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A131865 Partial sums of powers of 16. 42
1, 17, 273, 4369, 69905, 1118481, 17895697, 286331153, 4581298449, 73300775185, 1172812402961, 18764998447377, 300239975158033, 4803839602528529, 76861433640456465, 1229782938247303441, 19676527011956855057 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n) = if n=0 then 1 else a(n-1)+A001025(n);

for n>0: A131851(a(n))=n and abs(A131851(m))<n for m<a(n);

a(n) = A098704(n+2)/2.

16=2^4 is the growth measure for the Jacobsthal spiral (compare with phi^4 for the Fibonacci spiral). - Paul Barry, Mar 07 2008

Bisection of A115451. - Paul Curtz, May 20 2008

Second quadrisection of A115451. - Paul Curtz, May 21 2008

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=16, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=det(A). - Milan Janjic, Feb 21 2010

Partial sums are in A014899. Also, the sequence is related to A014931 by A014931(n+1) = (n+1)*a(n) - Sum_{i=1..n} a(i). - Bruno Berselli, Nov 07 2012

a(n) is the total number of holes in a certain box fractal (start with 16 boxes, 1 hole) after n iterations. See illustration in links. - Kival Ngaokrajang, Jan 28 2015

Except for 1 and 17, all terms are Brazilian repunits numbers in base 16, and so belong to A125134. All terms >= 273 are composite because a(n) = {[4^(n+1) -1] * [4^(n+1) - 1]}/15. - Bernard Schott, Jun 06 2017

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..800

Kival Ngaokrajang, Illustration of initial terms

Quynh Nguyen, Jean Pedersen, and Hien T. Vu, New Integer Sequences Arising From 3-Period Folding Numbers, Vol. 19 (2016), Article 16.3.1. See Table 1.

Index entries related to partial sums.

Index entries related to q-numbers.

Index entries for linear recurrences with constant coefficients, signature (17,-16).

FORMULA

a(n) = (16^(n+1)-1)/15. - - Bernard Schott, Jun 06 2017

a(n) = (A001025(n+1)-1)/15.

a(n) = 16*a(n-1)+1. - Paul Curtz, May 20 2008

G.f.: 1 / ( (16*x-1)*(x-1) ). - R. J. Mathar, Feb 06 2011

EXAMPLE

a(3) = 1+16+256+4096 = 4369 = in binary: 1000100010001.

a(4) = (16^5 - 1)/15 = (4^5 + 1) * (4^5 - 1)/15 = 1025 * 1023/15 = 205 * 341 = 69905 = (11111)_16. - Bernard Schott, Jun 06 2017

MAPLE

A131865:=n->(16^(n+1)-1)/15: seq(A131865(n), n=0..30); # Wesley Ivan Hurt, Apr 29 2017

MATHEMATICA

Table[(2^(4 n) - 1)/15, {n, 16}] (* Robert G. Wilson v, Aug 22 2007 *)

PROG

(Sage) [gaussian_binomial(n, 1, 16) for n in xrange(1, 18)] # Zerinvary Lajos, May 28 2009

(MAGMA) [(16^(n+1)-1)/15: n in [0..20]]; // Vincenzo Librandi, Sep 17 2011

(Maxima)

a[0]:0$

a[n]:=16*a[n-1]+1$

A131865(n):=a[n]$

makelist(A131865(n), n, 1, 30); /* Martin Ettl, Nov 05 2012 */

(PARI) A131865(n)=16^n\15  \\ M. F. Hasler, Nov 05 2012

CROSSREFS

Cf. A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A091045, A218721, A218722, A064108, A218724-A218734, A132469, A218736-A218753, A133853, A094028, A218723. - M. F. Hasler, Nov 05 2012

Sequence in context: A170650 A170698 A170736 * A179093 A029811 A113076

Adjacent sequences:  A131862 A131863 A131864 * A131866 A131867 A131868

KEYWORD

nonn,easy

AUTHOR

Reinhard Zumkeller, Jul 22 2007

STATUS

approved

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Last modified July 27 04:25 EDT 2017. Contains 289841 sequences.