Andrei Zabolotskii - cp's OEIS frontend

User: Andrei Zabolotskii

Andrei Zabolotskii has authored 3 sequences.

A385672 Irregular triangle read by rows: T(n, k) is the number of n-step walks on the square lattice having algebraic area k; n >= 0, 0 <= k <= floor(n^2/4).

1, 4, 12, 2, 40, 8, 4, 124, 42, 16, 6, 2, 416, 160, 92, 28, 16, 4, 4, 1348, 678, 362, 174, 88, 34, 22, 8, 6, 2, 4624, 2548, 1624, 756, 460, 200, 156, 56, 40, 20, 12, 4, 4, 15632, 10062, 6336, 3586, 2110, 1106, 742, 388, 278, 152, 82, 46, 34, 14, 8, 6, 2

Offset: 0

Andrei Zabolotskii, Aug 04 2025

Rows can be extended to negative k with T(n, -k) = T(n, k). Sums of such extended rows give 4^n.

The algebraic area is Integral y dx over the walk, which equals (Sum_{steps right} y) - (Sum_{steps left} y).

			The triangle begins:
     1
     4
    12,   2
    40,   8,   4
   124,  42,  16,   6,  2
   416, 160,  92,  28, 16,  4,  4
  1348, 678, 362, 174, 88, 34, 22, 8, 6, 2
   ...
T(3, 1) = 8: RUR (right, up, right), LUR, RDL, LDL, URU, URD, DLU, DLD.

Row lengths are A033638 = A002620 + 1.
A352838 is an analog that gives the number of closed walks.
Cf. A029552, A098613.

d = [{((0, 0), 0): 1}]
for _ in range(10):
    nd = {}
    for key, nw in d[-1].items():
        pos, ar = key
        x, y = pos
        for key in [
            ((x+1, y), ar + y),
            ((x-1, y), ar - y),
            ((x, y+1), ar),
            ((x, y-1), ar)
            ]:
            if key in nd:
                nd[key] += nw
            else:
                nd[key] = nw
    d.append(nd)
t = []
for nd in d:
    a = [0] * (max(ar for _, ar in nd) + 1)
    for key, nw in nd.items():
        _, ar = key
        if ar >= 0:
            a[ar] += nw
    t.append(a)
print(t)

It appears that T(2*n, n^2 - k) = 2 * A029552(k) for k < n and T(2*n+1, n^2+n - k) = 4 * A098613(k) for k < n.

A386406 Length of the preperiodic part of the decimal expansion of 1/n, including any leading zeros from the period.

Original entry on oeis.org

1, 0, 2, 1, 1, 0, 3, 0, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 3

Offset: 2

Andrei Zabolotskii, Jul 20 2025

See A114205.

			For n = 92, 1/n = 0.01(0869565217391304347826) = 0.010(8695652173913043478260), so the preperiodic part is "010" and has length a(92) = 3.

Index entries for sequences related to decimal expansion of 1/n.

Cf. A051628, A114205, A114206.

b[n_] := Block[{p,o}, {p,o} = RealDigits[1/n]; If[!IntegerQ[Last[p]], p = Join[Most[p],TakeWhile[Last[p],#==0&]]]; Length[p]-o];
Table[b[n], {n,2,100}]

a(n) = my(pre = max(valuation(n,2),valuation(n,5)), r = 10^pre % n); pre + if(r,logint(n\r,10)); \\ Kevin Ryde, Jul 22 2025

a(n) = p + (floor(log_10(1/f)) if f!=0), where p = A051628(n) and f = frac(10^p/n). - Kevin Ryde, Jul 22 2025

A352838 Irregular triangle read by rows: T(n, k) is the number of 2n-step closed walks on the square lattice having algebraic area k; n >= 0, 0 <= k <= floor(n^2/4).

Original entry on oeis.org

1, 4, 28, 4, 232, 72, 12, 2156, 1008, 308, 48, 8, 21944, 13160, 5540, 1560, 420, 80, 20, 240280, 168780, 87192, 33628, 11964, 3636, 1200, 264, 72, 12, 2787320, 2168544, 1291220, 610232, 262612, 101976, 40376, 13720, 4900, 1512, 420, 112, 28

Offset: 0

Andrei Zabolotskii, Apr 05 2022

Rows can be extended to negative k with T(n, -k) = T(n, k). Sums of such extended rows give A002894.

T(n, k) is the number of words of length 2n equal to z^k in the Heisenberg group, presented as , where z=[x,y]. In particular, T(n, 0) = A307468(n).

			The table begins:
       1
       4
      28,      4
     232,     72,    12
    2156,   1008,   308,    48,     8
   21944,  13160,  5540,  1560,   420,   80,   20
  240280, 168780, 87192, 33628, 11964, 3636, 1200, 264, 72, 12
     ...
T(2, 0) = 28: the 4-step walks enclosing algebraic area 0 include 16 walks of the form "some two steps, then two steps right back" and 12 walks of the form "some step, step back, a different step, step back".
T(2, 1) = 4: the 4-step walks enclosing algebraic area 1 are the walks around each of the 4 squares touching the origin in the positive direction; cf. A334756(2, 1) = 8, which also counts walks around these squares in the negative direction.

Andrei Zabolotskii, Table of n, a(n) for n = 0..1403 (rows 0..25)
Cédric Béguin, Alain Valette and Andrzej Zuk, On the spectrum of a random walk on the discrete Heisenberg group and the norm of Harper's operator, Journal of Geometry and Physics, 21 (1997), 337-356.
Li Gan, Algebraic Area of Lattice Random Walks and Exclusion Statistics, PhD thesis, Université Paris-Saclay, 2023. See Section 2.1.2, in particular Table 2.1 (divide terms in rows with nonzero A by 2 to get this table).
Stefan Mashkevich and Stéphane Ouvry, Area Distribution of Two-Dimensional Random Walks on a Square Lattice, J. Stat. Phys., 137 (2009), 71-78.
Morteza Mohammad-Noori, Enumeration of closed random walks in the square lattice according to their areas, arXiv:1012.3720 [math.CO], 2010. Published as: Morteza Mohammad-Noori, Enumeration of walks in the square lattice according to their areas, Journal of Combinatorial Mathematics and Combinatorial Computing, 91 (2014), 257-274.

Row lengths are A033638 = A002620 + 1.
Row n ends with 4 * A026741(n) for n > 0.
Row 16 is A178106.
Cf. A307468, A002894, A353091, A385672.
A334756 counts self-avoiding walks only.

z[0, 0, 0, 0] = 1;
z[-1, ] = z[, -1, _] = z[, , -1, ] = z[, , , -1] = 0;
z[m1_, m2_, l1_, l2_] := z[m1, m2, l1, l2] = Expand[z[m1, m2, l1 - 1, l2] + z[m1, m2, l1, l2 - 1] + q^(l2 - l1) z[m1 - 1, m2, l1, l2] + q^(l1 - l2) z[m1, m2 - 1, l1, l2]];
zN[n_] := Sum[z[m, m, n/2 - m, n/2 - m], {m, 0, n/2}];
walks[n_] := Module[{gf = zN[2 n], e}, e = Exponent[gf, q, Max]; CoefficientList[gf q^e, q][[e + 1 ;;]]];
Table[walks[n], {n, 0, 8}]

cp's OEIS Frontend

User: Andrei Zabolotskii

A385672 Irregular triangle read by rows: T(n, k) is the number of n-step walks on the square lattice having algebraic area k; n >= 0, 0 <= k <= floor(n^2/4).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Python

Formula

A386406 Length of the preperiodic part of the decimal expansion of 1/n, including any leading zeros from the period.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A352838 Irregular triangle read by rows: T(n, k) is the number of 2n-step closed walks on the square lattice having algebraic area k; n >= 0, 0 <= k <= floor(n^2/4).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica