A347902 a(n) = a(n-1) + a(n-3) + a(n-4) with initial values a(0) = 8, a(1)=5, a(2) = 13, a(3) = 30.

8, 5, 13, 30, 43, 61, 104, 177, 281, 446, 727, 1185, 1912, 3085, 4997, 8094, 13091, 21173, 34264, 55449, 89713, 145150, 234863, 380025, 614888, 994901, 1609789, 2604702, 4214491, 6819181, 11033672, 17852865, 28886537, 46739390, 75625927, 122365329

Offset

0,1

Comments

For n >= 3, a(n) is also the number of ways to tile this "central staircase" figure of length n with squares and dominoes; this is the picture for length n=10:

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Links

Table of n, a(n) for n=0..35.

Index entries for linear recurrences with constant coefficients, signature (1,0,1,1).

Formula

G.f.: (8 - 3*x + 8*x^2 + 9*x^3)/((1-x-x^2)*(1+x^2)).

a(n) = (7*Lucas(n+3) + 6*i^(n*(n+1))*(3-(-1)^n))/5 where i = sqrt(-1).

E.g.f.: (12*cos(x) - 24*sin(x) + 14*exp(x/2)*(2*cosh(sqrt(5)*x/2) + sqrt(5)*sinh(sqrt(5)*x/2)))/5. - Stefano Spezia, Sep 18 2021

5*a(n) = 7*A000032(n+3) - 12 *(-1)^floor((n-1)/2)*A000034(n). - R. J. Mathar, Sep 30 2021

From Greg Dresden, Mar 19 2024: (Start)

a(2*n) = (7*Lucas(2*n+3) + 12*(-1)^n)/5.

a(2*n+1) = (7*Lucas(2*n+4) - 24*(-1)^n)/5. (End)

Example

Here is one of the a(10)=727 tilings for n=10.
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Mathematica

LinearRecurrence[{1, 0, 1, 1}, {8, 5, 13, 30}, 33]

Crossrefs

Cf. A000032, A000034.

Sequence in context: A087462 A168204 A193681 * A253806 A348688 A199806

Adjacent sequences: A347899 A347900 A347901 * A347903 A347904 A347905

Keyword

nonn,easy

Author

Reeva Bohra and Greg Dresden, Sep 18 2021

Status

approved