A243399 a(0) = 1, a(1) = 19; for n > 1, a(n) = 19*a(n-1) + a(n-2).

1, 19, 362, 6897, 131405, 2503592, 47699653, 908796999, 17314842634, 329890807045, 6285240176489, 119749454160336, 2281524869222873, 43468721969394923, 828187242287726410, 15779026325436196713, 300629687425575463957, 5727743087411370011896

Offset

0,2

Comments

a(n+1)/a(n) tends to (19 + sqrt(365))/2.

a(n) equals the number of words of length n on alphabet {0,1,...,19} avoiding runs of zeros of odd lengths. - Milan Janjic, Jan 28 2015

From Michael A. Allen, May 03 2023: (Start)

Also called the 19-metallonacci sequence; the g.f. 1/(1-k*x-x^2) gives the k-metallonacci sequence.

a(n) is the number of tilings of an n-board (a board with dimensions n X 1) using unit squares and dominoes (with dimensions 2 X 1) if there are 19 kinds of squares available. (End)

Links

Bruno Berselli, Table of n, a(n) for n = 0..200

Michael A. Allen and Kenneth Edwards, Fence tiling derived identities involving the metallonacci numbers squared or cubed, Fib. Q. 60:5 (2022) 5-17.

Tanya Khovanova, Recursive Sequences.

Eric Weisstein's World of Mathematics, Fibonacci Polynomial.

Index entries for linear recurrences with constant coefficients, signature (19,1).

Formula

G.f.: 1/(1 - 19*x - x^2).

a(n) = (-1)^n*a(-n-2) = ((19 + sqrt(365))^(n+1)-(19 - sqrt(365))^(n+1))/(2^(n+1)*sqrt(365)).

a(n) = F(n+1, 19), the (n+1)-th Fibonacci polynomial evaluated at x = 19.

a(n)*a(n-2) - a(n-1)^2 = (-1)^n, with a(-2)=1, a(-1)=0.

Mathematica

RecurrenceTable[{a[n] == 19 a[n - 1] + a[n - 2], a[0] == 1, a[1] == 19}, a, {n, 0, 20}]

Prog

(PARI) v=vector(20); v[1]=1; v[2]=19; for(i=3, #v, v[i]=19*v[i-1]+v[i-2]); v
(Magma) [n le 2 select 19^(n-1) else 19*Self(n-1)+Self(n-2): n in [1..20]];
(Maxima) a[0]:1$ a[1]:19$ a[n]:=19*a[n-1]+a[n-2]$ makelist(a[n], n, 0, 20);
(Sage)
from sage.combinat.sloane_functions import recur_gen2
a = recur_gen2(1, 19, 19, 1)
[next(a) for i in (0..20)]

Crossrefs

Row n=19 of A073133, A172236 and A352361 and column k=19 of A157103.

Sequences with g.f. 1/(1-k*x-x^2) or x/(1-k*x-x^2): A000045 (k=1), A000129 (k=2), A006190 (k=3), A001076 (k=4), A052918 (k=5), A005668 (k=6), A054413 (k=7), A041025 (k=8), A099371 (k=9), A041041 (k=10), A049666 (k=11), A041061 (k=12), A140455 (k=13), A041085 (k=14), A154597 (k=15), A041113 (k=16), A178765 (k=17), A041145 (k=18), this sequence (k=19), A041181 (k=20). Also, many other sequences are in the OEIS with even k greater than 20 (denominators of continued fraction convergents to sqrt((k/2)^2+1)).

Sequence in context: A128360 A001029 A057685 * A041686 A263371 A023283

Adjacent sequences: A243396 A243397 A243398 * A243400 A243401 A243402

Keyword

nonn,easy

Author

Bruno Berselli, Jun 04 2014

Status

approved