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.

Showing 1-2 of 2 results.

A178987 a(n) = n*(n-3)*2^(n-2).

Original entry on oeis.org

0, -1, -2, 0, 16, 80, 288, 896, 2560, 6912, 17920, 45056, 110592, 266240, 630784, 1474560, 3407872, 7798784, 17694720, 39845888, 89128960, 198180864, 438304768, 964689920, 2113929216, 4613734400, 10032775168, 21743271936, 46976204800, 101200166912
Offset: 0

Views

Author

Paul Curtz, Jan 03 2011

Keywords

Comments

Binomial transform of 0, -1 followed by A005563.
The sequence defines an array by adding higher order differences in successive rows:
0, -1, -2, 0, 16, 80, 288, 896, 2560, 6912, 17920, 45056, 110592, ...
-1, -1, 2, 16, 64, 208, 608, 1664, 4352, 11008, 27136, 65536, ... A127276
0, 3, 14, 48, 144, 400, 1056, 2688, 6656, 16128, 38400, 90112, 208896, ... A176027
3, 11, 34, 96, 256, 656, 1632, 3968, 9472, 22272, 51712, 118784, ... A084266
8, 23, 62, 160, 400, 976, 2336, 5504, 12800, 29440, 67072, ...
The left column of the array (binomial transform of the sequence) is A067998.
For n>2, the sequence gives the number of permutations in the symmetric group S_{n+1} with peaks exactly in positions 2 and n-1. See Theorem 10 in [Billey-Burdzy-Sagan] reference.

Links

Crossrefs

Cf. A176027.

Programs

  • Magma
    [n*(n-3)*2^(n-2): n in [0..30]]; // Vincenzo Librandi, Aug 04 2011
  • Mathematica
    Table[n(n-3)2^(n-2),{n,0,30}] (* or *) LinearRecurrence[{6,-12,8},{0,-1,-2},30] (* Harvey P. Dale, Mar 24 2023 *)

Formula

a(n) = 16*A001793(n-3), n > 3.
a(n) = 8*A001788(n-2)-A052481(n-1). - R. J. Mathar, Jan 04 2011
a(n) = +6*a(n-1) -12*a(n-2) +8*a(n-3).
a(n+1)-a(n) = -A127276(n).
G.f.: -x*(-1+4*x)/(2*x-1)^3. - R. J. Mathar, Jan 04 2011
a(n) = Sum_{k=0..n-1} Sum_{i=0..n-1} (k-1) * C(n-1,i). - Wesley Ivan Hurt, Sep 20 2017
a(n) = Sum_{k=0..n} k^2 * (-1)^k * 3^(n-k) * binomial(n,k). - Seiichi Manyama, Apr 18 2025

A300451 a(n) = (3*n^2 - 3*n + 8)*2^(n - 3).

Original entry on oeis.org

1, 2, 7, 26, 88, 272, 784, 2144, 5632, 14336, 35584, 86528, 206848, 487424, 1134592, 2613248, 5963776, 13500416, 30343168, 67764224, 150470656, 332398592, 730857472, 1600126976, 3489660928, 7583301632, 16424894464, 35467034624, 76369887232, 164014063616
Offset: 0

Views

Author

Franck Maminirina Ramaharo, Mar 06 2018

Keywords

Comments

First difference yields A295288.
1 and 7 are the only odd terms.
a(n) gives the number of words of length n defined over the alphabet {a,b,c,d} such that letters from {a,b} are only used in pairs of at most one, and consist of (a,a), (a,b) and (b,a).

Examples

			a(4) = 88. The corresponding words are cccc, cccd, ccdc, ccdd, cdcc, cdcd, cddc, cddd, dccc, dccd, dcdc, dcdd, ddcc, ddcd, dddc, dddd, caac, caca, ccaa, caad, cada, caad, cabc, cacb, ccab, cabd, cadb, cabd, cbac, cbca, ccba, cbad, cbda, cbad, daac, daca, dcaa, daad, dada, daad, dabc, dacb, dcab, dabd, dadb, dabd, dbac, dbca, dcba, dbad, dbda, dbad, aacc, acac, acca, aacd, acad, acda, aadc, adac, adca, aadd, adad, adda, abcc, acbc, accb, abcd, acbd, acdb, abdc, adbc, adcb, abdd, adbd, addb, bacc, bcac, bcca, bacd, bcad, bcda, badc, bdac, bdca, badd, bdad, bdda.

References

  • Robert A. Beeler, How to Count: An Introduction to Combinatorics and Its Applications, Springer International Publishing, 2015.
  • Ian F. Blake, The Mathematical Theory of Coding, Academic Press, 1975.

Links

Crossrefs

Cf. A000079, A000292, A001788, A005448, A006003, A045943, A052481, A053730, A081908, A295288, A300184.

Programs

  • GAP
    List([0..30],n->(3*n^2-3*n+8)*2^(n-3)); # Muniru A Asiru, Mar 09 2018
  • Magma
    [(3*n^2-3*n+8)*2^(n-3): n in [0..30]]; // Vincenzo Librandi, Mar 10 2018
  • Maple
    A := n -> (3*n^2 - 3*n + 8)*2^(n - 3);
seq(A(n), n = 0 .. 70);
  • Mathematica
    Table[(3 n^2 - 3 n + 8) 2^(n - 3), {n, 0, 70}]
CoefficientList[Series[(1 - 4x + 7x^2)/(1 - 2x)^3, {x, 0, 30}], x] (* or *)
LinearRecurrence[{6, -12, 8}, {1, 2, 7}, 30] (* Robert G. Wilson v, Mar 07 2018 *)
  • Maxima
    makelist((3*n^2 - 3*n + 8)*2^(n - 3), n, 0, 70);
  • PARI
    a(n) = (3*n^2-3*n+8)*2^(n-3); \\ Altug Alkan, Mar 09 2018

Formula

G.f.: (1 - 4*x + 7*x^2)/(1 - 6*x + 12*x^2 - 8*x^3).
E.g.f: (1/2)*(3*x^2 + 2)*exp(2*x).
a(n) = ((3/4)*binomial(n, 2) + 1)*2^n.
a(n) = 2*a(n-1) + 3*(n - 1)*2^(n - 2), with a(0) = 1.
a(n) = 3*A001788(n) + A000079(n).
a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3), for n >= 3, with a(0) = 1, a(1) = 2 and a(2) = 7.
a(n) = A300184(n,2).
Showing 1-2 of 2 results.

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