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%I #110 Mar 31 2023 02:34:41
%S 1,17,273,4369,69905,1118481,17895697,286331153,4581298449,
%T 73300775185,1172812402961,18764998447377,300239975158033,
%U 4803839602528529,76861433640456465,1229782938247303441,19676527011956855057,314824432191309680913,5037190915060954894609
%N Partial sums of powers of 16.
%C 16 = 2^4 is the growth measure for the Jacobsthal spiral (compare with phi^4 for the Fibonacci spiral). - _Paul Barry_, Mar 07 2008
%C Second quadrisection of A115451. - _Paul Curtz_, May 21 2008
%C 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
%C 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
%C 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
%C 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
%C The sequence in binary is 1, 10001, 100010001, 1000100010001, 10001000100010001, ... cf. Plouffe link, A330135. - _Frank Ellermann_, Mar 05 2020
%H Vincenzo Librandi, <a href="/A131865/b131865.txt">Table of n, a(n) for n = 0..800</a>
%H A. Abdurrahman, <a href="https://arxiv.org/abs/1909.10889">CM Method and Expansion of Numbers</a>, arXiv:1909.10889 [math.NT], 2019.
%H Kival Ngaokrajang, <a href="/A131865/a131865.pdf">Illustration of initial terms</a>
%H Quynh Nguyen, Jean Pedersen, and Hien T. Vu, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Pedersen/pedersen2.html">New Integer Sequences Arising From 3-Period Folding Numbers</a>, Vol. 19 (2016), Article 16.3.1. See Table 1.
%H Simon Plouffe, <a href="http://plouffe.fr/simon/inspired3.pdf">Identities and approximations inspired from Ramanujan notebooks, III</a>, 2009.
%H <a href="/index/Par#partial">Index entries related to partial sums</a>.
%H <a href="/index/Q#q-numbers">Index entries related to q-numbers</a>.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (17,-16).
%F a(n) = if n=0 then 1 else a(n-1) + A001025(n).
%F for n > 0: A131851(a(n)) = n and abs(A131851(m)) < n for m < a(n).
%F a(n) = A098704(n+2)/2.
%F a(n) = (16^(n+1) - 1)/15. - _Bernard Schott_, Jun 06 2017
%F a(n) = (A001025(n+1) - 1)/15.
%F a(n) = 16*a(n-1) + 1. - _Paul Curtz_, May 20 2008
%F G.f.: 1 / ( (16*x-1)*(x-1) ). - _R. J. Mathar_, Feb 06 2011
%F E.g.f.: exp(x)*(16*exp(15*x) - 1)/15. - _Stefano Spezia_, Mar 06 2020
%e a(3) = 1 + 16 + 256 + 4096 = 4369 = in binary: 1000100010001.
%e 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
%p A131865:=n->(16^(n+1)-1)/15: seq(A131865(n), n=0..30); # _Wesley Ivan Hurt_, Apr 29 2017
%t Table[(2^(4 n) - 1)/15, {n, 16}] (* _Robert G. Wilson v_, Aug 22 2007 *)
%t Accumulate[16^Range[0,20]] (* or *) LinearRecurrence[{17,-16},{1,17},20] (* _Harvey P. Dale_, Jul 19 2019 *)
%o (Sage) [gaussian_binomial(n,1,16) for n in range(1,18)] # _Zerinvary Lajos_, May 28 2009
%o (Magma) [(16^(n+1)-1)/15: n in [0..20]]; // _Vincenzo Librandi_, Sep 17 2011
%o (Maxima)
%o a[0]:0$
%o a[n]:=16*a[n-1]+1$
%o A131865(n):=a[n]$
%o makelist(A131865(n),n,1,30); /* _Martin Ettl_, Nov 05 2012 */
%o (PARI) A131865(n)=16^n\15 \\ _M. F. Hasler_, Nov 05 2012
%o (Python)
%o def A131865(n): return (1<<(n+1<<2))//15 # _Chai Wah Wu_, Nov 10 2022
%Y 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
%K nonn,easy
%O 0,2
%A _Reinhard Zumkeller_, Jul 22 2007
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