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%I #56 Aug 28 2023 14:37:57
%S 1,2,7,22,67,202,607,1822,5467,16402,49207,147622,442867,1328602,
%T 3985807,11957422,35872267,107616802,322850407,968551222,2905653667,
%U 8716961002,26150883007,78452649022,235357947067,706073841202
%N a(0) = 1; a(n) = (5*3^(n-1) - 1)/2 for n > 0.
%C From Erich Friedman's math magic page 2nd paragraph under "Answers" section.
%C 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 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
%H Harry J. Smith, <a href="/A060816/b060816.txt">Table of n, a(n) for n = 0..200</a>
%H Erich Friedman, <a href="https://erich-friedman.github.io/mathmagic/1000.html">Math. Magic</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-3).
%F The following is a summary of formulas added over the past 18 years.
%F a(n) = 3*a(n-1) + 1; with a(0)=1, a(1)=2. - _Jason Earls_, Apr 29 2001
%F For n>0, a(n) = a(n-1)+5*3^(n-2) = (5*A003462(n)+1)/3 = a(n-1)+A005030(n-2). - _Henry Bottomley_, May 01 2001
%F From _Colin Barker_, Apr 24 2012: (Start)
%F a(n) = 4*a(n-1) - 3*a(n-2) for n > 2.
%F G.f.: (1-2*x+2*x^2)/((1-x)*(1-3*x)). (End)
%F a(n+1) = A134931(n) + 1. - _Philippe Deléham_, Apr 14 2013
%F For n > 0, A008343(a(n)) = 0. - _Dmitry Kamenetsky_, Feb 14 2017
%F For n > 0, a(n) = floor(3^n*5/6). - _M. F. Hasler_, Apr 06 2019
%F From _Klaus Purath_, Mar 31 2021: (Start)
%F a(n) = A017209(a(n-2)), n > 2.
%F a(n) = 2 + Sum_{i = 0..n-2} A005030(i).
%F a(n+1)*a(n+2) = a(n)*a(n+3) + 20*3^n, n > 1.
%F a(n) = 3^n - A007051(n-1). (End)
%F E.g.f.: (5*exp(3*x) - 3*exp(x) + 4)/6. - _Stefano Spezia_, Aug 28 2023
%t LinearRecurrence[{4,-3},{1,2,7},30] (* _Harvey P. Dale_, Nov 15 2022 *)
%o (PARI) { for (n=0, 200, if (n>1, a1=a=3*a1 + 1, if (n==0, a=1, a1=a=2)); write("b060816.txt", n, " ", a); ) } \\ _Harry J. Smith_, Jul 13 2009
%o (PARI) A060816(n)=if(n, 3^n*5\6, 1) \\ _M. F. Hasler_, Apr 06 2019
%Y Equals A057198 - 1.
%Y Cf. A005030 (first differences), A244762 (partial sums).
%Y Cf. A003215, A003462, A007051, A008343, A017209, A134931.
%K easy,nonn
%O 0,2
%A _Jason Earls_, Apr 29 2001
%E Edited by _M. F. Hasler_, Apr 06 2019 and by _N. J. A. Sloane_, Apr 09 2019
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