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A218724 a(n) = (21^n - 1)/20. 35
0, 1, 22, 463, 9724, 204205, 4288306, 90054427, 1891142968, 39714002329, 833994048910, 17513875027111, 367791375569332, 7723618886955973, 162195996626075434, 3406115929147584115, 71528434512099266416 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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).

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

MATHEMATICA

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

PROG

(PARI) A218724(n)=21^n\20

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

(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

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.

Sequence in context: A170655 A170703 A170741 * A276644 A139228 A240782

Adjacent sequences:  A218721 A218722 A218723 * A218725 A218726 A218727

KEYWORD

nonn,easy

AUTHOR

M. F. Hasler, Nov 04 2012

STATUS

approved

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Last modified October 19 07:21 EDT 2018. Contains 316337 sequences. (Running on oeis4.)