A185691 Fibonacci sequence with initial terms 10 and 21.

10, 21, 31, 52, 83, 135, 218, 353, 571, 924, 1495, 2419, 3914, 6333, 10247, 16580, 26827, 43407, 70234, 113641, 183875, 297516, 481391, 778907, 1260298, 2039205, 3299503, 5338708, 8638211, 13976919, 22615130, 36592049, 59207179, 95799228

Offset

0,1

Comments

From Wajdi Maaloul, Jun 23 2022: (Start)

For n>0, a(n) is the number of ways to tile the figure below with squares and dominoes (a strip of length n+2 that contains two vertical strip of height 3 in its first and third tiles).

_ _

|_| |_|

|_|_|_|_____ _

|_|_|_|_|_|_|...|_|

(End)

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

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

Formula

From G. C. Greubel, Jul 10 2017: (Start)

a(n+2) = a(n+1) + a(n) with a(0) = 10, a(1) = 21.

a(n) = 11*Fibonacci(n) + 10*Fibonacci(n+1).

a(n) = 9*Fibonacci(n+2) + Lucas(n+1).

G.f.: (10 + 11*x)/(1 - x - x^2).

E.g.f.: (1/sqrt(5))*exp(x/2)*(32*sinh(sqrt(5)*x/2) + 10*sqrt(5)*cosh(sqrt(5)*x/2)). (End)

Mathematica

Join[{a=10, b=21}, Table[c=a+b; a=b; b=c, {n, 60}]]
CoefficientList[Series[(10 + 11*x)/(1 - x - x^2), {x, 0, 50}], x] (* G. C. Greubel, Jul 10 2017 *)

Prog

(PARI) x='x+O('x^50); Vec((10 + 11*x)/(1 - x - x^2)) \\ G. C. Greubel, Jul 10 2017

Crossrefs

Cf. A000032, A000045.

Sequence in context: A245071 A256825 A190326 * A042291 A041194 A042871

Adjacent sequences: A185688 A185689 A185690 * A185692 A185693 A185694

Keyword

nonn

Author

Vladimir Joseph Stephan Orlovsky, Feb 28 2011

Status

approved