A209349 -id:A209349 - cp's OEIS frontend

A209350 Number of initially rising meander words, where each letter of the cyclic n-ary alphabet occurs twice.

1, 0, 1, 5, 9, 11, 16, 19, 25, 29, 36, 41, 49, 55, 64, 71, 81, 89, 100, 109, 121, 131, 144, 155, 169, 181, 196, 209, 225, 239, 256, 271, 289, 305, 324, 341, 361, 379, 400, 419, 441, 461, 484, 505, 529, 551, 576, 599, 625, 649, 676, 701, 729, 755, 784, 811, 841

Offset: 0

Alois P. Heinz, Mar 06 2012

In a meander word letters of neighboring positions have to be neighbors in the alphabet, where in a cyclic alphabet the first and the last letters are considered neighbors too.  The words are not considered cyclic here.
A word is initially rising if it is empty or if it begins with the first letter of the alphabet that can only be followed by the second letter in this word position.
a(n) is also the number of (2*n-1)-step walks on n-dimensional cubic lattice from (1,0,...,0) to (2,2,...,2) with positive unit steps in all dimensions such that the indices of dimensions used in consecutive steps differ by 1 or are in the set {1,n}.

			a(0) = 1: the empty word.
a(1) = 0 = |{ }|.
a(2) = 1 = |{abab}|.
a(3) = 5 = |{abacbc, abcabc, abcacb, abcbac, abcbca}|.
a(4) = 9 = |{ababcdcd, abadcbcd, abadcdcb, abcbadcd, abcbcdad, abcdabcd, abcdadcb, abcdcbad, abcdcdab}|.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Row n=2 of A209349.
First differences for n>2 give: A084964(n+1), A097065(n+3).
Cf. A245578.

a:= n-> `if`(n<3, (n-1)^2, (n/2+1)^2 -(n mod 2)*5/4):
seq(a(n), n=0..60);

LinearRecurrence[{2,0,-2,1},{1,0,1,5,9,11,16},60] (* Harvey P. Dale, Jan 02 2020 *)

G.f.: -(3*x^6-5*x^5-2*x^4+5*x^3+x^2-2*x+1) / ((x+1)*(x-1)^3).

a(n) = (n-1)^2 if n<3, a(n) = (n/2+1)^2 - (n mod 2)*5/4 else.

A209352 Number of initially rising meander words, where each letter of the cyclic 6-ary alphabet occurs n times.

Original entry on oeis.org

1, 1, 16, 484, 17956, 749956, 33779344, 1603842304, 79171327876, 4026836863204, 209730177700096, 11135960392243600, 600800844868633600, 32853035097265158400, 1817225079550242841600, 101519847275313821814784, 5720749624907993103318916, 324836041052683988251601956

Offset: 0

Alois P. Heinz, Mar 06 2012

In a meander word letters of neighboring positions have to be neighbors in the alphabet, where in a cyclic alphabet the first and the last letters are considered neighbors too.  The words are not considered cyclic here.
A word is initially rising if it is empty or if it begins with the first letter of the alphabet that can only be followed by the second letter in this word position.
a(n) is also the number of (6*n-1)-step walks on 6-dimensional cubic lattice from (1,0,...,0) to (n,n,...,n) with positive unit steps in all dimensions such that the indices of dimensions used in consecutive steps differ by 1 or are in the set {1,6}.

			a(0) =  1: the empty word.
a(1) =  1 = |{abcdef}|.
a(2) = 16 = |{ababcdcdefef, abafedcbcdef, abafefedcbcd, abafefedcdcb, abcbafedcdef, abcbafefedcd, abcbcdedefaf, abcbcdefafed, abcdcbafedef, abcdcbafefed, abcdcdefefab, abcdedcbafef, abcdefabcdef, abcdefafedcb, abcdefedcbaf, abcdefefabcd}|.

Alois P. Heinz, Table of n, a(n) for n = 0..500

Column k=6 of A209349.

Cf. A197657.

g:= proc(m, n, k) local h;
      h:= binomial(n-1, k);
      h^m +`if`(m<2, 0, h* g(m-1, n, n-k-2))
    end:
a:= n-> add(g(3, n, k), k=0..n)^2:
seq(a(n), n=0..30);

g[m_, n_, k_] := g[m, n, k] = With[{h = Binomial[n - 1, k]}, h^m + If[m < 2, 0, h g[m - 1, n, n - k - 2]]];
a[n_] := Sum[g[3, n, k], {k, 0, n}]^2;
a /@ Range[0, 30] (* Jean-François Alcover, May 14 2020, after Maple *)

a(n) = A197657(n-1)^2 for n>0, a(0) = 1.

a(n) ~ 3 * 2^(6*n - 4) / (Pi^2 * n^2). - Vaclav Kotesovec, May 14 2020

A209351 Number of initially rising meander words, where each letter of the cyclic 5-ary alphabet occurs n times.

Original entry on oeis.org

1, 1, 11, 182, 3542, 76258, 1753522, 42269392, 1055518634, 27091380674, 710781384782, 18986685114964, 514838929566304, 14138508923416768, 392518929926801888, 11000540560168582016, 310851652685562860378, 8848249504532000817058, 253498769515046174725606

Offset: 0

Alois P. Heinz, Mar 06 2012

In a meander word letters of neighboring positions have to be neighbors in the alphabet, where in a cyclic alphabet the first and the last letters are considered neighbors too.  The words are not considered cyclic here.
A word is initially rising if it is empty or if it begins with the first letter of the alphabet that can only be followed by the second letter in this word position.
a(n) is also the number of (5*n-1)-step walks on 5-dimensional cubic lattice from (1,0,...,0) to (n,n,...,n) with positive unit steps in all dimensions such that the indices of dimensions used in consecutive steps differ by 1 or are in the set {1,5}.

			a(0) =  1: the empty word.
a(1) =  1 = |{abcde}|.
a(2) = 11 = |{abaedcbcde, abaededcbc, abcbaedcde, abcbaededc, abcbcdeaed, abcbcdedea, abcdcbaede, abcdeabcde, abcdeaedcb, abcdedcbae, abcdedeabc}|.

Column k=5 of A209349.

A209353 Number of initially rising meander words, where each letter of the cyclic 7-ary alphabet occurs n times.

Original entry on oeis.org

1, 1, 19, 902, 53636, 3771698, 292587922, 24422343208, 2150227555324, 197404215493922, 18735386882618558, 1827242795963079380, 182291530666463403220, 18537924237935355619360

Offset: 0

Alois P. Heinz, Mar 06 2012

In a meander word letters of neighboring positions have to be neighbors in the alphabet, where in a cyclic alphabet the first and the last letters are considered neighbors too.  The words are not considered cyclic here.
A word is initially rising if it is empty or if it begins with the first letter of the alphabet that can only be followed by the second letter in this word position.
a(n) is also the number of (7*n-1)-step walks on 7-dimensional cubic lattice from (1,0,...,0) to (n,n,...,n) with positive unit steps in all dimensions such that the indices of dimensions used in consecutive steps differ by 1 or are in the set {1,7}.

			a(0) =  1: the empty word.
a(1) =  1 = |{abcdefg}|.
a(2) = 19 = |{abagfedcbcdefg, abagfgfedcbcde, abagfgfededcbc, abcbagfedcdefg, abcbagfgfedcde, abcbagfgfededc, abcbcdedefgagf, abcbcdedefgfga, abcbcdefgagfed, abcdcbagfedefg, abcdcbagfgfede, abcdedcbagfefg, abcdedcbagfgfe, abcdedefgfgabc, abcdefedcbagfg, abcdefgabcdefg, abcdefgagfedcb, abcdefgfedcbag, abcdefgfgabcde}|.

Column k=7 of A209349.

A240954 Number of initially rising meander words, where each letter of the cyclic n-ary alphabet occurs three times.

Original entry on oeis.org

1, 0, 1, 29, 100, 182, 484, 902, 2116, 4034, 8836, 17138, 36100, 70850, 145924, 288578, 586756

Offset: 0

Alois P. Heinz, Aug 04 2014

In a meander word letters of neighboring positions have to be neighbors in the alphabet, where in a cyclic alphabet the first and the last letters are considered neighbors too.  The words are not considered cyclic here.
A word is initially rising if it is empty or if it begins with the first letter of the alphabet that can only be followed by the second letter in this word position.
a(n) is also the number of (3*n-1)-step walks on n-dimensional cubic lattice from (1,0,...,0) to (3,3,...,3) with positive unit steps in all dimensions such that the indices of dimensions used in consecutive steps differ by 1 or are in the set {1,n}.

			a(0) = 1: the empty word.
a(1) = 0 = |{ }|.
a(2) = 1 = |{ababab}|.
a(3) = 29 = |{ababcacbc, ababcbcac, abacabcbc, abacacbcb, abacbacbc, abacbcabc, abacbcacb, abacbcbac, abacbcbca, abcabacbc, abcabcabc, abcabcacb, abcabcbac, abcabcbca, abcacabcb, abcacbabc, abcacbacb, abcacbcab, abcacbcba, abcbabcac, abcbacabc, abcbacacb, abcbacbac, abcbacbca, abcbcabac, abcbcabca, abcbcacab, abcbcacba, abcbcbaca}|.

Row n=3 of A209349.

a(2n) = A169721(n) for n>1.

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A209350 Number of initially rising meander words, where each letter of the cyclic n-ary alphabet occurs twice.

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A209352 Number of initially rising meander words, where each letter of the cyclic 6-ary alphabet occurs n times.

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A209351 Number of initially rising meander words, where each letter of the cyclic 5-ary alphabet occurs n times.

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A209353 Number of initially rising meander words, where each letter of the cyclic 7-ary alphabet occurs n times.

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A240954 Number of initially rising meander words, where each letter of the cyclic n-ary alphabet occurs three times.

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