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 A219788 Consider the succession rule (x, y, z) -> (z, y+z, x+y+z). Sequence gives z values starting at (0, 1, 2). 1
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 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) 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). 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 EXAMPLE 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. 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 *) PROG (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 CROSSREFS Sequence in context: A182889 A256169 A298405 * A099965 A319566 A294450 Adjacent sequences:  A219785 A219786 A219787 * A219789 A219790 A219791 KEYWORD nonn,easy AUTHOR Andrew Pharo, Nov 27 2012 STATUS approved

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