A137726 -id:A137726 - cp's OEIS frontend

A124353 Number of (directed) Hamiltonian circuits on the n-antiprism graph.

6, 18, 32, 58, 112, 220, 450, 938, 1982, 4220, 9022, 19332, 41472, 89022, 191150, 410506, 881656, 1893634, 4067256, 8735972, 18763898, 40302866, 86566390, 185935764, 399371142, 857808780, 1842486536, 3957474934, 8500256470, 18257692546, 39215680080, 84231321290, 180920373632, 388598695916

Offset: 1

Eric W. Weisstein, Oct 27 2006

nonn

The antiprism graph is defined for n>=3; extended to n=1 using the closed form.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Max A. Alekseyev, Gérard P. Michon, Making Walks Count: From Silent Circles to Hamiltonian Cycles, arXiv:1602.01396 [math.CO], 2016-2017.
Mordecai J. Golin and Yiu Cho Leung, Unhooking Circulant Graphs: A Combinatorial Method for Counting Spanning Trees, Hamiltonian Cycles and other Parameters, Technical report HKUST-TCSC-2004-02. See also.
Eric Weisstein's World of Mathematics, Antiprism Graph
Eric Weisstein's World of Mathematics, Hamiltonian Cycle
Index entries for linear recurrences with constant coefficients, signature (3,-1,-2,0,1).

Cf. A124352.

I:=[6,18,32,58,112]; [n le 5 select I[n] else 3*Self(n-1) - Self(n-2) - 2*Self(n-3) + Self(n-5): n in [1..35]]; // Vincenzo Librandi, Feb 04 2016

Table[2 (2 n + RootSum[-1 - 2 # - #^2 + #^3 &, #^n &]), {n, 20}]
LinearRecurrence[{3, -1, -2, 0, 1}, {6, 18, 32, 58, 112}, 50] (* Vincenzo Librandi, Feb 04 2016 *)
Join[{6, 18}, Rest[Rest[Rest[CoefficientList[Series[-18*x^2 - 6*x - 6 + (4*x^2 + 4*x - 6)/(x^3 + 2*x^2 + x - 1) + 4/(x - 1)^2 + 4/(x - 1), {x, 0, 50}], x]]]]] (* G. C. Greubel, Apr 27 2017 *)

x='x+O('x^50); concat([6,18], Vec(-18*x^2-6*x-6+(4*x^2+4*x-6)/(x^3+2*x^2+x-1)+4/(x-1)^2+4/(x-1))) \\ G. C. Greubel, Apr 27 2017

a(n) = 3*a(n-1) - a(n-2) - 2*a(n-3) + a(n-5).
a(n) = 2*a(n-1) + a(n-2) - a(n-3) - a(n-4) - 12.
O.g.f.: -18*x^2-6*x-6+(4*x^2+4*x-6)/(x^3+2*x^2+x-1)+4/(x-1)^2+4/(x-1) . - R. J. Mathar, Feb 10 2008
a(n) = 2*(n + 3*A000930(2*n) - 2*A000930(2*n-1)) = A137725(2*n) = 2*A137726(2*n).

Formulas and further terms from Max Alekseyev, Feb 08 2008

Typo in formula corrected by Max Alekseyev, Nov 03 2010

A137725 Number of sequences of length n with elements {-2,-1,+1,+2}, such that the sum of elements of the whole sequence but of no proper subsequence equals 0 modulo n. For n>=4, the number of Hamiltonian (directed) circuits on the circulant graph C_n(1,2).

Original entry on oeis.org

4, 4, 16, 18, 24, 32, 46, 58, 82, 112, 158, 220, 316, 450, 650, 938, 1364, 1982, 2892, 4220, 6170, 9022, 13206, 19332, 28314, 41472, 60760, 89022, 130446, 191150, 280120, 410506, 601600, 881656, 1292102, 1893634, 2775226, 4067256, 5960822, 8735972, 12803156, 18763898, 27499794, 40302866, 59066684

Offset: 1

Max Alekseyev, Feb 08 2008

nonn

Number of 1-D walks with jumps to next-nearest neighbors with n steps, starting at 0 and ending at -2n, -n, 0, n, or 2n, such that every point is visited at most once and every pair of points at the distance n contains at least one unvisited point (not counting the ending visit). Cf. A092765.

For n>1, the number of circular permutations (counted up to rotations) of {0, 1,...,n-1} such that the distance between every two adjacent elements is -2,-1,1,or 2 modulo n. Cf. A003274.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Mordecai J. Golin and Yiu Cho Leung, Unhooking Circulant Graphs: A Combinatorial Method for Counting Spanning Trees, Hamiltonian Cycles and other Parameters, Technical report HKUST-TCSC-2004-02. See also
Eric Weisstein's World of Mathematics, Circulant Graph
Eric Weisstein's World of Mathematics, Hamiltonian Cycle
Index entries for linear recurrences with constant coefficients, signature (2,0,-1,0,-1,1).

Cf. A124353, A137726.

CoefficientList[Series[-2*x^2-2*x-6-1/(x+1)+2/(x-1)^2+1/(x-1)+(4*x-6)/(x^3+x-1), {x, 0, 50}], x] (* G. C. Greubel, Apr 28 2017 *)

my(x='x+O('x^50)); Vec(-2*x^2-2*x-6-1/(x+1)+2/(x-1)^2+1/(x-1)+(4*x-6)/(x^3+x-1)) \\ G. C. Greubel, Apr 28 2017

For even n>=4, a(n) = 2*(n + 3*A000930(n) - 2*A000930(n-1)); for odd n>=3, a(n) = 2*(n + 1 + 3*A000930(n) - 2*A000930(n-1)).
For n>8, a(n) = 2*a(n-1) - a(n-3) - a(n-5) + a(n-6) or a(n) = a(n-1) + a(n-2) - a(n-5) - 4.
O.g.f.: -2*x^2-2*x-6-1/(x+1)+2/(x-1)^2+1/(x-1)+(4*x-6)/(x^3+x-1). - R. J. Mathar, Feb 10 2008

Typo in formulas corrected by Max Alekseyev, Nov 03 2010

A226509 Expansion of (3-2*x)/(1-x-x^3)+x/(1-x)^2+x/(1-x^2).

Original entry on oeis.org

3, 3, 3, 8, 9, 12, 16, 23, 29, 41, 56, 79, 110, 158, 225, 325, 469, 682, 991, 1446, 2110, 3085, 4511, 6603, 9666, 14157, 20736, 30380, 44511, 65223, 95575, 140060, 205253, 300800, 440828, 646051, 946817, 1387613, 2033628, 2980411

Offset: 0

N. J. A. Sloane, Jun 11 2013

Z. Skupien, Sums of Powered Characteristic Roots Count Distance-Independent Circular Sets, Discussiones Mathematicae Graph Theory. Volume 33, Issue 1, Pages 217-229, ISSN (Print) 2083-5892, DOI: 10.7151/dmgt.1658, April 2013.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-1,0,-1,1).

CoefficientList[Series[(3 - 2 x) / (1 - x - x^3) + x / (1 - x)^2 + x / (1 - x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 19 2013 *)

a(n) = A137726(n), for n>2 and a(0) = a(1) = a(2) = 3. [Bruno Berselli, Jun 17 2013]

cp's OEIS Frontend

A124353 Number of (directed) Hamiltonian circuits on the n-antiprism graph.

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A137725 Number of sequences of length n with elements {-2,-1,+1,+2}, such that the sum of elements of the whole sequence but of no proper subsequence equals 0 modulo n. For n>=4, the number of Hamiltonian (directed) circuits on the circulant graph C_n(1,2).

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Mathematica

PARI

Formula

Extensions

A226509 Expansion of (3-2*x)/(1-x-x^3)+x/(1-x)^2+x/(1-x^2).

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