A002898 -id:A002898 - cp's OEIS frontend

A337907 The number of walks of n steps on the hexagonal lattice that start at the origin and end at the non-adjacent vertex (3/2,sqrt(3)/2).

2, 6, 48, 220, 1320, 6930, 39200, 215208, 1208340, 6754440, 38076192, 214939296, 1218641424, 6925848930, 39477746880, 225542306704, 1291514481972, 7410367503396, 42599109627360, 245305128355560, 1414839151645920, 8172376003368720, 47270088643265280, 273766119948648000

Offset: 2

R. J. Mathar, Sep 29 2020

			There are a(2)=2 paths with 2 steps: RU and UR, where R=(1,0), L=(-1,0), U=(1/2,sqrt(3)/2), u=(-1/2,sqrt(3)/2), D=(1/2,-sqrt(3)/2), d=(-1/2,-sqrt(3)/2).
There are a(3)=6 paths with 3 steps: UUD, UDU, DUU, RRu, RuR, uRR.

Cf. A002898 (returns to origin), A337905, A337906.

Maple
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# see A337905
```

D-finite with recurrence -(n-2)*(n+3)*(n+2)*(n+1)*a(n) +n*(n+2)*(n^2+n+12)*a(n-1) +24*n*(n-1)*(n^2+3*n-1)*a(n-2) +36*n*(n-1)*(n-2)*(n+4)*a(n-3)=0.

A094060 Number of walks of length n on hexagonal grid that start and end at the origin. Intermediate returns to the origin are not permitted.

Original entry on oeis.org

1, 0, 6, 12, 54, 216, 1032, 4896, 24606, 125040, 651348, 3432168, 18331992, 98814816, 537343632, 2942475552, 16214888286, 89835783264, 500116783740, 2795958732024, 15690597591636, 88354191756816, 499060719941616, 2826794871554112, 16052536475622792

Offset: 0

Gareth McCaughan, Jun 10 2004

Alois P. Heinz, Table of n, a(n) for n = 0..1291 (first 97 terms from Ludovic Schwob)

Cf. A002898, A054474.

b:= proc(n) option remember; `if`(n<3, [1,0,6][n+1], ((n-1)*
      n*b(n-1) +24*(n-1)^2*b(n-2) +36*(n-1)*(n-2)*b(n-3))/n^2)
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      b(n)-add(a(n-i)*b(i), i=1..n-1))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Dec 07 2020

b[n_] := b[n] = If[n<3, {1, 0, 6}[[n+1]], ((n-1)n b[n-1] + 24(n-1)^2* b[n-2] + 36(n-1)(n-2) b[n-3])/n^2];
a[n_] := a[n] = If[n==0, 1, b[n] - Sum[a[n-i] b[i], {i, 1, n-1}]];
a /@ Range[0, 23] (* Jean-François Alcover, Jan 14 2021, after Alois P. Heinz *)

seq(n)={my(g=sum(m=0, n, (3*m)!/m!^3*x^(2*m)*(1+2*x)^m, O(x*x^n))); Vec(2-1/g)} \\ Andrew Howroyd, Aug 09 2025

INVERTi transform of A002898. - R. J. Mathar, Sep 29 2020

A242743 Decimal expansion of an Ising constant related to the random coloring problem.

Original entry on oeis.org

9, 2, 9, 6, 9, 5, 3, 9, 8, 3, 4, 1, 6, 1, 0, 2, 1, 4, 9, 8, 5, 3, 8, 4, 9, 7, 3, 6, 6, 5, 8, 7, 8, 1, 2, 1, 7, 6, 5, 8, 8, 2, 3, 7, 5, 1, 5, 1, 8, 0, 2, 1, 6, 7, 5, 8, 2, 3, 4, 3, 1, 4, 2, 9, 7, 0, 1, 9, 2, 0, 8, 4, 7, 4, 5, 7, 2, 5, 0, 8, 1, 2, 5, 6, 1, 8, 5, 3, 1, 2, 1, 0, 4, 4, 7, 7, 4, 6, 1, 5, 8, 8, 9, 4, 6

Offset: 0

Jean-François Alcover, Jun 02 2014

In Ising model on the 2D square lattice, the negated ratio of free energy per node to the temperature at the critical point. - Andrey Zabolotskiy, Sep 12 2017

			0.929695398341610214985384973665878...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.22 Lenz-Ising constants, p. 399.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Lars Onsager, Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition, Phys. Rev. 65, 117 (1944) [see eq. (121)].

Cf. A002898, A006752, A218387, A245592.

SetDefaultRealField(RealField(100)); R:=RealField(); Log(2)/2 + 2*Catalan(R)/Pi(R); // G. C. Greubel, Aug 25 2018

RealDigits[Log[2]/2 + 2*Catalan/Pi, 10, 105] // First

default(realprecision, 100); log(2)/2 + 2*Catalan/Pi \\ G. C. Greubel, Aug 25 2018

log(2)/2 + 2*G/Pi = log(2)/2 + A218387/2, where G is Catalan's constant.

A292881 Number of n-step closed paths on the E6 lattice.

Original entry on oeis.org

1, 0, 72, 1440, 54216, 2134080, 93993120, 4423628160, 219463602120, 11341793393280

Offset: 0

Samuel Savitz, Sep 26 2017

Calculated by brute force computational enumeration.

The moments of the imaginary part of the suitably normalized E6 lattice Green's function.

			The 2-step walks consist of hopping to one of the 72 minimal vectors of the E6 lattice and then back to the origin.

S. Savitz and M. Bintz, Exceptional Lattice Green's Functions, arXiv:1710.10260 [math-ph], 2017.

Cf. A126869 (Linear A1 lattice), A002898 (Hexagonal A2), A002899 (FCC A3), A271432 (D4), A271650 (D5), A271651 (D6), A292882 (E7), A271670 (D7), A292883 (E8), A271671 (D8).

Summed combinatorial expressions and recurrence relations for this sequence exist, but have not been determined. These would allow one to write a differential equation or hypergeometric expression for the E6 lattice Green's function.

A292882 Number of n-step closed paths on the E7 lattice.

Original entry on oeis.org

1, 0, 126, 4032, 228690, 14394240, 1020623940, 78353170560, 6393827197170

Offset: 0

Samuel Savitz, Sep 26 2017

Calculated by brute force computational enumeration.

The moments of the imaginary part of the suitably normalized E7 lattice Green's function.

			The 2-step walks consist of hopping to one of the 126 minimal vectors of the E7 lattice and then back to the origin.

S. Savitz and M. Bintz, Exceptional Lattice Green's Functions, arXiv:1710.10260 [math-ph], 2017.

Cf. A126869 (Linear A1 lattice), A002898 (Hexagonal A2), A002899 (FCC A3), A271432 (D4), A271650 (D5), A292881 (E6), A271651 (D6), A271670 (D7), A292883 (E8), A271671 (D8).

Summed combinatorial expressions and recurrence relations for this sequence exist, but have not been determined. These would allow one to write a differential equation or hypergeometric expression for the E7 lattice Green's function.

A292883 Number of n-step closed paths on the E8 lattice.

Original entry on oeis.org

1, 0, 240, 13440, 1260720, 137813760, 17141798400, 2336327078400, 341350907713200

Offset: 0

Samuel Savitz, Sep 26 2017

Calculated by brute force computational enumeration.

The moments of the imaginary part of the suitably normalized E8 lattice Green's function.

			The 2-step walks consist of hopping to one of the 240 minimal vectors of the E8 lattice and then back to the origin.

S. Savitz and M. Bintz, Exceptional Lattice Green's Functions, arXiv:1710.10260 [math-ph], 2017.

Cf. A126869 (Linear A1 lattice), A002898 (Hexagonal A2), A002899 (FCC A3), A271432 (D4), A271650 (D5), A292881 (E6), A271651 (D6), A292882 (E7), A271670 (D7), A271671 (D8).

Summed combinatorial expressions and recurrence relations for this sequence exist, but have not been determined. These would allow one to write a differential equation or hypergeometric expression for the E8 lattice Green's function.

A328735 Constant term in the expansion of (x + y + z + 1/x + 1/y + 1/z + xy + yz + zx + 1/(xy) + 1/(yz) + 1/(zx) + xyz + 1/(xyz))^n.

Original entry on oeis.org

1, 0, 14, 72, 882, 8400, 95180, 1060080, 12389650, 146472480, 1767391164, 21581516880, 266718438756, 3327025429728, 41849031952728, 530135326392672, 6757845419895570, 86619827323917888, 1115719258312182524, 14434274832755201424, 187477238295444829732

Offset: 0

Seiichi Manyama, Oct 26 2019

nonn

Column k=4 of A328748.
Sum_{i=0..n} (-2)^(n-i)*binomial(n,i)*Sum_{j=0..i} binomial(i,j)^m: A126869 (m=2), A002898 (m=3), this sequence (m=4), A328751 (m=5).
Cf. A002899, A005260, A328725.

Table[Sum[(-2)^(n-i)*Binomial[n,i] * Sum[Binomial[i,j]^4, {j,0,i}], {i,0,n}], {n,0,20}] (* Vaclav Kotesovec, Mar 20 2023 *)

{a(n) = polcoef(polcoef(polcoef((-2+(1+x)*(1+y)*(1+z)+(1+1/x)*(1+1/y)*(1+1/z))^n, 0), 0), 0)}

{a(n) = sum(i=0, n, (-2)^(n-i)*binomial(n, i)*sum(j=0, i, binomial(i, j)^4))}

a(n) = Sum_{i=0..n} (-2)^(n-i)*binomial(n,i)*Sum_{j=0..i} binomial(i,j)^4.
From Vaclav Kotesovec, Mar 20 2023: (Start)
Recurrence: n^3*a(n) = 2*(n-1)*n*(2*n - 1)*a(n-1) + 112*(n-1)^3*a(n-2) + 184*(n-2)*(n-1)*(2*n - 3)*a(n-3) + 336*(n-3)*(n-2)*(n-1)*a(n-4).
a(n) ~ 2^(n-4) * 7^(n + 3/2) / (Pi^(3/2) * n^(3/2)). (End)

A337906 The number of walks of n steps on the hexagonal lattice that start at the origin and end at the non-adjacent vertex (2,0).

Original entry on oeis.org

1, 6, 34, 200, 1095, 6230, 34636, 195552, 1099140, 6218520, 35210736, 200099328, 1139401263, 6504768270, 37211008120, 213311576192, 1225053737622, 7047867363108, 40612374024300, 234376628718960, 1354498970905080, 7838134441085520, 45412879702456800, 263417461793120000

Offset: 2

R. J. Mathar, Sep 29 2020

			There is a(2)=1 path with 2 steps: RR, where R=(1,0), L=(-1,0), U=(1/2,sqrt(3)/2), u=(-1/2,sqrt(3)/2), D=(1/2,-sqrt(3)/2), d=(-1/2,-sqrt(3)/2).
There are a(3)=6 paths with 3 steps: RUD, RDU, DRU, DUR, URD, UDR.

Cf. A002898 (returns to origin), A337905, A337907.

Maple
```
# see A337905
```

HexLat[n_, finx_, finy_] := Module[{a = 0, L, R}, For[L = 0, L <= n, L++, For[R = Mod[n + finy - L, 2], R <= n - L , R += 2, a = a + Binomial[n, L]*Binomial[n - L, R]*Binomial[n - L - R, n/2 + L/2 - 3*R/2 + finx]*Binomial[n - L - R, (n - L - R - finy)/2]]]; a];
Table[HexLat[n, 2, 0], {n, 2, 25}] (* Jean-François Alcover, Jun 25 2023, after R. J. Mathar in A337905 *)

D-finite with recurrence (n-2)*(3*n^2-5*n-20)*(n+2)^2*a(n) -n*(3*n^4-2*n^3+n^2-130*n-208)*a(n-1) -24*n*(n-1)*(n-3)*(3*n^2+7*n-2)*a(n-2) -36*n*(n-1)*(n-2)*(3*n^2+n-22)*a(n-3)=0.

a(n) ~ 2^(n-1) * 3^(n + 1/2) / (Pi*n). - Vaclav Kotesovec, Apr 30 2024

A057648 Number of excursions of length n on the upper-right part of the hexagonal lattice.

Original entry on oeis.org

1, 0, 2, 2, 13, 34, 158, 594, 2665, 11558, 53320, 247488, 1181266, 5708884, 28049474, 139417402, 701063005, 3559326294, 18233244530, 94140532624, 489573775236, 2562613997512, 13493827469116, 71441865994904

Offset: 0

Cyril Banderier, Oct 12 2000

nonn

Excursions = walks from the origin to the origin.

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice. - Sean A. Irvine, Jun 22 2022

Robert Israel, Table of n, a(n) for n = 0..1000
C. Banderier, Analytic combinatorics of random walks and planar maps, PhD Thesis, 2001.

Cf. A002898, A057647.

gf:=(1-2*x)*hypergeom([-1/2, 1/2],[2],16*x^2/(1-2*x)^2)/(4*x^2) - (2*x+1)*((1-6*x)*hypergeom([1/3, 2/3],[2],27*x^2*(2*x+1))+1/2)/(6*x^2):
S:= series(gf,x,103):
seq(coeff(S,x,j),j=0..100); # Robert Israel, Dec 08 2014

G.f.: (1-2*x)*hypergeom([-1/2, 1/2],[2],16*x^2/(1-2*x)^2)/(4*x^2) - (2*x+1)*((1-6*x)*hypergeom([1/3, 2/3],[2],27*x^2*(2*x+1))+1/2)/(6*x^2). - Mark van Hoeij, Dec 08 2014

a(n) ~ (2*sqrt(3) - 3) * 2^n * 3^(n+2) / (Pi*n^3). - Vaclav Kotesovec, Apr 30 2024

Title corrected by Sean A. Irvine, Jun 22 2022

A198802 Number of closed paths of length n whose steps are 18th roots of unity, U_18(n).

Original entry on oeis.org

1, 0, 18, 36, 918, 5400, 82800, 801360, 10907190, 132053040, 1802041668, 24199809480, 340640607384, 4834708246368, 70229958125184, 1032223723667136, 15391538570569590, 231935110984687968, 3531542904056225916, 54244559313713885688, 839979883121036697468

Offset: 0

Simon Plouffe, Oct 30 2011

nonn

U_18(n), comment in article: For each m >= 1, the sequence (U_m(N)), N >= 0 is P-recursive but is not algebraic when m > 2.

Andrew Howroyd, Table of n, a(n) for n = 0..200
Gilbert Labelle and Annie Lacasse, Closed paths whose steps are roots of unity, in FPSAC 2011, Reykjavik, Iceland DMTCS proc. AO, 2011, 599-610.

Cf. A002898, A198808.

seq(n)={Vec(serlaplace(sum(k=0, n, if(k,2,1)*(x^k*besseli(k, 2*x + O(x^(n-k+1)))/k!)^3)^3))} \\ Andrew Howroyd, Nov 01 2018

E.g.f.: g(x)^3 where g(x) is the e.g.f. of A002898.

a(n) ~ 2^(n-3) * 3^(2*n + 9/2) / (Pi^3 * n^3). - Vaclav Kotesovec, Apr 30 2024

cp's OEIS Frontend

A337907 The number of walks of n steps on the hexagonal lattice that start at the origin and end at the non-adjacent vertex (3/2,sqrt(3)/2).

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A094060 Number of walks of length n on hexagonal grid that start and end at the origin. Intermediate returns to the origin are not permitted.

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A242743 Decimal expansion of an Ising constant related to the random coloring problem.

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A292881 Number of n-step closed paths on the E6 lattice.

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A292882 Number of n-step closed paths on the E7 lattice.

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A292883 Number of n-step closed paths on the E8 lattice.

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A328735 Constant term in the expansion of (x + y + z + 1/x + 1/y + 1/z + x*y + y*z + z*x + 1/(x*y) + 1/(y*z) + 1/(z*x) + x*y*z + 1/(x*y*z))^n.

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A337906 The number of walks of n steps on the hexagonal lattice that start at the origin and end at the non-adjacent vertex (2,0).

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Maple

A328735 Constant term in the expansion of (x + y + z + 1/x + 1/y + 1/z + xy + yz + zx + 1/(xy) + 1/(yz) + 1/(zx) + xyz + 1/(xyz))^n.