A218721 a(n) = (18^n-1)/17.

0, 1, 19, 343, 6175, 111151, 2000719, 36012943, 648232975, 11668193551, 210027483919, 3780494710543, 68048904789775, 1224880286215951, 22047845151887119, 396861212733968143, 7143501829211426575, 128583032925805678351

Offset: 0

Author

M. F. Hasler, Nov 04 2012

Keywords

• nonn
• easy

Comments

Partial sums of powers of 18 (A001027), q-integers for q=18: diagonal k=1 in triangle A022182.

Partial sums are in A014901. Also, the sequence is related to A014935 by A014935(n) = n*a(n) - Sum_{i=0..n-1} a(i), for n>0. - Bruno Berselli, Nov 06 2012

From Bernard Schott, May 06 2017: (Start)

Except for 0, 1 and 19, all terms are Brazilian repunits numbers in base 18, and so belong to A125134. From n = 3 to n = 8286, all terms are composite. See link "Generalized repunit primes".

As explained in the extensions of A128164, a(25667) = (18^25667 - 1)/17 would be (is) the smallest prime in base 18. (End)

Examples

			a(3) = (18^3 - 1)/17 = 343 = 7 * 49; a(6) = (18^6 - 1)/17 = 2000719 = 931 * 2149. - _Bernard Schott_, May 01 2017
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..800
• Harvey Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930.
• Index entries related to partial sums
• Index entries for linear recurrences with constant coefficients, signature (19,-18).

Crossrefs

Cf. A000225, A001027, A002275, A002450, A002452, A003462, A003463, A003464, A014901, A014935, A016123, A016125, A022182, A023000, A023001, A064108, A091030, A091045, A094028, A125134, A128164, A131865, A135518, A135519, A218722, A218724, A218733, A218743, A218752.

Programs

Magma

[n le 2 select n-1 else 19*Self(n-1)-18*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

Mathematica

LinearRecurrence[{19, -18}, {0, 1}, 40] (* Vincenzo Librandi, Nov 07 2012 *)
Join[{0},Accumulate[18^Range[0,20]]] (* Harvey P. Dale, Nov 08 2012 *)

Maxima

A218721(n):=(18^n-1)/17$ makelist(A218721(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

PARI

A218721(n)=18^n\17

Formula

a(n) = floor(18^n/17).

G.f.: x/((1-x)*(1-18*x)). - Bruno Berselli, Nov 06 2012

a(n) = 19*a(n-1) - 18*a(n-2). - Vincenzo Librandi, Nov 07 2012

E.g.f.: exp(x)*(exp(17*x) - 1)/17. - Stefano Spezia, Mar 11 2023