A131865 Partial sums of powers of 16.

1, 17, 273, 4369, 69905, 1118481, 17895697, 286331153, 4581298449, 73300775185, 1172812402961, 18764998447377, 300239975158033, 4803839602528529, 76861433640456465, 1229782938247303441, 19676527011956855057, 314824432191309680913, 5037190915060954894609

Offset: 0

Author

Reinhard Zumkeller, Jul 22 2007

Keywords

• nonn
• easy

Comments

16 = 2^4 is the growth measure for the Jacobsthal spiral (compare with phi^4 for the Fibonacci spiral). - Paul Barry, Mar 07 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=0..n-1} a(i) for n>0. - 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

The sequence in binary is 1, 10001, 100010001, 1000100010001, 10001000100010001, ... cf. Plouffe link, A330135. - Frank Ellermann, Mar 05 2020

Examples

			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
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..800
• A. Abdurrahman, CM Method and Expansion of Numbers, arXiv:1909.10889 [math.NT], 2019.
• 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.
• Simon Plouffe, Identities and approximations inspired from Ramanujan notebooks, III, 2009.
• Index entries related to partial sums.
• Index entries related to q-numbers.
• Index entries for linear recurrences with constant coefficients, signature (17,-16).

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

Programs

Magma

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

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 *)
Accumulate[16^Range[0,20]] (* or *) LinearRecurrence[{17,-16},{1,17},20] (* Harvey P. Dale, Jul 19 2019 *)

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

Python

def A131865(n): return (1<<(n+1<<2))//15 # Chai Wah Wu, Nov 10 2022

Sage

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

Formula

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.

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

E.g.f.: exp(x)*(16*exp(15*x) - 1)/15. - Stefano Spezia, Mar 06 2020