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 for linear recurrences with constant coefficients, signature (17,-16).
FORMULA
a(n) = if n=0 then 1 else a(n-1) + A001025(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
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
KEYWORD
nonn,easy
AUTHOR
Reinhard Zumkeller, Jul 22 2007
STATUS
approved