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A101622 A Horadam-Jacobsthal sequence. 9
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.

a(n)=a(n-1)+2a(n-2)+5; a(n)=2a(n-1)+a(n-2)-2a(n-3); a(n)=A000079(n+1)-A010693(n). Note a(n+1)=A141722(n)+5=A141722(n)+A010716(n). a(2n+1)-a(2n)=1,7,31,=A083420. a(2n+1)-2a(2n)=1. a(2n)=A002446=6*A002450, a(2n+1)=A141725. [Paul Curtz, Jan 01 2009]

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

REFERENCES

S. 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. 34, 40-54, 1996.

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

S. Wolfram, A New Kind of Science

Wolfram Research, Wolfram Atlas of Simple Programs

Index entries for sequences related to cellular automata

Index to 2D 5-Neighbor Cellular Automata

Index to Elementary Cellular Automata]

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

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} C(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 A147330

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 February 16 02:39 EST 2019. Contains 320140 sequences. (Running on oeis4.)