cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A169720 a(n) = (3*2^(n-1)-1)*(3*2^n-1).

Original entry on oeis.org

1, 10, 55, 253, 1081, 4465, 18145, 73153, 293761, 1177345, 4713985, 18865153, 75479041, 301953025, 1207885825, 4831690753, 19327057921, 77308821505, 309236465665, 1236948221953, 4947797606401, 19791199862785, 79164818325505, 316659311050753, 1266637319700481
Offset: 0

Views

Author

Alice V. Kleeva (alice27353(AT)gmail.com), Jan 19 2010

Keywords

Comments

A subsequence of the triangular numbers A000217.

Links

Crossrefs

Cf. A169721-A169727, A033484, A000217.

Programs

  • Magma
    I:=[1, 10, 55]; [n le 3 select I[n] else 7*Self(n-1)-14*Self(n-2)+8*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Dec 03 2012
  • Mathematica
    CoefficientList[Series[(1 + 3*x - x^2)/((1-x)*(1-2*x)*(1-4*x)), {x, 0, 30}], x] (* or *) LinearRecurrence[{7, -14, 8}, {1, 10, 55}, 30] (* Vincenzo Librandi, Dec 03 2012 *)
  • PARI
    a(n)=polcoeff((1+3*x-x^2)/((1-x)*(1-2*x)*(1-4*x)+x*O(x^n)),n) \\ Paul D. Hanna, Apr 29 2010

Formula

G.f.: (1 + 3*x - x^2)/((1-x)*(1-2*x)*(1-4*x)). - Paul D. Hanna, Apr 29 2010
a(n) = A000217(A033484(n)). - Mitch Harris, Dec 02 2012
a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3). - Vincenzo Librandi, Dec 03 2012
a(n) = (3*A169726(n)-1)/2. - L. Edson Jeffery, Dec 03 2012
a(n) = A006095(n+2) +3*A006095(n+1) - A006905(n). - R. J. Mathar, Dec 04 2016

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

Views

Author

Alois P. Heinz, Aug 04 2014

Keywords

Comments

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}.

Examples

			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}|.

Crossrefs

Row n=3 of A209349.

Formula

a(2n) = A169721(n) for n>1.
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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