A124353 -id:A124353 - cp's OEIS frontend

A124352 Number of directed Hamiltonian paths on the n-antiprism graph.

240, 816, 2400, 6480, 16660, 41440, 100836, 241520, 571692, 1340832, 3121456, 7222040, 16622220, 38085312, 86918688, 197677368, 448182640, 1013320480, 2285339532, 5142429512, 11547488652, 25881229248, 57906534100, 129352490384, 288522099360, 642668803056, 1429687270740, 3176714397960

Offset: 3

Eric W. Weisstein, Oct 27 2006

nonn

Andrew Howroyd, Table of n, a(n) for n = 3..50
Eric Weisstein's World of Mathematics, Antiprism Graph
Eric Weisstein's World of Mathematics, Hamiltonian Path

Cf. A124353 (numbers of directed Hamiltonian cycles).

Conjectures from Colin Barker, Dec 21 2015: (Start)
a(n) = 5*a(n-1)-6*a(n-2)-4*a(n-3)+7*a(n-4)+5*a(n-5)-5*a(n-6)-3*a(n-7)+a(n-8)+a(n-9) for n>11.
G.f.: 4*x^3*(60-96*x-60*x^2+84*x^3+61*x^4-73*x^5-41*x^6+15*x^7+14*x^8) / ((1-x)^3*(1-x-2*x^2-x^3)^2).
(End)
Equivalent conjecture: a(n) = 2*a(n-1) +3*a(n-2) -2*a(n-3) -6*a(n-4) -4*a(n-5) -a(n-6) +672*n -1376 -72*n^2 if n>=9. - R. J. Mathar, Jan 25 2016

a(6)-a(10) from Eric W. Weisstein, Apr 03 2008
a(11)-a(18) from Eric W. Weisstein, Dec 16 2013
a(19)-a(30) from Andrew Howroyd, Dec 20 2015

A137726 Number of sequences of length n with elements {-2,-1,+1,+2}, counted up to simultaneous reversal and negation, 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 (undirected) cycles on the circulant graph C_n(1,2).

Original entry on oeis.org

2, 2, 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, 4367986, 6401578, 9381949, 13749897, 20151433, 29533342

Offset: 1

Max Alekseyev, Feb 08 2008

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

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.
Spoj, Problem 7709: Jumping Zippy
Eric Weisstein's World of Mathematics, Circulant Graph
Index entries for linear recurrences with constant coefficients, signature (2,0,-1,0,-1,1).

Cf. A069241, A124353, A137725.

Rest[CoefficientList[Series[-x*(x^7 + 2*x^5 - 4*x^4 - 5*x^3 + 4*x^2 - 2*x + 2)/((x - 1)^2*(x + 1)*(x^3 + x - 1)), {x, 0, 50}], x]] (* G. C. Greubel, Apr 27 2017 *)

x='x+O('x^50); Vec(-x*(x^7 + 2*x^5 - 4*x^4 - 5*x^3 + 4*x^2 - 2*x + 2)/((x - 1)^2*(x + 1)*(x^3 + x - 1))) \\ G. C. Greubel, Apr 27 2017

For even n>=4, a(n) = n + 3*A000930(n) - 2*A000930(n-1); for odd n>=3, a(n) = 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) - 2.
a(n) = A137725(n) / 2.
G.f.: -x*(x^7+2*x^5-4*x^4-5*x^3+4*x^2-2*x+2)/((x-1)^2*(x+1)*(x^3+x-1)). - Colin Barker, Aug 22 2012

Formulae corrected by Max Alekseyev, Nov 03 2010

A338154 a(n) is the number of acyclic orientations of the edges of the n-antiprism.

Original entry on oeis.org

426, 4968, 50640, 486930, 4547088, 41796168, 380789562, 3451622904, 31194607488, 281440825122, 2536622917920, 22848990484344, 205743704494026, 1852238413383048, 16673036119790640, 150072652217086770, 1350735146332489008, 12157047307392618408

Offset: 3

Peter Kagey, Oct 13 2020

nonn

Conjectured linear recurrence and g.f. confirmed by Kagey's formula. - Ray Chandler, Mar 10 2024

			For n = 3, the 3-antiprism is the octahedron (3-dimensional cross-polytope), so a(3) = A033815(3) = 426.

Peter Kagey, Table of n, a(n) for n = 3..1000
Eric Weisstein's World of Mathematics, Antiprism Graph
Wikipedia, Acyclic orientation
Index entries for linear recurrences with constant coefficients, signature (17, -88, 153, -81).

Cf. A077263, A124352, A124353, A192742, A284699, A284700, A284701, A287988, A294152, A297384.

Cf. A033815 (cross-polytope), A058809 (wheel), A334247 (hypercube), A338152 (demihypercube), A338153 (prism).

A338154[n_] := Round[-2^(1-n)*((7 - Sqrt[13])^n + (7 + Sqrt[13])^n) + 9^n + 5] (* Peter Kagey, Nov 15 2020 *)

Conjectures from Colin Barker, Oct 13 2020: (Start)
G.f.: 6*x^3*(71 - 379*x + 612*x^2 - 324*x^3) / ((1 - x)*(1 - 9*x)*(1 - 7*x + 9*x^2)).
a(n) = 17*a(n-1) - 88*a(n-2) + 153*a(n-3) - 81*a(n-4) for n>6.
(End)
a(n) = -2^(1-n)*((7-sqrt(13))^n + (7+sqrt(13))^n) + 9^n + 5. - Peter Kagey, Nov 15 2020

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

A287988 Number of (undirected) paths in the n-antiprism graph.

Original entry on oeis.org

2, 56, 396, 2040, 9130, 37944, 151172, 586608, 2235618, 8407640, 31292844, 115494312, 423283562, 1542120664, 5589611460, 20170172896, 72499928322, 259692909048, 927342338956, 3302291258200, 11730149911914, 41572470711288, 147031327493572, 519029653663056

Offset: 1

Eric W. Weisstein, Jun 03 2017

nonn

Sequence extrapolated to n=1 using recurrence. - Andrew Howroyd, Jun 05 2017

Andrew Howroyd, Table of n, a(n) for n = 1..200
Eric Weisstein's World of Mathematics, Antiprism Graph
Eric Weisstein's World of Mathematics, Graph Path
Index entries for linear recurrences with constant coefficients, signature (10, -37, 64, -58, 36, -26, 16, -5, 2, -1).

Cf. A077263, A124352, A124353, A287992.

Table[n RootSum[-1 - # - 3 #^2 + #^3 &, 23 #^n + 32 #^(n + 1) + 5 #^(n + 2) &]/44 - 7 n - 3 n^2 - 2 n^3, {n, 20}]
LinearRecurrence[{10, -37, 64, -58, 36, -26, 16, -5, 2, -1}, {2, 56, 396, 2040, 9130, 37944, 151172, 586608, 2235618, 8407640}, 20]
CoefficientList[Series[(2 (2 x^6 + 4 x^5 + x^4 + 24 x^3 - 4 x^2 + 20 x + 1) (1 - 2 x - x^2))/((1 - x)^4 (1 - 3 x - x^2 - x^3)^2), {x, 0, 20}], x]

Vec(2*(2*x^6+4*x^5+x^4+24*x^3-4*x^2+20*x+1)*(1-2*x-x^2)/((1-x)^4*(1-3*x-x^2-x^3)^2) + O(x^20)) \\ Andrew Howroyd, Jun 05 2017

From Andrew Howroyd, Jun 05 2017 (Start)
a(n) = 10*a(n-1)-37*a(n-2)+64*a(n-3) -58*a(n-4)+36*a(n-5)-26*a(n-6) +16*a(n-7)-5*a(n-8) +2*a(n-9)-a(n-10) for n>10.
G.f.: 2*x*(2*x^6+4*x^5+x^4+24*x^3-4*x^2+20*x+1) * (1-2*x-x^2) / ((1-x)^4 * (1-3*x-x^2-x^3)^2).
(End)

a(1)-a(2) and a(14)-a(24) from Andrew Howroyd, Jun 05 2017

A306447 Number of (undirected) Hamiltonian cycles in the n-antiprism graph.

Original entry on oeis.org

3, 9, 16, 29, 56, 110, 225, 469, 991, 2110, 4511, 9666, 20736, 44511, 95575, 205253, 440828, 946817, 2033628, 4367986, 9381949, 20151433, 43283195, 92967882, 199685571, 428904390, 921243268, 1978737467, 4250128235, 9128846273, 19607840040, 42115660645

Offset: 1

Eric W. Weisstein, May 04 2019

Extended to a(1)-a(2) using the formula/recurrence.

Colin Barker, Table of n, a(n) for n = 1..1000
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).

LinearRecurrence[{3,-1,-2,0,1},{3,9,16,29,56},40] (* Harvey P. Dale, Aug 06 2020 *)

Vec(x*(3 - 8*x^2 - 4*x^3 + 3*x^4) / ((1 - x)^2*(1 - x - 2*x^2 - x^3)) + O(x^35)) \\ Colin Barker, Jul 12 2020

a(n) = A124353(n)/2.
From Colin Barker, Jul 12 2020: (Start)
G.f.: x*(3 - 8*x^2 - 4*x^3 + 3*x^4) / ((1 - x)^2*(1 - x - 2*x^2 - x^3)).
a(n) = 3*a(n-1) - a(n-2) - 2*a(n-3) + a(n-5) for n>5.
(End)

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A124352 Number of directed Hamiltonian paths on the n-antiprism graph.

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A338154 a(n) is the number of acyclic orientations of the edges of the n-antiprism.

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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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A287988 Number of (undirected) paths in the n-antiprism graph.

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A306447 Number of (undirected) Hamiltonian cycles in the n-antiprism graph.

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