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%I #42 Mar 23 2023 17:16:49
%S 0,1,21,421,8421,168421,3368421,67368421,1347368421,26947368421,
%T 538947368421,10778947368421,215578947368421,4311578947368421,
%U 86231578947368421,1724631578947368421,34492631578947368421,689852631578947368421,13797052631578947368421,275941052631578947368421
%N a(n) = (20^n - 1)/19.
%C Partial sums of powers of 20 (A009964), q-integers for q=20: diagonal k=1 in triangle A022184.
%C 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
%C 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
%H M. F. Hasler, <a href="/A064108/b064108.txt">Table of n, a(n) for n = 0..100</a>
%H Kival Ngaokrajang, <a href="/A064108/a064108.pdf">Illustration of initial terms</a>
%H <a href="/index/Par#partial">Index entries related to partial sums</a>
%H <a href="/index/Q#q-numbers">Index entries related to q-numbers</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (21,-20).
%F a(n) = 20*a(n-1) + 1, with a(0)=0. - _Vincenzo Librandi_, Aug 07 2010
%F a(0)=0, a(1)=1, a(n) = 21*a(n-1) - 20*a(n-2). - _Harvey P. Dale_, Oct 04 2012
%F a(n) = floor(20^n/19). - _M. F. Hasler_, Nov 04 2012
%F G.f.: x/((1 - x)*(1 - 20*x)). - _Bruno Berselli_, Nov 06 2012
%F E.g.f.: exp(x)*(exp(19*x) - 1)/19. - _Stefano Spezia_, Mar 23 2023
%e 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:
%e ..........1
%e .........2
%e ........4
%e .......8
%e .....16
%e ....32
%e ...64
%e .128
%e 256
%e -----------
%e 26947368421
%p a:=n->sum(20^(n-j), j=0..n): seq(a(n), n=0..15); # _Zerinvary Lajos_, Feb 11 2007
%t (20^Range[20]-1)/19 (* or *) NestList[20#+1&,1,20] (* _Harvey P. Dale_, Oct 04 2012 *)
%o (Sage) [gaussian_binomial(n,1,20) for n in range(1,17)] # _Zerinvary Lajos_, May 29 2009
%o (PARI) for (n=0, 100, write("b064108.txt", n, " ", (20^n - 1)/19)) \\ _Harry J. Smith_, Sep 07 2009
%o (PARI) A064108(n)=20^n\19 \\ _M. F. Hasler_, Nov 04 2012
%o (Maxima) A064108(n):=(20^n-1)/19$ makelist(A064108(n),n,1,30); /* _Martin Ettl_, Nov 05 2012 */
%Y Cf. A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A131865, A091045, A218722, A064108, A218724, ..., A218733, ..., A218743, ..., A218752, A094028.
%Y Cf. also A249604.
%Y Cf. A009964, A014904, A014937, A022184.
%K nonn,easy
%O 0,3
%A _Jason Earls_, Sep 17 2001
%E Edited and extended to offset 0 by _M. F. Hasler_, Nov 04 2012
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