A153372 -id:A153372 - cp's OEIS frontend

A153368 Number of zig-zag paths from top to bottom of a rectangle of width 11 with n rows.

11, 20, 38, 72, 138, 264, 508, 976, 1882, 3624, 6996, 13488, 26054, 50264, 97124, 187440, 362250, 699240, 1351492, 2609008, 5042950, 9735768, 18818772, 36332016, 70229066, 135588200, 262091348, 506012592, 978124038, 1888445784, 3650380228

Offset: 1

Joseph Myers, Dec 24 2008

Heuristically, a(n) = +6*a(n-2) -9*a(n-4) +2*a(n-6). - R. J. Mathar, Jun 16 2011

Number of words of length n using a 11 symbol alphabet where neighboring letters are neighbors in the alphabet. - Andrew Howroyd, Apr 17 2017

Joseph Myers, BMO 2008--2009 Round 1 Problem 1---Generalisation
Index entries for linear recurrences with constant coefficients, signature (0, 6, 0, -9, 0, 2).

Column 11 of A220062.

Cf. A153369, A153370, A153371, A153372 (bisection), A153373.

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, Jul 01 2018, after Alois P. Heinz *)

Empirical G.f.: x*(11+20*x-28*x^2-48*x^3+9*x^4+12*x^5)/((1-2*x^2)*(1-4*x^2+x^4)). - Colin Barker, Apr 17 2012

a(n) = A153369(n) + A153370(n). - Andrew Howroyd, Apr 17 2017

A153373 Number of zig-zag paths from top to bottom of a rectangle of width 11 with 2n-1 rows whose color is not that of the top right corner.

Original entry on oeis.org

5, 18, 66, 244, 906, 3372, 12566, 46860, 174810, 652252, 2433942, 9083004, 33897050, 126503148, 472111446, 1761934444, 6575609946, 24540472572, 91586214806, 341804255580, 1275630545370, 4760717401612, 17767238012502

Offset: 1

Joseph Myers, Dec 24 2008

Joseph Myers, BMO 2008--2009 Round 1 Problem 1---Generalisation
Index entries for linear recurrences with constant coefficients, signature (6, -9, 2).

Cf. A153368-A153372.

Empirical G.f.: x*(5-12*x+3*x^2)/(1-6*x+9*x^2-2*x^3). - Colin Barker, Jan 04 2012

A153369 Number of zig-zag paths from top to bottom of a rectangle of width 11 with n rows whose color is that of the top right corner.

Original entry on oeis.org

6, 10, 20, 36, 72, 132, 264, 488, 976, 1812, 3624, 6744, 13488, 25132, 50264, 93720, 187440, 349620, 699240, 1304504, 2609008, 4867884, 9735768, 18166008, 36332016, 67794100, 135588200, 253006296, 506012592, 944222892, 1888445784, 3523868888

Offset: 1

Joseph Myers, Dec 24 2008

Joseph Myers, BMO 2008--2009 Round 1 Problem 1---Generalisation
Index entries for linear recurrences with constant coefficients, signature (0, 6, 0, -9, 0, 2).

A153368, A153370, A153371, A153372, A153373

LinearRecurrence[{0,6,0,-9,0,2},{6,10,20,36,72,132},40] (* Harvey P. Dale, Sep 25 2024 *)

Empirically, g.f. -2*x*(3+5*x-8*x^2-12*x^3+3*x^4+3*x^5) / ( (2*x^2-1)*(x^4-4*x^2+1) ) with a(n)= +6*a(n-2) -9*a(n-4) +2*a(n-6). - R. J. Mathar, Jun 16 2011

A153370 Number of zig-zag paths from top to bottom of a rectangle of width 11 with n rows whose color is not that of the top right corner.

Original entry on oeis.org

5, 10, 18, 36, 66, 132, 244, 488, 906, 1812, 3372, 6744, 12566, 25132, 46860, 93720, 174810, 349620, 652252, 1304504, 2433942, 4867884, 9083004, 18166008, 33897050, 67794100, 126503148, 253006296, 472111446, 944222892, 1761934444, 3523868888

Offset: 1

Joseph Myers, Dec 24 2008

Joseph Myers, BMO 2008--2009 Round 1 Problem 1---Generalisation
Index entries for linear recurrences with constant coefficients, signature (0, 6, 0, -9, 0, 2).

A153368, A153369, A153372. Bisections: A153371, A153373.

Empirical: G.f. -x*(2*x+1)*(3*x^4-12*x^2+5) / ( (2*x^2-1)*(x^4-4*x^2+1) ) and a(n)= +6*a(n-2) -9*a(n-4) +2*a(n-6). - R. J. Mathar, Jun 16 2011

A153371 Number of zig-zag paths from top to bottom of a rectangle of width 11 with 2n rows whose color is that of the top right corner.

Original entry on oeis.org

10, 36, 132, 488, 1812, 6744, 25132, 93720, 349620, 1304504, 4867884, 18166008, 67794100, 253006296, 944222892, 3523868888, 13151219892, 49080945144, 183172429612, 683608511160, 2551261090740, 9521434803224, 35534476025004

Offset: 1

Joseph Myers, Dec 24 2008

Joseph Myers, BMO 2008--2009 Round 1 Problem 1---Generalisation
Index entries for linear recurrences with constant coefficients, signature (6, -9, 2).

A153368, A153369, A153370, A153372, A153373

Empirical: G.f. -2*x*(5-12*x+3*x^2) / ( (2*x-1)*(x^2-4*x+1) ) with a(n)= +6*a(n-1) -9*a(n-2) +2*a(n-3) and a(n) = (2^n+4*A001075(n+1))/3. - R. J. Mathar, Jun 16 2011

cp's OEIS Frontend

A153368 Number of zig-zag paths from top to bottom of a rectangle of width 11 with n rows.

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A153373 Number of zig-zag paths from top to bottom of a rectangle of width 11 with 2n-1 rows whose color is not that of the top right corner.

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A153369 Number of zig-zag paths from top to bottom of a rectangle of width 11 with n rows whose color is that of the top right corner.

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A153370 Number of zig-zag paths from top to bottom of a rectangle of width 11 with n rows whose color is not that of the top right corner.

Views

Author

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A153371 Number of zig-zag paths from top to bottom of a rectangle of width 11 with 2n rows whose color is that of the top right corner.

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Author

Keywords

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