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 A122983 a(n) = (2 + (-1)^n + 3^n)/4. 18
 1, 1, 3, 7, 21, 61, 183, 547, 1641, 4921, 14763, 44287, 132861, 398581, 1195743, 3587227, 10761681, 32285041, 96855123, 290565367, 871696101, 2615088301, 7845264903, 23535794707, 70607384121, 211822152361, 635466457083 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Old definition was: "Binomial transform of aeration of A081294". Binomial transform is A063376. A122983 = (1,1,3,7,1,1,3,7,...) mod 10. - M. F. Hasler, Feb 25 2008 Equals row sums of triangle A158301. - Gary W. Adamson, Mar 15 2009 a(n) = the number of ternary sequences of length n where the numbers of (0's, 1's) are both even. A015518 covers the (odd, even) and (even, odd) cases, and A081251 covers (odd, odd). - Toby Gottfried, Apr 18 2010 This sequence also describes the number of moves of the k-th disk solving (non-optimally) the [RED ; NEUTRAL ; BLUE] pre-colored Magnetic Tower of Hanoi (MToH) puzzle. The sequence A183119 is the partial sums of the sequence in question (obviously describing the total number of moves associated with the specific solution algorithm). For other MToH-related sequences, Cf. A183111 - A183125. Let B=[1,sqrt(2),0; sqrt(2),1,sqrt(2); 0,sqrt(2),1] be a 3 X 3 matrix. Then a(n)=[B^n]_(1,1), n=0,1,2,.... - L. Edson Jeffery, Dec 21 2011 Also the domination number of the n-Hanoi graph. - Eric W. Weisstein, Jun 16 2017 Also the matching number of the n-Sierpinski sieve graph. - Eric W. Weisstein, Jun 17 2017 Let M = [1,1,1,0; 1,1,0,1; 1,0,1,1; 0,1,1,1], a 4 X 4 matrix. Then a(n) is the upper left entry in M^n. - Philippe Deléham, Aug 23 2020 LINKS M. F. Hasler, Table of n, a(n) for n = 0..199. Ji Young Choi, A Generalization of Collatz Functions and Jacobsthal Numbers, J. Int. Seq., Vol. 21 (2018), Article 18.5.4. Alexander Diaz-Lopez, Pamela E. Harris, Erik Insko, Darleen Perez-Lavin, Peaks Sets of Classical Coxeter Groups, arXiv preprint, arXiv:1505.04479 [math.GR], 2015. A. M. Hinz, S. Klavžar, U. Milutinović, C. Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 99. Book's website Uri Levy, The Magnetic Tower of Hanoi, arXiv:1003.0225 [math.CO], 2010. Eric Weisstein's World of Mathematics, Domination Number Eric Weisstein's World of Mathematics, Matching Number Eric Weisstein's World of Mathematics, Hanoi Graph Eric Weisstein's World of Mathematics, Sierpinski Sieve Graph Index entries for linear recurrences with constant coefficients, signature (3,1,-3). FORMULA From Paul Barry, Jun 14 2007: (Start) G.f.: (1-2*x-x^2)/((1-x)*(1+x)*(1-3*x)); a(n) = 3^n/4+(-1)^n/4+1/2; E.g.f.: cosh(x)^2*exp(x). (End) a(n) = 3*a(n-1) + a(n-2) - 3*a(n-3); a(0)=1, a(1)=1, a(2)=3. - Harvey P. Dale, Sep 03 2013 E.g.f.: Q(0)/2, where Q(k) = 1 + 3^k/( 2 - 2*(-1)^k/( 3^k + (-1)^k - 2*x*3^k/( 2*x + (k+1)*(-1)^k/Q(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Dec 22 2013 a(2*n) = 3*a(2*n-1); a(2*n+1) = 3*a(2*n) - 2. - Philippe Deléham, Aug 23 2020 MAPLE A122983 := n -> ceil(3^n/4); 'A122983(n)' \$ n=0..22; # M. F. Hasler, Feb 25 2008 a[ -1]:=1:a:=1:a:=3:for n from 2 to 50 do a[n]:=2*a[n-1]+3*a[n-2]-2 od: seq(a[n], n=-1..25); # Zerinvary Lajos, Apr 28 2008 MATHEMATICA CoefficientList[Series[(1 - 2 x - x^2)/((1 - x) (1 + x) (1 - 3 x)), {x, 0, 40}], x] (* Harvey P. Dale, Sep 03 2013 *) LinearRecurrence[{3, 1, -3}, {1, 1, 3}, 40] (* Harvey P. Dale, Sep 03 2013 *) Table[(2 + (-1)^n + 3^n)/4, {n, 0, 20}] (* Eric W. Weisstein, Jun 16 2017 *) Table[Floor[3^n/4] + 1, {n, 0, 20}] (* Eric W. Weisstein, Jan 17 2018 *) Floor[3^Range[0, 20]/4] + 1 (* Eric W. Weisstein, Jan 17 2018 *) PROG (PARI) A122983(n)=3^n\4+1 \\ M. F. Hasler, Feb 25 2008 CROSSREFS Cf. a(j+1) = A137822(2^j) and these are the record values of A137822. Cf. A054879 (bisection), A066443 (bisection). Row sums of A158303. Sequence in context: A056779 A183113 A102877 * A005355 A182399 A025235 Adjacent sequences:  A122980 A122981 A122982 * A122984 A122985 A122986 KEYWORD easy,nonn AUTHOR Paul Barry, Sep 22 2006 EXTENSIONS Extended and corrected (existing Maple code) by M. F. Hasler, Feb 25 2008 Description changed to formula by Eric W. Weisstein, Jun 16 2017 STATUS approved

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Last modified May 11 02:34 EDT 2021. Contains 343784 sequences. (Running on oeis4.)