A327871 -id:A327871 - cp's OEIS frontend

A327872 Total number of nodes in all self-avoiding planar walks starting at (0,0), ending at (n,n), remaining in the first quadrant and using steps (0,1), (-1,1), and (1,-1) with the restriction that (-1,1) and (1,-1) are always immediately followed by (0,1).

1, 4, 21, 148, 980, 6444, 41888, 270088, 1730079, 11023480, 69930146, 441988260, 2784820519, 17499028820, 109701885600, 686313858480, 4285914086100, 26721615383496, 166361793070466, 1034375862301240, 6423778211164860, 39850734775066644, 246976735839649218

Offset: 0

Alois P. Heinz, Sep 28 2019

Alois P. Heinz, Table of n, a(n) for n = 0..1276
Alois P. Heinz, Animation of A327871(5) = 369 walks with a(5) = 6444 nodes
Wikipedia, Lattice path
Wikipedia, Self-avoiding walk

Cf. A327871.

b:= proc(x, y, t) option remember; (p-> p+[0, p[1]])(`if`(
       min(x, y)<0, 0, `if`(max(x, y)=0, [1, 0], b(x-1, y, 1)+
      `if`(t=1, b(x-1, y+1, 0)+b(x+1, y-1, 0), 0))))
    end:
a:= n-> b(n$2, 0)[2]:
seq(a(n), n=0..25);

b[x_, y_, t_] := b[x, y, t] = Function[p, p + {0, p[[1]]}][If[Min[x, y] < 0, {0, 0}, If[Max[x, y] == 0, {1, 0}, b[x - 1, y, 1] + If[t == 1, b[x - 1, y + 1, 0] + b[x + 1, y - 1, 0], 0]]]];
a[n_] := b[n, n, 0][[2]];
a /@ Range[0, 25] (* Jean-François Alcover, May 13 2020, after Maple *)

a(n) ~ sqrt(113 - 179/sqrt(13)) * (70 + 26*sqrt(13))^n * sqrt(n) / (sqrt(Pi) * 2^(3/2) * 3^(3*n + 3/2)). - Vaclav Kotesovec, Oct 12 2019

A344394 a(n) = binomial(n, n/2 - 1/4 + (-1)^n/4)*hypergeom([-n/4 - 1/8 + (-1)^n/8, -n/4 + 3/8 + (-1)^n/8], [n/2 + 7/4 + (-1)^n/4], 4).

Original entry on oeis.org

1, 1, 2, 5, 9, 25, 44, 133, 230, 726, 1242, 4037, 6853, 22737, 38376, 129285, 217242, 740554, 1239980, 4266830, 7123765, 24701425, 41141916, 143567173, 238637282, 837212650, 1389206210, 4896136845, 8112107475, 28703894775, 47495492400, 168640510725, 278722764954

Offset: 0

Peter Luschny, May 19 2021

nonn

Related to the Motzkin triangle A064189 counting certain lattice paths.

Cf. A026300, A064189, A026302 (even bisection), A344396 (odd bisection), A327871.

alias(C=binomial):
a := n -> add(C(n, j)*(C(n - j, j + n/2 - 1/4 + (-1)^n/4) - C(n - j, j + n/2 + 7/4 + (-1)^n/4)), j = 0..n): seq(a(n), n = 0..32);

a[n_] := Binomial[n, n/2 - 1/4 + (-1)^n/4] Hypergeometric2F1[-n/4 - 1/8 + (-1)^n/8, -n/4 + 3/8 + (-1)^n/8, n/2 + 7/4 + (-1)^n/4, 4];
Table[a[n], {n, 0, 32}]

a(n) = Sum_{j = 0..n} C(n, j)*(C(n - j, j + n/2 - 1/4 + (-1)^n/4) - C(n - j, j + n/2 + 7/4 + (-1)^n/4)).
a(n) = A064189(n, floor(n/2)), the middle column of the Motzkin triangle.
a(n) = A026300(n, ceiling(n/2)).

A344396 a(n) = binomial(2n + 1, n)hypergeom([-(n + 1)/2, -n/2], [n + 2], 4).

Original entry on oeis.org

1, 5, 25, 133, 726, 4037, 22737, 129285, 740554, 4266830, 24701425, 143567173, 837212650, 4896136845, 28703894775, 168640510725, 992671051482, 5853000551090, 34562387229046, 204368928058958, 1209916827501876, 7170955214476509, 42543879586512435, 252638095187722437

Offset: 0

Peter Luschny, May 19 2021

nonn

Related to the Motzkin triangle A064189 counting certain lattice paths.

Cf. A027908, A064189, A026300, A344394, A327871.

alias(C=binomial):
a := n -> add(C(2*n + 1, j)*(C(2*n + 1 - j, j + n) - C(2*n + 1 - j, j + n + 2)), j = 0..2*n+1): seq(a(n), n=0..23);

a[n_] := Binomial[2 n + 1, n] Hypergeometric2F1[-(n + 1)/2, -n/2, n + 2, 4];
Table[a[n], {n, 0, 23}]

a(n) = Sum_{j=0..2*n+1} C(2*n + 1, j)*(C(2*n + 1 - j, j + n) - C(2*n + 1 - j, j + n + 2)).
a(n) = A064189(2*n+1, n).
a(n) = A026300(2*n+1, n+1).
a(n) ~ sqrt((5242 + 18674/sqrt(13))/2187) * ((70 + 26*sqrt(13))/27)^n / sqrt(Pi*n). - Vaclav Kotesovec, May 19 2021
From Peter Bala, Aug 03 2023: (Start)
P-recursive: 3*(13*n - 4)*(3*n + 2)*(3*n + 1)*(n + 1)*a(n) = 2*(2*n + 1)*(455*n^3 + 315*n^2 - 44*n - 24)*a(n-1) + 36*(13*n + 9)*(2*n + 1)*(2*n - 1)*n*a(n-2) with a(0) = 1 and a(1) = 5.
a(n) = (1/2)*A027908(n+1). (End)

A344395 a(n) = binomial(4n - 1, 2n - 1)hypergeom([-n, -n + 1/2], [2n + 1], 4).

Original entry on oeis.org

1, 5, 133, 4037, 129285, 4266830, 143567173, 4896136845, 168640510725, 5853000551090, 204368928058958, 7170955214476509, 252638095187722437, 8931025389858103602, 316640855103349347725, 11254413331736554364987, 400893874585938826203909, 14307778459379093347171266

Offset: 0

Peter Luschny, May 19 2021

nonn

Related to the Motzkin triangle A064189 counting certain lattice paths.

Cf. A064189, A344394, A327871.

alias(C=binomial):
a := n -> `if`(n = 0, 1, add(C(4*n - 1, j)*(C(4*n - 1 - j, j + 2*n - 1) - C(4*n - 1 - j, j + 2*n + 1)), j = 0..4*n-1)): seq(a(n), n = 0..17);

a[n_] := Binomial[4 n - 1, 2 n - 1] Hypergeometric2F1[-n, -n + 1/2, 2 n + 1, 4];
Table[a[n], {n, 0, 19}]

a(n) = Sum_{j=0..4*n-1} C(4*n-1, j)*(C(4*n-1-j, j+2*n-1) - C(4*n-1-j, j+2*n+1)) for n >= 1.
a(n) = A064189(4*n - 1, 2*n - 1) for n >= 1.
a(n) = A344394(4*n - 1) for n >= 1.
a(n) ~ sqrt(1014 + 156*sqrt(13)) * (13688 + 3640*sqrt(13))^n / (52 * sqrt(Pi*n) * 3^(6*n+1)). - Vaclav Kotesovec, Feb 18 2024
D-finite with recurrence +9*n*(6*n-1)*(3*n-1)*(3835115277622*n -6057563812695) *(2*n-1)*(3*n-2) *(6*n-5)*a(n) +2*(776430552534185648*n^7 -13254965233720706112*n^6 +77698256107321929944*n^5 -233839293644869788720*n^4 +406279253239920624227*n^3 -412808144693534857728*n^2 +228023561050132883751*n -52874097275943488160)*a(n-1) -108*(4*n-5)*(4*n-7) *(51631651831183544*n^5 -528937515408392660*n^4 +2125620894576233062*n^3 -4194554621940993427*n^2 +4055650255694760927*n -1531029729082241880)*a(n-2) +402408*(4*n-11)*(n-2) *(4*n-5)*(4*n-9)*(330342177838*n -391995025711)*(2*n-5) *(4*n-7)*a(n-3)=0. - R. J. Mathar, Mar 25 2024

cp's OEIS Frontend

A327872 Total number of nodes in all self-avoiding planar walks starting at (0,0), ending at (n,n), remaining in the first quadrant and using steps (0,1), (-1,1), and (1,-1) with the restriction that (-1,1) and (1,-1) are always immediately followed by (0,1).

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A344394 a(n) = binomial(n, n/2 - 1/4 + (-1)^n/4)*hypergeom([-n/4 - 1/8 + (-1)^n/8, -n/4 + 3/8 + (-1)^n/8], [n/2 + 7/4 + (-1)^n/4], 4).

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A344396 a(n) = binomial(2*n + 1, n)*hypergeom([-(n + 1)/2, -n/2], [n + 2], 4).

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A344395 a(n) = binomial(4*n - 1, 2*n - 1)*hypergeom([-n, -n + 1/2], [2*n + 1], 4).

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Maple

Mathematica

Formula

A344396 a(n) = binomial(2n + 1, n)hypergeom([-(n + 1)/2, -n/2], [n + 2], 4).

A344395 a(n) = binomial(4n - 1, 2n - 1)hypergeom([-n, -n + 1/2], [2n + 1], 4).