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 A131865 Partial sums of powers of 16. 45
 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 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=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 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). 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 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

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Last modified June 7 05:20 EDT 2020. Contains 334837 sequences. (Running on oeis4.)