A218724 a(n) = (21^n - 1)/20.

0, 1, 22, 463, 9724, 204205, 4288306, 90054427, 1891142968, 39714002329, 833994048910, 17513875027111, 367791375569332, 7723618886955973, 162195996626075434, 3406115929147584115, 71528434512099266416, 1502097124754084594737, 31544039619835776489478

Offset: 0

Author

M. F. Hasler, Nov 04 2012

Keywords

• nonn
• easy

Comments

Partial sums of powers of 21 (A009965); q-integers for q=21: diagonal k=1 in triangle A022185.

Partial sums are in A014905. Also, the sequence is related to A014938 by A014938(n) = n*a(n) - Sum_{i=0..n-1} a(i) for n > 0. - Bruno Berselli, Nov 06 2012

For n >= 1, 4*a(n) is the total number of holes in a certain box fractal (start with 21 boxes, 4 holes) after n iterations. See illustration in links. - Kival Ngaokrajang, Jan 27 2015

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..700
• Kival Ngaokrajang, Illustration of initial terms
• Index entries related to partial sums
• Index entries for linear recurrences with constant coefficients, signature (22,-21).

Crossrefs

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, A218722, A064108, A218725-A218734, A132469, A218736-A218753, A133853, A094028, A218723.

Cf. A009965, A014905, A014938, A022185.

Programs

Magma

[n le 2 select n-1 else 22*Self(n-1) - 21*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 07 2012

Mathematica

LinearRecurrence[{22, -21}, {0, 1}, 40] (* Vincenzo Librandi, Nov 07 2012 *)

Maxima

A218724(n):=(21^n-1)/20$ makelist(A218724(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

PARI

A218724(n)=21^n\20

Formula

a(n) = floor(21^n/20).

G.f.: x/((1-x)*(1-21*x)). - Bruno Berselli, Nov 06 2012

a(n) = 22*a(n-1) - 21*a(n-2). - Vincenzo Librandi, Nov 07 2012

a(n) = 21*a(n-1) + 1. - Kival Ngaokrajang, Jan 27 2015

a(n) = a(n-1) + 21^(n-1), n >= 1, a(0) = 0. - Wolfdieter Lang, Feb 02 2015

E.g.f.: exp(11*x)*sinh(10*x)/10. - Elmo R. Oliveira, Aug 29 2024