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A122983 a(n) = (2 + (-1)^n + 3^n)/4.

%I A122983 #94 Jun 17 2025 22:23:56
%S A122983 1,1,3,7,21,61,183,547,1641,4921,14763,44287,132861,398581,1195743,
%T A122983 3587227,10761681,32285041,96855123,290565367,871696101,2615088301,
%U A122983 7845264903,23535794707,70607384121,211822152361,635466457083
%N A122983 a(n) = (2 + (-1)^n + 3^n)/4.
%C A122983 Old definition was: "Binomial transform of aeration of A081294".
%C A122983 Binomial transform is A063376.
%C A122983 A122983 = (1,1,3,7,1,1,3,7,...) mod 10. - _M. F. Hasler_, Feb 25 2008
%C A122983 Equals row sums of triangle A158301. - _Gary W. Adamson_, Mar 15 2009
%C A122983 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
%C A122983 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.
%C A122983 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
%C A122983 Also the domination number of the n-Hanoi graph. - _Eric W. Weisstein_, Jun 16 2017
%C A122983 Also the matching number of the n-Sierpinski gasket graph. - _Eric W. Weisstein_, Jun 17 2017
%C A122983 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
%C A122983 Also the lower matching number (=independent domination number) of the n-Hanoi graph. - _Eric W. Weisstein_, Aug 01 2023
%H A122983 M. F. Hasler, <a href="/A122983/b122983.txt">Table of n, a(n) for n = 0..199</a>.
%H A122983 Ji Young Choi, <a href="https://www.emis.de/journals/JIS/VOL21/Choi/choi10.html">A Generalization of Collatz Functions and Jacobsthal Numbers</a>, J. Int. Seq., Vol. 21 (2018), Article 18.5.4.
%H A122983 Alexander Diaz-Lopez, Pamela E. Harris, Erik Insko, and Darleen Perez-Lavin, <a href="http://arxiv.org/abs/1505.04479">Peaks Sets of Classical Coxeter Groups</a>, arXiv preprint, arXiv:1505.04479 [math.GR], 2015.
%H A122983 A. M. Hinz, S. Klavžar, U. Milutinović, and C. Petr, <a href="http://dx.doi.org/10.1007/978-3-0348-0237-6">The Tower of Hanoi - Myths and Maths</a>, Birkhäuser 2013. See page 99. <a href="http://tohbook.info">Book's website</a>
%H A122983 Uri Levy, <a href="http://arxiv.org/abs/1003.0225">The Magnetic Tower of Hanoi</a>, arXiv:1003.0225 [math.CO], 2010.
%H A122983 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DominationNumber.html">Domination Number</a>.
%H A122983 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HanoiGraph.html">Hanoi Graph</a>.
%H A122983 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LowerIndependenceNumber.html">Lower Independence Number</a>.
%H A122983 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MatchingNumber.html">Matching Number</a>.
%H A122983 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SierpinskiGasketGraph.html">Sierpiński Gasket Graph</a>.
%H A122983 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,1,-3).
%F A122983 From _Paul Barry_, Jun 14 2007: (Start)
%F A122983 G.f.: (1-2*x-x^2)/((1-x)*(1+x)*(1-3*x));
%F A122983 a(n) = 3^n/4+(-1)^n/4+1/2;
%F A122983 E.g.f.: cosh(x)^2*exp(x). (End)
%F A122983 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
%F A122983 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
%F A122983 a(2*n) = 3*a(2*n-1); a(2*n+1) = 3*a(2*n) - 2. - _Philippe Deléham_, Aug 23 2020
%p A122983 A122983 := n -> ceil(3^n/4); 'A122983(n)' $ n=0..22; # _M. F. Hasler_, Feb 25 2008
%p A122983 a[ -1]:=1:a[0]:=1:a[1]:=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
%t A122983 CoefficientList[Series[(1 - 2 x - x^2)/((1 - x) (1 + x) (1 - 3 x)), {x, 0, 40}], x] (* _Harvey P. Dale_, Sep 03 2013 *)
%t A122983 LinearRecurrence[{3, 1, -3}, {1, 1, 3}, 40] (* _Harvey P. Dale_, Sep 03 2013 *)
%t A122983 Table[(2 + (-1)^n + 3^n)/4, {n, 0, 20}] (* _Eric W. Weisstein_, Jun 16 2017 *)
%t A122983 Table[Floor[3^n/4] + 1, {n, 0, 20}] (* _Eric W. Weisstein_, Jan 17 2018 *)
%t A122983 Floor[3^Range[0, 20]/4] + 1 (* _Eric W. Weisstein_, Jan 17 2018 *)
%o A122983 (PARI) A122983(n)=3^n\4+1 \\ _M. F. Hasler_, Feb 25 2008
%o A122983 (Python)
%o A122983 def A122983(n): return (1 if n&1 else 3)+3**n>>2 # _Chai Wah Wu_, Apr 12 2023
%Y A122983 Cf. a(j+1) = A137822(2^j) and these are the record values of A137822.
%Y A122983 Cf. A054879 (bisection), A066443 (bisection). Row sums of A158303.
%K A122983 easy,nonn
%O A122983 0,3
%A A122983 _Paul Barry_, Sep 22 2006
%E A122983 Extended and corrected (existing Maple code) by _M. F. Hasler_, Feb 25 2008
%E A122983 Description changed to formula by _Eric W. Weisstein_, Jun 16 2017