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 A101622 A Horadam-Jacobsthal sequence. 8
 0, 1, 6, 13, 30, 61, 126, 253, 510, 1021, 2046, 4093, 8190, 16381, 32766, 65533, 131070, 262141, 524286, 1048573, 2097150, 4194301, 8388606, 16777213, 33554430, 67108861, 134217726, 268435453, 536870910, 1073741821, 2147483646, 4294967293, 8589934590 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Companion sequence to A084639. 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 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 Named after the Australian mathematician Alwyn Francis Horadam (1923-2016) and the German mathematician Ernst Jacobsthal (1882-1965). - Amiram Eldar, Jun 10 2021 REFERENCES Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 A. F. Horadam, Jacobsthal Representation Numbers, Fib Quart., Vol. 34, No. 1 (1996), pp. 40-54. Wolfdieter Lang, Notes on certain inhomogeneous three term recurrences. [Wolfdieter Lang, Oct 18 2010] N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015. Eric Weisstein's World of Mathematics, Elementary Cellular Automaton. Stephen Wolfram, A New Kind of Science. Wolfram Research, Wolfram Atlas of Simple Programs. Index entries for linear recurrences with constant coefficients, signature (2,1,-2). FORMULA a(n) = (2^(n+2) + (-1)^n - 5)/2. G.f.: x*(1+4*x)/((1-x)*(1+x)*(1-2*x)). a(n) = (A014551(n+2)-5)/2. (1, 6, 13, 30, 61, ...) are the row sums of A131953. - Gary W. Adamson, Jul 31 2007 From Paul Curtz, Jan 01 2009: (Start) a(n) = a(n-1) + 2*a(n-2) + 5. a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3). a(n) = A000079(n+1) - A010693(n). a(n+1) = A141722(n) + 5 = A141722(n) + A010716(n). a(2n+1) - a(2n) = 1, 7, 31, ... = A083420. a(2n+1) - 2*a(2n) = 1. a(2n) = A002446 = 6*A002450, a(2n+1) = A141725. (End) a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n>2. - Colin Barker, Mar 28 2017 a(n) = (1/2) * Sum_{k=1..n} binomial(n+1,k) * (2+(-1)^k). - Wesley Ivan Hurt, Sep 23 2017 MATHEMATICA LinearRecurrence[{2, 1, -2}, {0, 1, 6}, 40] (* Harvey P. Dale, Jul 08 2014 *) PROG (Magma) [(2^(n+2)+(-1)^n-5)/2: n in [0..35]]; // Vincenzo Librandi, Aug 12 2011 (PARI) concat(0, Vec(x*(1+4*x)/((1-x)*(1+x)*(1-2*x)) + O(x^30))) \\ Colin Barker, Mar 28 2017 CROSSREFS Cf. A131953. Sequence in context: A086652 A159694 A145976 * A256871 A192304 A343544 Adjacent sequences:  A101619 A101620 A101621 * A101623 A101624 A101625 KEYWORD easy,nonn AUTHOR Paul Barry, Dec 10 2004 STATUS approved

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Last modified September 26 09:30 EDT 2022. Contains 356993 sequences. (Running on oeis4.)