A048248 Possible traces of n-step walks on 1-D lattice, ignoring translations.

1, 2, 4, 7, 14, 23, 45, 72, 137, 217, 402, 635, 1149, 1816, 3221, 5101, 8898, 14127, 24305, 38688, 65809, 105009, 176962, 282995, 473269, 758312, 1260253, 2022661, 3344354, 5375207

Offset

1,2

Comments

Translations discounted, so the sequence of visits <0,1,2,1,0> has trace (2,2), as do <0,-1,0,1,0>, <0,-1,-2,-1,0>, etc.

Trace as used here refers to the number of times an edge is used in the walk. - Sean A. Irvine, Jun 06 2021

Links

Table of n, a(n) for n=1..30.

Sean A. Irvine, Java program (github)

Example

a(4)=7 since a walk of 4 steps can leave traces (1,1,1,1), (1,1,2), (2,1,1), (2,2), (1,3), (3,1) and (4). Note that (1,2,1) is impossible.

Mathematica

For[size = 1, size < 10, size++, traces = {}; For[i = 0, i < 2^ size, i++, thePath = IntegerDigits[i, 2, size]*2 - 1; loc = size + 2; theTrace = Table[0, {z, -size - 1, size + 1}]; For[j = 1, j <= size, j++, loc += thePath[[j]]; If[thePath[[j]] == 1, theTrace[[loc - 1]]++, theTrace[[loc]]++ ]; ]; theTrace = Select[theTrace, # > 0 &]; If[ ! MemberQ[traces, theTrace], traces = Append[traces, theTrace]]; ]; Print[Length[traces]]]; - Sam Handler (sam_5_5_5_0(AT)yahoo.com), Aug 17 2006

Crossrefs

Sequence in context: A034426 A000075 A285903 * A370636 A056180 A000076

Adjacent sequences: A048245 A048246 A048247 * A048249 A048250 A048251

Keyword

nonn,more

Author

Tony Bartoletti

Extensions

a(19)-a(21) from Sam Handler (sam_5_5_5_0(AT)yahoo.com), Aug 17 2006

a(22)-a(30) from Sean A. Irvine, Jun 06 2021

Status

approved