A337142 a(n) is the number of words of length n over the alphabet {0,1,2} with at least two 1's and exactly one occurrence of the subword 22.

0, 0, 0, 0, 3, 18, 69, 216, 597, 1518, 3633, 8304, 18306, 39192, 81906, 167736, 337623, 669522, 1310559, 2536224, 4858719, 9224262, 17370693, 32472816, 60302340, 111305040, 204307620, 373111680, 678188235, 1227359874, 2212281369, 3972626952, 7108762953

Offset

0,5

Links

Table of n, a(n) for n=0..32.

Maria Juliana Mantilla Morales, Segundo trabajo matemática computacional (in Spanish).

Index entries for linear recurrences with constant coefficients, signature (4,-2,-8,5,8,-2,-4,-1).

Formula

a(n) = (((1+sqrt(5))/2)^n)*(-3*sqrt(5)/125 + ((135+13*sqrt(5))/500)*n + ((-45-9*sqrt(5))/250)*n^2 + ((3+sqrt(5))/100)*n^3) + (((1-sqrt(5))/2)^n)*(3*sqrt(5)/125 + ((135-13*sqrt(5))/500)*n + ((-45+9*sqrt(5))/250)*n^2 + ((3-sqrt(5))/100)*n^3).

G.f.: 3*x^4*(x+1)^2/(x^2+x-1)^4. - Alois P. Heinz, Sep 14 2020

E.g.f.: (cosh(x/2) + sinh(x/2))*(5*x*(6 - 15*x + 11*x^2)*cosh(sqrt(5)*x/2) + sqrt(5)*(-12 + 30*x - 27*x^2 + 25*x^3)*sinh(sqrt(5)*x/2))/250. - Stefano Spezia, Sep 16 2020

a(n) = (n-3)*((n^2+3*n+2)*F(n) + 3*(n-3)*n*F(n+1))/50, where F(n) is the n-th Fibonacci number. - Vaclav Kotesovec, Sep 16 2020

Example

a(4) = 3: 1122, 1221, 2211.
a(5) = 18: 01122, 10122, 11022, 21122, 12122, 02211, 22011, 22101, 22110, 22112, 22121, 01221, 10221, 12201, 12210, 21221, 12212, 11220.
The word 11222 is not included because the subword 22 occurs more than once (exactly twice).

Crossrefs

Cf. A000045.

Sequence in context: A027333 A026576 A048899 * A107583 A373651 A157535

Adjacent sequences: A337139 A337140 A337141 * A337143 A337144 A337145

Keyword

nonn,easy

Author

Maria Juliana Mantilla Morales, Sep 14 2020

Extensions

a(16)-a(32) from Alois P. Heinz, Sep 14 2020

Status

approved