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

0,3

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

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).

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

Example

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

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 *)

Prog

(Sage) [gaussian_binomial(n, 1, 20) for n in range(1, 17)] # Zerinvary Lajos, May 29 2009
(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
(Maxima) A064108(n):=(20^n-1)/19$ makelist(A064108(n), n, 1, 30); /* Martin Ettl, Nov 05 2012 */

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.

Sequence in context: A170654 A170702 A170740 * A353144 A067895 A215856

Adjacent sequences: A064105 A064106 A064107 * A064109 A064110 A064111

Keyword

nonn,easy

Author

Jason Earls, Sep 17 2001

Extensions

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

Status

approved