904, 947, 991, 1036, 1082, 1129, 1177, 1226, 1276, 1327, 1379, 1432, 1486

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

For the meaning of Skolem number of a graph, see Definitions 1.4 and 1.5 in Carrigan and Green.

LINKS

Table of n, a(n) for n=1..55.

Braxton Carrigan and Garrett Green, Skolem Number of Subgraphs on the Triangular Lattice, Communications on Number Theory and Combinatorial Theory 2 (2021), Article 2.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

O.g.f.: x*(1 - x + 2*x^2 - 3*x^3 + 3*x^4 - x^5)/(1 - x)^3.

E.g.f.: exp(x)*(2 + x^2)/2 - 1 + x^3/6.

a(n) = 3*a(n-1) - 3*a(n-2) - a(n-3) for n > 6.

Except for a(3) = 5:

a(n) = 1 + n*(n - 1)/2 (see Theorem 2.5 in Carrigan and Green).

a(n) = 1 + A161680(n).

a(n) = A152947(n-1).