OFFSET
0,1
COMMENTS
This sequence is the subject of the 4th problem of the 12th British Mathematical Olympiad in 1976 (see the link BMO).
Proposition: a(n) is never a prime number.
Proof:
If n is even, 3 divides a(n),
if n is odd with n = 4*k+1, 13 divides a(n), and
if n is odd with n = 4*k+3, 5 divides a(n).
REFERENCES
A. Gardiner, The Mathematical Olympiad Handbook: An Introduction to Problem Solving, Oxford University Press, 1997, reprinted 2011, Problem 4 pp. 70 and 216-217 (1991).
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
British Mathematical Olympiad, 1976 - Problem 4
Index entries for linear recurrences with constant coefficients, signature (9,-8).
FORMULA
a(n) = 19 * A001018(n) + 17.
From Colin Barker, Feb 25 2020: (Start)
G.f.: (36 - 155*x) / ((1 - x)*(1 - 8*x)).
a(n) = 9*a(n-1) - 8*a(n-2) for n>1.
(End)
E.g.f.: exp(x)*(17 + 19*exp(7*x)). - Stefano Spezia, Feb 25 2020
EXAMPLE
a(4) = 19 * 8^4 + 17 = 77841 = 3 * 25947.
a(5) = 19 * 8^5 + 17 = 622609 = 13 * 47893.
a(7) = 19 * 8^7 + 17 = 39845905 = 5 * 7969181.
MAPLE
B:=seq(19*8^n+17, n=0..40);
MATHEMATICA
Table[19 * 8^n + 17, {n, 0, 19}] (* Amiram Eldar, Feb 23 2020 *)
PROG
(PARI) Vec((36 - 155*x) / ((1 - x)*(1 - 8*x)) + O(x^20)) \\ Colin Barker, Feb 25 2020
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Bernard Schott, Feb 23 2020
STATUS
approved
