A104743 a(n) = 3^n + n.

1, 4, 11, 30, 85, 248, 735, 2194, 6569, 19692, 59059, 177158, 531453, 1594336, 4782983, 14348922, 43046737, 129140180, 387420507, 1162261486, 3486784421, 10460353224, 31381059631, 94143178850, 282429536505, 847288609468

Offset: 0

Author

Zak Seidov, Mar 23 2005

Keywords

• nonn
• easy

Comments

This sequence also describes the total number of moves solving (non-optimally) the [RED ; NEUTRAL ; NEUTRAL] or [NEUTRAL ; NEUTRAL ; BLUE] pre-colored Magnetic Tower of Hanoi puzzle (see the "CROSSREFS" in A183121). For other Magnetic Tower of Hanoi related sequences, cf. A183111 - A183125.

Form an infinite array with m(0,k) = 2*k+1 and m(n,k) = Sum_{r=0..n-1} m(r,k) + Sum_{c=0..k-1} m(n,c), being the sum of the terms above m(n,k) plus those terms to the left of m(n,k). The first row is A005408. The second row is A033484. The first column is A011782. The sum of the terms in the n-th antidiagonal is a(n). - J. M. Bergot, Jun 07 2013

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Uri Levy, The Magnetic Tower of Hanoi, arXiv:1003.0225 [math.CO], 2010.
• Index entries for linear recurrences with constant coefficients, signature (5,-7,3).

Crossrefs

Cf. A103537.

Programs

GAP

List([0..40], n-> 3^n +n ); # G. C. Greubel, May 21 2019

Magma

[3^n +n: n in [0..40]]; // G. C. Greubel, May 21 2019

Mathematica

Table[3^n +n, {n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Jan 18 2009, modified by G. C. Greubel, May 21 2019 *)
LinearRecurrence[{5,-7,3},{1,4,11},30] (* Harvey P. Dale, Aug 01 2020 *)

PARI

{a(n) = 3^n + n}; \\ G. C. Greubel, May 21 2019

Sage

[3^n +n for n in (0..40)] # G. C. Greubel, May 21 2019

Formula

a(n) = n + 3^n.

From Colin Barker, Jan 25 2012: (Start)

a(n) = 5*a(n-1) - 7*a(n-2) + 3*a(n-3).

G.f.: (1+x)*(1-2*x) / ((1-3*x)*(1-x)^2). (End)

E.g.f.: x*exp(x) + exp(3*x). - G. C. Greubel, May 21 2019

Extensions

More terms from Vladimir Joseph Stephan Orlovsky, Jan 18 2009