A220062
Number A(n,k) of n length words over k-ary alphabet, where neighboring letters are neighbors in the alphabet; square array A(n,k), n>=0, k>=0, read by antidiagonals.
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 2, 0, 0, 1, 4, 4, 2, 0, 0, 1, 5, 6, 6, 2, 0, 0, 1, 6, 8, 10, 8, 2, 0, 0, 1, 7, 10, 14, 16, 12, 2, 0, 0, 1, 8, 12, 18, 24, 26, 16, 2, 0, 0, 1, 9, 14, 22, 32, 42, 42, 24, 2, 0, 0, 1, 10, 16, 26, 40, 58, 72, 68, 32, 2, 0, 0
Offset: 0
A(5,3) = 12: there are 12 words of length 5 over 3-ary alphabet {a,b,c}, where neighboring letters are neighbors in the alphabet: ababa, ababc, abcba, abcbc, babab, babcb, bcbab, bcbcb, cbaba, cbabc, cbcba, cbcbc.
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, 7, ...
0, 0, 2, 4, 6, 8, 10, 12, ...
0, 0, 2, 6, 10, 14, 18, 22, ...
0, 0, 2, 8, 16, 24, 32, 40, ...
0, 0, 2, 12, 26, 42, 58, 74, ...
0, 0, 2, 16, 42, 72, 104, 136, ...
0, 0, 2, 24, 68, 126, 188, 252, ...
Columns k=0, 2-10 give:
A000007,
A040000,
A029744(n+2) for n>0,
A006355(n+3) for n>0,
A090993(n+1) for n>0,
A090995(n-1) for n>2,
A129639,
A153340,
A153362,
A153360.
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b:= proc(n, i, k) option remember; `if`(n=0, 1,
`if`(i=0, add(b(n-1, j, k), j=1..k),
`if`(i>1, b(n-1, i-1, k), 0)+
`if`(i b(n, 0, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);
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b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i == 0, Sum[b[n-1, j, k], {j, 1, k}], If[i>1, b[n-1, i-1, k], 0] + If[iJean-François Alcover, Jan 19 2015, after Alois P. Heinz *)
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TransferGf(m,u,t,v,z)=vector(m,i,u(i))*matsolve(matid(m)-z*matrix(m,m,i,j,t(i,j)),vectorv(m,i,v(i)));
ColGf(m,z)=1+z*TransferGf(m, i->1, (i,j)->abs(i-j)==1, j->1, z);
a(n,k)=Vec(ColGf(k,x) + O(x^(n+1)))[n+1];
for(n=0, 7, for(k=0, 7, print1( a(n,k), ", ") ); print(); );
\\ Andrew Howroyd, Apr 17 2017
A208666
Number of 2n-bead necklaces labeled with numbers 1..n allowing reversal, with neighbors differing by exactly 1.
Original entry on oeis.org
0, 1, 4, 14, 44, 152, 514, 1866, 6884, 26137, 100442, 390592, 1526272, 5989223, 23548688, 92727898, 365445200, 1441195226, 5686268314, 22444465311, 88622259788, 350040069245, 1383007946774, 5465854718664, 21607909105528, 85444555330132, 337962745845558, 1337094537703089
Offset: 1
All solutions for n=3:
..1....1....1....2
..2....2....2....3
..1....3....1....2
..2....2....2....3
..3....3....1....2
..2....2....2....3
A208667
Number of 2n-bead necklaces labeled with numbers 1..4 allowing reversal, with neighbors differing by exactly 1.
Original entry on oeis.org
3, 5, 8, 14, 24, 47, 89, 187, 396, 881, 1990, 4645, 10935, 26211, 63320, 154378, 378444, 933023, 2308957, 5735372, 14286908, 35683815, 89324138, 224057919, 563033979, 1417210457, 3572641304, 9018885122, 22796905056, 57692673963, 146167385345, 370710166435
Offset: 1
All solutions for n=4:
..1....2....2....2....1....1....1....1....1....3....1....1....2....2
..2....3....3....3....2....2....2....2....2....4....2....2....3....3
..1....2....2....2....1....1....3....3....3....3....3....1....4....4
..2....3....3....3....2....2....4....2....2....4....2....2....3....3
..1....4....2....2....3....1....3....1....3....3....3....3....2....4
..2....3....3....3....4....2....4....2....4....4....2....2....3....3
..1....4....2....4....3....3....3....3....3....3....3....3....4....4
..2....3....3....3....2....2....2....2....2....4....2....2....3....3
A208668
Number of 2n-bead necklaces labeled with numbers 1..5 allowing reversal, with neighbors differing by exactly 1.
Original entry on oeis.org
4, 7, 12, 23, 44, 97, 212, 512, 1260, 3251, 8540, 23035, 62780, 173453, 482692, 1353077, 3811364, 10785235, 30625196, 87239999, 249174236, 713416601, 2046945140, 5884580074, 16946835092, 48883925867, 141217957620, 408519816611, 1183291934300, 3431535849813
Offset: 1
All solutions for n=3:
..3....3....1....1....1....2....2....3....1....2....2....4
..4....4....2....2....2....3....3....4....2....3....3....5
..5....3....1....1....3....2....4....3....3....2....4....4
..4....4....2....2....2....3....5....4....4....3....3....5
..5....5....3....1....3....2....4....3....3....4....4....4
..4....4....2....2....2....3....3....4....2....3....3....5
A208669
Number of 2n-bead necklaces labeled with numbers 1..6 allowing reversal, with neighbors differing by exactly 1.
Original entry on oeis.org
5, 9, 16, 32, 65, 152, 360, 937, 2512, 7034, 20124, 58899, 174408, 522453, 1576468, 4787425, 14607763, 44758541, 137607258, 424343533, 1311982883, 4065937220, 12627227963, 39290663343, 122470374536, 382359843141, 1195524220166, 3743191124783
Offset: 1
All solutions for n=3:
..3....5....2....3....3....2....4....4....2....1....1....2....1....3....4....1
..4....6....3....4....4....3....5....5....3....2....2....3....2....4....5....2
..5....5....2....3....3....4....4....4....4....1....3....2....3....5....6....1
..6....6....3....4....4....3....5....5....5....2....4....3....2....4....5....2
..5....5....4....5....3....4....6....4....4....1....3....2....3....5....6....3
..4....6....3....4....4....3....5....5....3....2....2....3....2....4....5....2
A208670
Number of 2n-bead necklaces labeled with numbers 1..7 allowing reversal, with neighbors differing by exactly 1.
Original entry on oeis.org
6, 11, 20, 41, 86, 208, 514, 1398, 3934, 11576, 34850, 107303, 334396, 1053851, 3345320, 10685570, 34292064, 110498897, 357256018, 1158492478, 3766458404, 12274037845, 40082339406, 131144365904, 429838330172, 1411104048106, 4639351259122, 15273992343065
Offset: 1
All solutions for n=3:
..1....5....1....5....3....4....1....6....4....1....2....2....4....5....3....4
..2....6....2....6....4....5....2....7....5....2....3....3....5....6....4....5
..1....5....3....7....3....4....1....6....6....3....4....2....6....5....5....4
..2....6....2....6....4....5....2....7....5....4....3....3....7....6....4....5
..3....5....3....7....5....6....1....6....6....3....4....2....6....7....5....4
..2....6....2....6....4....5....2....7....5....2....3....3....5....6....4....5
..
..2....2....3....3
..3....3....4....4
..2....4....3....5
..3....5....4....6
..4....4....3....5
..3....3....4....4
Showing 1-6 of 6 results.
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