A014903 -id:A014903 - cp's OEIS frontend

A218722 a(n) = (19^n-1)/18.

0, 1, 20, 381, 7240, 137561, 2613660, 49659541, 943531280, 17927094321, 340614792100, 6471681049901, 122961939948120, 2336276859014281, 44389260321271340, 843395946104155461, 16024522975978953760, 304465936543600121441

Offset: 0

M. F. Hasler, Nov 04 2012

Partial sums of powers of 19 (A001029); q-integers for q=19: diagonal k=1 in triangle A022183.

Partial sums are in A014903. Also, the sequence is related to A014936 by A014936(n) = n*a(n)-sum(a(i), i=0..n-1) for n>0. - Bruno Berselli, Nov 06 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..700
Index entries related to partial sums
Index entries for linear recurrences with constant coefficients, signature (20,-19).

Cf. similar sequences of the form (k^n-1)/(k-1): A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218721, A064108, A218724-A218734, A132469, A218736-A218753, A133853, A094028, A218723.

Cf. A001029, A014903, A014936, A022183.

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

LinearRecurrence[{20, -19}, {0, 1}, 40] (* Vincenzo Librandi, Nov 07 2012 *)

A218722(n):=(19^n-1)/18$ makelist(A218722(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

PARI
```
A218722(n)=19^n\18
```

a(n) = floor(19^n/18).
G.f.: x/((1-x)*(1-19*x)). - Bruno Berselli, Nov 06 2012
a(n) = 20*a(n-1) - 19*a(n-2). - Vincenzo Librandi, Nov 07 2012
E.g.f.: exp(10*x)*sinh(9*x)/9. - Stefano Spezia, Mar 11 2023

A014869 Numbers k that divide s(k), where s(1)=1, s(j)=19*s(j-1)+j.

Original entry on oeis.org

1, 3, 9, 19, 27, 57, 81, 171, 243, 381, 513, 729, 1143, 1539, 2187, 3429, 4617, 4707, 6561, 7239, 10287, 13053, 13149, 13851, 14121, 19683, 21717, 30861, 39159, 39447, 41553, 42363, 48387, 59049, 65151, 89433, 92583, 117477, 118341, 124659, 127089, 145161, 177147

Offset: 1

N. J. A. Sloane, Olivier Gérard

nonn

s(n) = A014903(n).

nxt[{n_,a_}]:={n+1,19a+n+1}; Transpose[Select[NestList[nxt,{1,1},70000], Divisible[Last[#],First[#]]&]][[1]] (* Harvey P. Dale, Jan 21 2015 *)

lista(nn) = my(s); for(k=1, nn, s=19*s+k; if(s%k==0, print1(k, ", "))); \\ Jinyuan Wang, Aug 08 2021

More terms from Jinyuan Wang, Aug 08 2021

cp's OEIS Frontend

A218722 a(n) = (19^n-1)/18.

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