A094309 -id:A094309 - cp's OEIS frontend

A209240 Triangular array read by rows. T(n,k) is the number of ternary length-n words in which the longest run of consecutive 0's is exactly k; n>=0, 0<=k<=n.

1, 2, 1, 4, 4, 1, 8, 14, 4, 1, 16, 44, 16, 4, 1, 32, 132, 58, 16, 4, 1, 64, 384, 200, 60, 16, 4, 1, 128, 1096, 668, 214, 60, 16, 4, 1, 256, 3088, 2180, 740, 216, 60, 16, 4, 1, 512, 8624, 6992, 2504, 754, 216, 60, 16, 4, 1, 1024, 23936, 22128, 8332, 2576, 756, 216, 60, 16, 4, 1

Offset: 0

Geoffrey Critzer, Jan 13 2013

Row sums are 3^n.
Column k=0 is A000079.
Column k=1 is A094309.
Limit of reversed rows gives A120926.

			1;
2,   1;
4,   4,    1;
8,   14,   4,    1;
16,  44,   16,   4,   1;
32,  132,  58,   16,  4,   1;
64,  384,  200,  60,  16,  4,  1;
128, 1096, 668,  214, 60,  16, 4,  1;
256, 3088, 2180, 740, 216, 60, 16, 4,  1;

Alois P. Heinz, Rows n = 0..140, flattened

Cf. A048004.

nn=10;f[list_]:=Select[list,#>0&];Map[f,Transpose[Table[CoefficientList[ Series[(1-x^k)/(1-3x+2x^(k+1))-(1-x^(k-1))/(1-3x+2x^k),{x,0,nn}],x],{k,1,nn+1}]]]//Grid

O.g.f. for column k: (1-x)^2*x^k/(1-3*x+2*x^(k+1))/(1-3*x+2*x^(k+2)).

A176758 a(n) = Sum_{k=0..floor((n-1)/2)} (3^k-1)binomial(n, 2k+1).

Original entry on oeis.org

2, 8, 28, 88, 264, 768, 2192, 6176, 17248, 47872, 132288, 364416, 1001600, 2748416, 7532800, 20627968, 56452608, 154423296, 422276096, 1154447360, 3155544064, 8624177152, 23567831040, 64400793600, 175970803712, 480810303488

Offset: 3

Roger L. Bagula, Apr 25 2010

G. C. Greubel, Table of n, a(n) for n = 3..1000
Index entries for linear recurrences with constant coefficients, signature (4,-2,-4).

Cf. A002605, A094706.

I:=[2,8,28]; [n le 3 select I[n] else 4*Self(n-1) - 2*Self(n-2) +4*Self(n-3): n in [1..31]]; // G. C. Greubel, Sep 17 2021

(* First program *)
a[n_, q_]:= Sum[(q^((m-1)/2) - 1)*Binomial[n, m], {m,1,n,2}];
Table[a[n, 3], {n, 3, 30}]
(* Second program *)
A002605[n_]:= (-I*Sqrt[2])^(n-1)*ChebyshevU[n-1, I/Sqrt[2]];
Table[(Boole[n==0] - 2^n)/2 + A002605[n], {n, 3, 30}] (* G. C. Greubel, Sep 17 2021 *)

[(-i*sqrt(2))^(n-1)*chebyshev_U(n-1, i/sqrt(2)) - 2^(n-1) for n in (3..30)] # G. C. Greubel, Sep 17 2021

From R. J. Mathar, Jan 29 2012: (Start)
a(n) = A002605(n) - 2^(n-1) = 2*A094309(n).
G.f.: 2*x^3/( (1-2*x)*(1-2*x-2*x^2) ). (End)

cp's OEIS Frontend

A209240 Triangular array read by rows. T(n,k) is the number of ternary length-n words in which the longest run of consecutive 0's is exactly k; n>=0, 0<=k<=n.

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A176758 a(n) = Sum_{k=0..floor((n-1)/2)} (3^k-1)binomial(n, 2k+1).

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cp's OEIS Frontend

A209240 Triangular array read by rows. T(n,k) is the number of ternary length-n words in which the longest run of consecutive 0's is exactly k; n>=0, 0<=k<=n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A176758 a(n) = Sum_{k=0..floor((n-1)/2)} (3^k-1)*binomial(n, 2*k+1).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

A176758 a(n) = Sum_{k=0..floor((n-1)/2)} (3^k-1)binomial(n, 2k+1).