login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A191757 Number of n-step four-sided prudent self-avoiding walks ending on the top side of their box. 3
1, 3, 7, 19, 49, 129, 333, 865, 2233, 5763, 14825, 38087, 97641, 249961, 638861, 1630681, 4156737, 10583483, 26916167, 68383509, 173565889, 440133159, 1115145081, 2823128197, 7141682287, 18053470305, 45606579731, 115137581735, 290498368253 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

REFERENCES

E. Duchi, On some classes of prudent walks, in: FPSAC'05, Taormina, Italy, 2005.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..100

M. Bousquet-Mélou, Families of prudent self-avoiding walks, DMTCS proc. AJ, 2008, 167-180.

EXAMPLE

a(3) = 19: EEE, EEN, ENE, ENN, ENW, NEE, NEN, NNE, NNN, NNW, NWN, NWW, WNE, WNN, WNW, WWN, WWW, SEN, SWN.

MAPLE

b:= proc(d, i, n, x, y, w, s)

      if w<x then b([3, 2, 1, 4][d], i, n, w, y, x, s)

    else b(d, i, n, x, y, w, s):=

         `if`(y>n, 0, `if`(n=0, `if`(y=0, 1, 0),

         `if`(d in [0, 1] or d<>3 and (x=0 or i),

              b(1, evalb(x=0), n-1, max(x-1, 0), y, w+1, s), 0) +

         `if`(d in [0, 2] or d<>4 and (y=0 or i),

              b(2, evalb(y=0), n-1, x, max(y-1, 0), w, s+1), 0) +

         `if`(d in [0, 3] or d<>1 and (w=0 or i),

              b(3, evalb(w=0), n-1, x+1, y, max(w-1, 0), s), 0) +

         `if`(d in [0, 4] or d<>2 and (s=0 or i),

              b(4, evalb(s=0), n-1, x, y+1, w, max(s-1, 0)), 0)))

      fi

    end:

a:= n-> b(0, false, n, 0, 0, 0, 0):

seq(a(n), n=0..30);

MATHEMATICA

b[d_, i_, n_, x_, y_, w_, s_] := b[d, i, n, x, y, w, s] = If[w < x,  b[{3, 2, 1, 4}[[d]], i, n, w, y, x, s], b[d, i, n, x, y, w, s] = If[y > n, 0, If[n == 0, If[y == 0, 1, 0], If[MemberQ[{0, 1}, d] || d != 3 && (x == 0 || i), b[1, x == 0, n - 1, Max[x - 1, 0], y, w + 1, s], 0] + If[MemberQ[{0, 2}, d] || d != 4 && (y == 0 || i), b[2, y == 0, n - 1, x, Max[y - 1, 0], w, s + 1], 0] + If[MemberQ[{0, 3}, d] || d != 1 && (w == 0 || i), b[3, w == 0, n - 1, x + 1, y, Max[w - 1, 0], s], 0] + If[MemberQ[{0, 4}, d] || d != 2 && (s == 0 || i), b[4, s == 0, n - 1, x, y + 1, w, Max[s - 1, 0]], 0]]]];

a[n_] :=  b[0, False, n, 0, 0, 0, 0];

Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 23 2017, translated from Maple *)

CROSSREFS

Cf. A191756, A191758.

Sequence in context: A073063 A007288 A191824 * A061646 A017926 A017927

Adjacent sequences:  A191754 A191755 A191756 * A191758 A191759 A191760

KEYWORD

nonn,walk

AUTHOR

Alois P. Heinz, Jun 15 2011

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 8 18:37 EST 2019. Contains 329865 sequences. (Running on oeis4.)