A064108 a(n) = (20^n - 1)/19.

0, 1, 21, 421, 8421, 168421, 3368421, 67368421, 1347368421, 26947368421, 538947368421, 10778947368421, 215578947368421, 4311578947368421, 86231578947368421, 1724631578947368421, 34492631578947368421, 689852631578947368421, 13797052631578947368421, 275941052631578947368421

Offset: 0

Author

Jason Earls, Sep 17 2001

Keywords

• nonn
• easy

Comments

Partial sums of powers of 20 (A009964), q-integers for q=20: diagonal k=1 in triangle A022184.

Partial sums are in A014904. Also, the sequence is related to A014937 by A014937(n) = n*a(n)-Sum_{i=0..n-1} a(i), for n>0. - Bruno Berselli, Nov 06 2012

For n >= 1, a(n) is the total number of holes in a certain box fractal (start with 20 boxes, 1 hole) after n iterations. See illustration in links. - Kival Ngaokrajang, Jan 28 2015

Examples

			From _N. J. A. Sloane_, Nov 04 2014: Can also be obtained by writing powers of 2 in a staggered array and adding them (cf. A249604). For example, a(9) is:
..........1
.........2
........4
.......8
.....16
....32
...64
.128
256
-----------
26947368421
		

Links

• M. F. Hasler, Table of n, a(n) for n = 0..100
• Kival Ngaokrajang, Illustration of initial terms
• Index entries related to partial sums
• Index entries related to q-numbers
• Index entries for linear recurrences with constant coefficients, signature (21,-20).

Crossrefs

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

Cf. also A249604.

Cf. A009964, A014904, A014937, A022184.

Programs

Maple

a:=n->sum(20^(n-j), j=0..n): seq(a(n), n=0..15); # Zerinvary Lajos, Feb 11 2007

Mathematica

(20^Range[20]-1)/19 (* or *) NestList[20#+1&,1,20] (* Harvey P. Dale, Oct 04 2012 *)

Maxima

A064108(n):=(20^n-1)/19$ makelist(A064108(n),n,1,30); /* Martin Ettl, Nov 05 2012 */

PARI

for (n=0, 100, write("b064108.txt", n, " ", (20^n - 1)/19))  \\ Harry J. Smith, Sep 07 2009

PARI

A064108(n)=20^n\19  \\ M. F. Hasler, Nov 04 2012

Sage

[gaussian_binomial(n,1,20) for n in range(1,17)] # Zerinvary Lajos, May 29 2009

Formula

a(n) = 20*a(n-1) + 1, with a(0)=0. - Vincenzo Librandi, Aug 07 2010

a(0)=0, a(1)=1, a(n) = 21*a(n-1) - 20*a(n-2). - Harvey P. Dale, Oct 04 2012

a(n) = floor(20^n/19). - M. F. Hasler, Nov 04 2012

G.f.: x/((1 - x)*(1 - 20*x)). - Bruno Berselli, Nov 06 2012

E.g.f.: exp(x)*(exp(19*x) - 1)/19. - Stefano Spezia, Mar 23 2023

Extensions

Edited and extended to offset 0 by M. F. Hasler, Nov 04 2012