This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A060816 #76 Jul 27 2025 18:54:03
%S A060816 1,2,7,22,67,202,607,1822,5467,16402,49207,147622,442867,1328602,
%T A060816 3985807,11957422,35872267,107616802,322850407,968551222,2905653667,
%U A060816 8716961002,26150883007,78452649022,235357947067,706073841202
%N A060816 a(0) = 1; a(n) = (5*3^(n-1) - 1)/2 for n > 0.
%C A060816 From Erich Friedman's math magic page 2nd paragraph under "Answers" section.
%C A060816 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
%C A060816 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
%C A060816 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
%H A060816 Harry J. Smith, <a href="/A060816/b060816.txt">Table of n, a(n) for n = 0..200</a>
%H A060816 Erich Friedman, <a href="https://erich-friedman.github.io/mathmagic/1000.html">Math. Magic</a>
%H A060816 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-3).
%F A060816 The following is a summary of formulas added over the past 18 years.
%F A060816 a(n) = A057198(n) - 1.
%F A060816 a(n) = 3*a(n-1) + 1; with a(0)=1, a(1)=2. - _Jason Earls_, Apr 29 2001
%F A060816 From _Henry Bottomley_, May 01 2001: (Start)
%F A060816 For n>0, a(n) = a(n-1)+5*3^(n-2) = a(n-1)+A005030(n-2).
%F A060816 For n>0, a(n) = (5*A003462(n)+1)/3. (End)
%F A060816 From _Colin Barker_, Apr 24 2012: (Start)
%F A060816 a(n) = 4*a(n-1) - 3*a(n-2) for n > 2.
%F A060816 G.f.: (1-2*x+2*x^2)/((1-x)*(1-3*x)). (End)
%F A060816 a(n+1) = A134931(n) + 1. - _Philippe Deléham_, Apr 14 2013
%F A060816 For n > 0, A008343(a(n)) = 0. - _Dmitry Kamenetsky_, Feb 14 2017
%F A060816 For n > 0, a(n) = floor(3^n*5/6). - _M. F. Hasler_, Apr 06 2019
%F A060816 From _Klaus Purath_, Mar 31 2021: (Start)
%F A060816 a(n) = A017209(a(n-2)), n > 2.
%F A060816 a(n) = 2 + Sum_{i = 0..n-2} A005030(i).
%F A060816 a(n+1)*a(n+2) = a(n)*a(n+3) + 20*3^n, n > 1.
%F A060816 a(n) = 3^n - A007051(n-1). (End)
%F A060816 E.g.f.: (5*exp(3*x) - 3*exp(x) + 4)/6. - _Stefano Spezia_, Aug 28 2023
%t A060816 LinearRecurrence[{4,-3},{1,2,7},30] (* _Harvey P. Dale_, Nov 15 2022 *)
%o A060816 (PARI) A060816(n)=if(n, 3^n*5\6, 1) \\ _M. F. Hasler_, Apr 06 2019
%Y A060816 Cf. A005030 (first differences), A244762 (partial sums).
%Y A060816 Cf. A003215, A003462, A007051, A008343, A017209, A057198, A134931.
%K A060816 easy,nonn
%O A060816 0,2
%A A060816 _Jason Earls_, Apr 29 2001
%E A060816 Edited by _M. F. Hasler_, Apr 06 2019 and by _N. J. A. Sloane_, Apr 09 2019