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A101622 A Horadam-Jacobsthal sequence. 8

%I #51 Feb 06 2022 10:06:55

%S 0,1,6,13,30,61,126,253,510,1021,2046,4093,8190,16381,32766,65533,

%T 131070,262141,524286,1048573,2097150,4194301,8388606,16777213,

%U 33554430,67108861,134217726,268435453,536870910,1073741821,2147483646,4294967293,8589934590

%N A Horadam-Jacobsthal sequence.

%C Companion sequence to A084639.

%C This is the sequence A(0,1;1,2;5) of the family of sequences [a,b:c,d:k] considered by G. Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. - _Wolfdieter Lang_, Oct 18 2010

%C Except for the initial three terms, the decimal representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 961", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. - _Robert Price_, Mar 27 2017

%C Named after the Australian mathematician Alwyn Francis Horadam (1923-2016) and the German mathematician Ernst Jacobsthal (1882-1965). - _Amiram Eldar_, Jun 10 2021

%D Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

%H Vincenzo Librandi, <a href="/A101622/b101622.txt">Table of n, a(n) for n = 0..1000</a>

%H A. F. Horadam, <a href="http://www.fq.math.ca/Scanned/34-1/horadam2.pdf">Jacobsthal Representation Numbers</a>, Fib Quart., Vol. 34, No. 1 (1996), pp. 40-54.

%H Wolfdieter Lang, <a href="/A101622/a101622.pdf">Notes on certain inhomogeneous three term recurrences</a>. [_Wolfdieter Lang_, Oct 18 2010]

%H N. J. A. Sloane, <a href="http://arxiv.org/abs/1503.01168">On the Number of ON Cells in Cellular Automata</a>, arXiv:1503.01168 [math.CO], 2015.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>.

%H Stephen Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>.

%H Wolfram Research, <a href="http://atlas.wolfram.com/">Wolfram Atlas of Simple Programs</a>.

%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>

%H <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>

%H <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>]

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2).

%F a(n) = (2^(n+2) + (-1)^n - 5)/2.

%F G.f.: x*(1+4*x)/((1-x)*(1+x)*(1-2*x)).

%F a(n) = (A014551(n+2)-5)/2.

%F (1, 6, 13, 30, 61, ...) are the row sums of A131953. - _Gary W. Adamson_, Jul 31 2007

%F From _Paul Curtz_, Jan 01 2009: (Start)

%F a(n) = a(n-1) + 2*a(n-2) + 5.

%F a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3).

%F a(n) = A000079(n+1) - A010693(n).

%F a(n+1) = A141722(n) + 5 = A141722(n) + A010716(n).

%F a(2n+1) - a(2n) = 1, 7, 31, ... = A083420.

%F a(2n+1) - 2*a(2n) = 1.

%F a(2n) = A002446 = 6*A002450, a(2n+1) = A141725. (End)

%F a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n>2. - _Colin Barker_, Mar 28 2017

%F a(n) = (1/2) * Sum_{k=1..n} binomial(n+1,k) * (2+(-1)^k). - _Wesley Ivan Hurt_, Sep 23 2017

%t LinearRecurrence[{2,1,-2},{0,1,6},40] (* _Harvey P. Dale_, Jul 08 2014 *)

%o (Magma) [(2^(n+2)+(-1)^n-5)/2: n in [0..35]]; // _Vincenzo Librandi_, Aug 12 2011

%o (PARI) concat(0, Vec(x*(1+4*x)/((1-x)*(1+x)*(1-2*x)) + O(x^30))) \\ _Colin Barker_, Mar 28 2017

%Y Cf. A131953.

%K easy,nonn

%O 0,3

%A _Paul Barry_, Dec 10 2004

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)