A101622 A Horadam-Jacobsthal sequence.

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

Offset: 0

Author

Paul Barry, Dec 10 2004

Keywords

• easy
• nonn

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 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).

Crossrefs

Cf. A131953.

Programs

Magma

[(2^(n+2)+(-1)^n-5)/2: n in [0..35]]; // Vincenzo Librandi, Aug 12 2011

Mathematica

LinearRecurrence[{2,1,-2},{0,1,6},40] (* Harvey P. Dale, Jul 08 2014 *)

PARI

concat(0, Vec(x*(1+4*x)/((1-x)*(1+x)*(1-2*x)) + O(x^30))) \\ Colin Barker, Mar 28 2017

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