0,2

From Erich Friedman's math magic page 2nd paragraph under "Answers" section.

Let A be the Hessenberg matrix of order n, defined by: A[1,j] = 1, A[i,i] = 2,(i>1), A[i,i-1] = -1, and A[i,j] = 0 otherwise. Then, for n >= 1, a(n) = (-1)^n*charpoly(A,-1). - Milan Janjic, Jan 26 2010

If n > 0 and H = hex number (A003215), then 9*H(a(n)) - 2 = H(a(n+1)), for example 9*H(2) - 2 = 9*19 - 2 = 169 = H(7). For n > 2, this is a subsequence of A017209, see formula. - Klaus Purath, Mar 31 2021

Consider the Tower of Hanoi puzzle of order n (i.e., with n rings to be moved). Label each ring with a distinct symbol from an alphabet of size n. Construct words by performing moves according to the standard rules of the puzzle, recording the corresponding symbol each time a ring is moved. To ensure finiteness, we forbid returning to any previously encountered state. Additionally, we impose the constraint that the same ring cannot be moved twice in succession. Then, a(n) is the number of distinct words that can be formed under these rules. - Thomas Baruchel, Jul 22 2025

Harry J. Smith, Table of n, a(n) for n = 0..200

Erich Friedman, Math. Magic

Index entries for linear recurrences with constant coefficients, signature (4,-3).

The following is a summary of formulas added over the past 18 years.

a(n) = A057198(n) - 1.

a(n) = 3*a(n-1) + 1; with a(0)=1, a(1)=2. - Jason Earls, Apr 29 2001

From Henry Bottomley, May 01 2001: (Start)

For n>0, a(n) = a(n-1)+5*3^(n-2) = a(n-1)+A005030(n-2).

For n>0, a(n) = (5*A003462(n)+1)/3. (End)

From Colin Barker, Apr 24 2012: (Start)

a(n) = 4*a(n-1) - 3*a(n-2) for n > 2.

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

a(n+1) = A134931(n) + 1. - Philippe Deléham, Apr 14 2013

For n > 0, A008343(a(n)) = 0. - Dmitry Kamenetsky, Feb 14 2017

For n > 0, a(n) = floor(3^n*5/6). - M. F. Hasler, Apr 06 2019

From Klaus Purath, Mar 31 2021: (Start)

a(n) = A017209(a(n-2)), n > 2.

a(n) = 2 + Sum_{i = 0..n-2} A005030(i).

a(n+1)*a(n+2) = a(n)*a(n+3) + 20*3^n, n > 1.

a(n) = 3^n - A007051(n-1). (End)

E.g.f.: (5*exp(3*x) - 3*exp(x) + 4)/6. - Stefano Spezia, Aug 28 2023

LinearRecurrence[{4, -3}, {1, 2, 7}, 30] (* Harvey P. Dale, Nov 15 2022 *)

(PARI) A060816(n)=if(n, 3^n*5\6, 1) \\ M. F. Hasler, Apr 06 2019

Cf. A005030 (first differences), A244762 (partial sums).

Cf. A003215, A003462, A007051, A008343, A017209, A057198, A134931.

Sequence in context: A088211 A071684 A290917 * A171847 A037552 A308113

Adjacent sequences: A060813 A060814 A060815 * A060817 A060818 A060819

easy,nonn

Jason Earls, Apr 29 2001

Edited by M. F. Hasler, Apr 06 2019 and by N. J. A. Sloane, Apr 09 2019

approved