This site is supported by donations to The OEIS Foundation.

 Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing. Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A060647 Number of alpha-beta evaluations in a tree of depth n and branching factor b=3. 4
 1, 3, 5, 11, 17, 35, 53, 107, 161, 323, 485, 971, 1457, 2915, 4373, 8747, 13121, 26243, 39365, 78731, 118097, 236195, 354293, 708587, 1062881, 2125763, 3188645, 6377291, 9565937, 19131875, 28697813, 57395627, 86093441, 172186883 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 REFERENCES P. H. Winston, Artificial Intelligence, Addison-Wesley, 1977, pp. 115-122, (alpha-beta technique). LINKS Harry J. Smith, Table of n, a(n) for n = 0..500 FORMULA a(2n) = 2*(3^n) - 1, a(2n+1) = 3^n + 3^(n+1) - 1. Formula for b branches: a(2n) = 2*(b^n)-1, a(2n+1) = b^n+b^(n+1)-1. a(n) = A068911(n+1) - 1. G.f.: (1+2*z-z^2)/((1-z)*(1-3*z^2)). - Emeric Deutsch, Nov 18 2002 a(n) = (sqrt(3))^n(1+2/sqrt(3))+(1-2/sqrt(3))(-sqrt(3))^n-1. - Paul Barry, Apr 17 2004 a(2n+1) = 3*a(2n-1) + 2;  a(2n) = (a(2n-1) + a(2n+1))/2, with a(1)= 1. See A062318 for case where a(1)= 0. a(n) = (2^((1+(-1)^n)/2))*(b^((2*n-1+(-1)^n)/4))+((1-(-1)^n)/2)*(b^((2*n+1-(-1)^n)/4))-1, with b=3. - Luce ETIENNE, Aug 30 2014 EXAMPLE a(2n+1) = 2*a(2n) + 1, a(15) = a(2*7+1) = 2*a(14) + 1 = 2*4373 + 1 = 8747. MAPLE A060647 := proc(n, b) option remember: if n mod 2 = 0 then RETURN(2*b^(n/2)-1) else RETURN(b^((n-1)/2) +b^((n+1)/2)-1) fi: end: for n from 0 to 60 do printf(`%d, `, A060647(n, 3)) od: a[0]:=1:a[1]:=3:for n from 2 to 100 do a[n]:=3*a[n-2]+2 od: seq(a[n], n=0..33); # Zerinvary Lajos, Mar 17 2008 MATHEMATICA f[n_] := Simplify[Sqrt[3]^n(1 + 2/Sqrt[3]) + (1 - 2/Sqrt[3])(-Sqrt[3])^n - 1]; Table[ f[n], {n, 0, 34}] (* or *) f[n_] := If[ EvenQ[n], 2(3^(n/2)) - 1, 3^((n - 1)/2) + 3^((n + 1)/2) - 1]; Table[ f[n], {n, 0, 34}] (* or *) CoefficientList[ Series[(1 + 2x - x^2)/((1 - x)(1 - 3x^2)), {x, 0, 35}], x] (* Robert G. Wilson v, Nov 17 2005 *) PROG (PARI) { for (n=0, 500, if (n%2==0, a=2*(3^(n/2)) - 1, m=(n - 1)/2; a=3^m + 3^(m + 1) - 1); write("b060647.txt", n, " ", a); ) } \\ Harry J. Smith, Jul 09 2009 CROSSREFS For b=2 see A052955. Cf. A068911. Sequence in context: A006171 A261674 A319632 * A320353 A155989 A125557 Adjacent sequences:  A060644 A060645 A060646 * A060648 A060649 A060650 KEYWORD easy,nonn AUTHOR Frank Ellermann, Apr 17 2001 EXTENSIONS More terms from James A. Sellers, Apr 19 2001 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 5 20:41 EST 2019. Contains 329777 sequences. (Running on oeis4.)