A190995 Fibonacci sequence beginning 9, 7.

9, 7, 16, 23, 39, 62, 101, 163, 264, 427, 691, 1118, 1809, 2927, 4736, 7663, 12399, 20062, 32461, 52523, 84984, 137507, 222491, 359998, 582489, 942487, 1524976, 2467463, 3992439, 6459902, 10452341, 16912243, 27364584, 44276827, 71641411, 115918238, 187559649

Offset: 0

Author

Vladimir Joseph Stephan Orlovsky, Jun 07 2011

Keywords

• nonn
• easy

Comments

From Wajdi Maaloul, Jun 20 2022: (Start)

For n>0, 2*a(n) is the number of ways to tile this figure below with squares and dominoes (a strip of length n+1 that begins with a length 3 vertical strip and length 4 one).

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(End)

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,1).

Crossrefs

Cf. A000032, A000045, A190994.

Programs

Magma

[n le 2 select 11-2*n else Self(n-1)+Self(n-2): n in [1..50]]; // Vincenzo Librandi, Feb 15 2012

Maple

a:= n-> (<<0|1>, <1|1>>^n. <<9, 7>>)[1, 1]:
seq(a(n), n=0..36);  # Alois P. Heinz, Oct 26 2022

Mathematica

LinearRecurrence[{1, 1}, {9, 7}, 100]

PARI

a(n)=7*fibonacci(n)+9*fibonacci(n-1) \\ Charles R Greathouse IV, Jun 08 2011

SageMath

[7*fibonacci(n) + 9*fibonacci(n-1) for n in range(51)] # G. C. Greubel, Oct 26 2022

Formula

a(n) = ((9+sqrt(5))/2)*((1+sqrt(5))/2)^n + ((9-sqrt(5))/2)*((1-sqrt(5))/2)^n. - Antonio Alberto Olivares

G.f.: (9-2*x)/(1-x-x^2). - Colin Barker, Jan 11 2012

a(n) = 7*Fibonacci(n) + 9*Fibonacci(n-1) = 7*Fibonacci(n+1) + 2*Fibonacci(n-1) = 7*Lucas(n) - 5*Fibonacci(n-1) for n>0. - Wajdi Maaloul, Jun 20 2022