A330770 a(n) = 19 * 8^n + 17 for n >= 0.

36, 169, 1233, 9745, 77841, 622609, 4980753, 39845905, 318767121, 2550136849, 20401094673, 163208757265, 1305670058001, 10445360463889, 83562883710993, 668503069687825, 5348024557502481, 42784196460019729, 342273571680157713, 2738188573441261585

Offset: 0

Author

Bernard Schott, Feb 23 2020

Keywords

• nonn
• easy

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).

Examples

			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.
		

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 to sequences related to Olympiads
• Index entries for linear recurrences with constant coefficients, signature (9,-8).

Crossrefs

Cf. A001018 (8^n).

Programs

Maple

B:=seq(19*8^n+17, n=0..40);

Mathematica

Table[19 * 8^n + 17, {n, 0, 19}] (* Amiram Eldar, Feb 23 2020 *)

PARI

Vec((36 - 155*x) / ((1 - x)*(1 - 8*x)) + O(x^20)) \\ Colin Barker, Feb 25 2020

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