A014935 -id:A014935 - cp's OEIS frontend

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

M. F. Hasler, Nov 04 2012

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)

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

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).

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.

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

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

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

PARI
```
A218721(n)=18^n\17
```

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

A014901 a(1)=1, a(n) = 18*a(n-1) + n.

Original entry on oeis.org

1, 20, 363, 6538, 117689, 2118408, 38131351, 686364326, 12354557877, 222382041796, 4002876752339, 72051781542114, 1296932067758065, 23344777219645184, 420205989953613327, 7563707819165039902, 136146740744970718253, 2450641333409472928572, 44111544001370512714315

Offset: 1

N. J. A. Sloane, Olivier Gérard

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (20,-37,18).

Row n=18 of A126885.

Cf. A014935.

I:=[1, 20, 363]; [n le 3 select I[n] else 20*Self(n-1) - 37*Self(n-2) + 18*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Oct 20 2012

LinearRecurrence[{20, -37, 18}, {1, 20, 363}, 20] (* Vincenzo Librandi, Oct 20 2012 *)
nxt[{n_,a_}]:={n+1,18a+n+1}; NestList[nxt,{1,1},20][[All,2]] (* Harvey P. Dale, Oct 08 2017 *)

a[1]:1$
a[2]:20$
a[3]:363$
a[n]:=20*a[n-1]-37*a[n-2]+18*a[n-3]$
A014901(n):=a[n]$
makelist(A014901(n),n,1,30); /* Martin Ettl, Nov 06 2012 */

a(1)=1, a(2)=20, a(3)=363, a(n) = 20*a(n-1) - 37*a(n-2) + 18*a(n-3). - Vincenzo Librandi, Oct 20 2012
From Elmo R. Oliveira, Mar 29 2025: (Start)
G.f.: x/((1-18*x)*(1-x)^2).
E.g.f.: exp(x)*(18*exp(17*x) - 17*x - 18)/289.
a(n) = (18^(n+1) - 17*n - 18)/289. (End)
a(-1-n) = A014935(n)/18^n for all n in Z. - Michael Somos, Mar 29 2025

cp's OEIS Frontend

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

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