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

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

Offset: 0

Author

Paul Barry, Sep 22 2006

Keywords

• easy
• nonn

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 gasket 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

Also the lower matching number (=independent domination number) of the n-Hanoi graph. - Eric W. Weisstein, Aug 01 2023

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, and Darleen Perez-Lavin, Peaks Sets of Classical Coxeter Groups, arXiv preprint, arXiv:1505.04479 [math.GR], 2015.
• A. M. Hinz, S. Klavžar, U. Milutinović, and 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, Hanoi Graph.
• Eric Weisstein's World of Mathematics, Lower Independence Number.
• Eric Weisstein's World of Mathematics, Matching Number.
• Eric Weisstein's World of Mathematics, Sierpiński Gasket Graph.
• Index entries for linear recurrences with constant coefficients, signature (3,1,-3).

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.

Programs

Maple

A122983 := n -> ceil(3^n/4); 'A122983(n)' $ n=0..22; # M. F. Hasler, Feb 25 2008
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

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 *)

PARI

A122983(n)=3^n\4+1 \\ M. F. Hasler, Feb 25 2008

Python

def A122983(n): return (1 if n&1 else 3)+3**n>>2 # Chai Wah Wu, Apr 12 2023

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

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