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 A183112 Magnetic Tower of Hanoi, total number of moves, optimally solving the [RED ; BLUE ; NEUTRAL] or [NEUTRAL ; RED ; BLUE] pre-colored puzzle. 2
 0, 1, 4, 13, 38, 113, 336, 1001, 2994, 8965, 26868, 80565, 241630, 724793, 2174232, 6522465, 19567050, 58700621, 176101052, 528301933, 1584903926, 4754708929, 14264122464, 42792360793, 128377072354, 385131201813, 1155393582212, 3466180711333, 10398542080270 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS The Magnetic Tower of Hanoi puzzle is described in link 1 listed below. The Magnetic Tower is pre-colored. Pre-coloring is [RED ; BLUE ; NEUTRAL] or [NEUTRAL ; RED ; BLUE], given in [Source ; Intermediate ; Destination] order. The solution algorithm producing the sequence is optimal (the sequence presented gives the minimum number of moves to solve the puzzle for the given pre-coloring configurations). Optimal solutions are discussed and their optimality is proved in link 2 listed below. Number of moves to solve the given puzzle, for large N, is close to 0.5*(10/11)*3^N ~ 0.5*0.909*3^(N). Series designation: S909(N). REFERENCES "The Magnetic Tower of Hanoi", Uri Levy, Journal of Recreational Mathematics, Volume 35 Number 3 (2006),  2010, pp 173. LINKS U. Levy, The Magnetic Tower of Hanoi, arXiv:1003.0225 Uri Levy, Magnetic Towers of Hanoi and their Optimal Solutions, arXiv:1011.3843 Web applet to play The Magnetic Tower of Hanoi Index entries for linear recurrences with constant coefficients, signature (4,-2,-2,-5,6). FORMULA Recurrence Relations (a(n)=S909(n) as in the referenced papers): a(n) = a(n-2) + a(n-3) + 3^(n-1) + 3^(n-3) + 2; n >= 3 ; a(0)  = 0 Closed-Form Expression: Define: λ1 = [1+sqrt(26/27)]^(1/3) +  [1-sqrt(26/27)]^(1/3) λ2 = -0.5* λ1 + 0.5*i*{[sqrt(27)+sqrt(26)]^(1/3)- [sqrt(27)-sqrt(26)]^(1/3)} λ3 = -0.5* λ1 - 0.5*i*{[sqrt(27)+sqrt(26)]^(1/3)- [sqrt(27)-sqrt(26)]^(1/3)} AS = [(7/11)* λ2* λ3 - (10/11)*(λ2 + λ3) + (19/11)]/[( λ2 - λ1)*( λ3 - λ1)] BS = [(7/11)* λ1* λ3 - (10/11)*(λ1 + λ3) + (19/11)]/[( λ1 - λ2)*( λ3 - λ2)] CS = [(7/11)* λ1* λ2 - (10/11)*(λ1 + λ2) + (19/11)]/[( λ2 - λ3)*( λ1 - λ3)] For any n > 0: a(n) = (5/11)*3^n + AS* λ1^(n-1) + BS* λ2^(n-1) + CS* λ3^(n-1) - 1. G.f.: x*(1-x^2-4*x^3)/((1-x)*(1-3*x)*(1-x^2-2*x^3)); a(n)=4*a(n-1)-2*a(n-2)-2*a(n-3)-5*a(n-4)+6*a(n-5) with n>5. - Bruno Berselli, Dec 29 2010 MATHEMATICA LinearRecurrence[{4, -2, -2, -5, 6}, {0, 1, 4, 13, 38}, 21] (* Jean-François Alcover, Dec 14 2018 *) CROSSREFS A183111 is an "original" sequence describing the number of moves of disk number k, optimally solving the pre-colored puzzle at hand. The integer sequence listed above is the partial sums of the A183111 original sequence. A003462 "Partial sums of A000244" is the sequence (also) describing the total number of moves solving [RED ; BLUE ; BLUE] or [RED ; RED ; BLUE] pre-colored Magnetic Tower of Hanoi puzzle. Cf. A183111 - A183125. Sequence in context: A159036 A058693 A027076 * A266429 A105693 A215404 Adjacent sequences:  A183109 A183110 A183111 * A183113 A183114 A183115 KEYWORD nonn AUTHOR Uri Levy, Dec 25 2010 STATUS approved

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Last modified December 4 12:21 EST 2020. Contains 338923 sequences. (Running on oeis4.)