A219788 Consider the succession rule (x, y, z) -> (z, y+z, x+y+z). Sequence gives z values starting at (0, 1, 2).

2, 3, 8, 17, 39, 87, 196, 440, 989, 2222, 4993, 11219, 25209, 56644, 127278, 285991, 642616, 1443945, 3244515, 7290359, 16381288, 36808420, 82707769, 185842670, 417584689, 938304279, 2108350577, 4737420744, 10644887786, 23918845739, 53745158520, 120764274993

Offset: 1

Author

Andrew Pharo, Nov 27 2012

Keywords

• nonn
• easy

Comments

The rule can be generalized for any number of starting terms s: (xs, ..., x2, x1) -> (x1, x1 + x2, ..., x1 + x2 + ... + xs), using (0, 1, ..., s-1) as seed values. This sequence is s=3, and s=2 yields the Fibonacci series.

For s=3 the ratio of S1 (the first in the sub-series) to S3 (the 3rd in the sub-series) converges on 2.2469796 and the ration of S2 (the 2nd in the sub-series) to S3 converges on 1.2469796 thus the difference, S2-S3, converges on 1 regardless of the seed values used.

For s=20 the series is: 19, 190, 2660, 33915, 445949, ....

a(n-2) is the top left entry of the n-th power of the 3 X 3 matrix [0, 1, 1; 1, 1, 1; 1, 0, 1] or of the 3 X 3 matrix [0, 1, 1; 1, 1, 0; 1, 1, 1]. - R. J. Mathar, Feb 03 2014

From Andrew Pharo, Jun 02 2014: (Start)

For s=2 the ratio of successive terms is 1.6180339887... or phi (or phi(2));

for s=3 this ratio is 2.24697960412319..., phi(3) = 4*cos(Pi/7)^2-1 (see Falbo link);

for s=4 this ratio is 3.5133370918694...;

for s=20 this ratio is 13.0538985560545... and so on.

We can define a function phi(s) which approximates to:

phi(s) ~ phi(2) + theta*(s-2) where theta ~ 0.636264133.

(End)

Examples

			The seed values are (0,1,2), giving a(1) = 2. (2, 2+1, 2+1+0) is the next triple, giving a(2) = 2+1+0 = 3. (3, 6, 8) is next, yielding a(3) = 8. The triples that follow begin (8,14,17), (17,31,39), etc.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..2844
• Clement Falbo, The Golden Ratio - A Contrary Viewpoint, Vol. 36, No. 2, March 2005, The College Mathematics Journal.
• Y-h. Guo, Some n-Color Compositions, J. Int. Seq. 15 (2012) 12.1.2, eq. (10) and Theorem 8.
• Brian Hopkins, Hua Wang, Restricted Color n-color Compositions, arXiv:2003.05291 [math.CO], 2020.
• R. Sachdeva and A. K. Agarwal, Combinatorics of certain restricted n-color composition functions, Discrete Mathematics, 340, (2017), 361-372.
• Index entries for linear recurrences with constant coefficients, signature (2,1,-1).

Programs

Mathematica

Rest@ CoefficientList[Series[-x (-2 + x)/(1 - 2 x - x^2 + x^3), {x, 0, 32}], x] (* Michael De Vlieger, Jun 17 2020 *)
sr[{x_,y_,z_}]:={z,y+z,x+y+z}; NestList[sr,{0,1,2},40][[All,3]] (* Harvey P. Dale, Aug 18 2020 *)

PARI

first(n)=my(x=0,y=1,z=2,v=List([z])); for(i=2, n, [x,y,z]=[z, y+z, x+y+z]; listput(v,c)); Vec(v) \\ Charles R Greathouse IV, Nov 28 2012

Formula

a(n) = 2a(n-1) + a(n-2) - a(n-3). - Charles R Greathouse IV, Nov 28 2012

The essentially identical sequence 1,0,2,3,8,17,39,... with offset 0 is defined by a(n) = 2a(n-1) + a(n-2) - a(n-3) with initial terms a(0)=1, a(1)=0, a(2)=2. - N. J. A. Sloane, Jan 16 2017

G.f.: -x*(-2+x) / ( 1-2*x-x^2+x^3 ). - R. J. Mathar, Feb 03 2014

a(n) = 2*A006054(n+1)-A006054(n). - R. J. Mathar, Aug 22 2016