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

0,2

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

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).

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

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

Prog

(Sage) [gaussian_binomial(n, 1, 16) for n in range(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
(Python)
def A131865(n): return (1<<(n+1<<2))//15 # Chai Wah Wu, Nov 10 2022

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 * A298373 A179093 A029811

Adjacent sequences: A131862 A131863 A131864 * A131866 A131867 A131868

Keyword

nonn,easy

Author

Reinhard Zumkeller, Jul 22 2007

Status

approved