Search: id:A101265
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%I A101265
%S A101265 1,2,6,21,77,286,1066,3977,14841,55386,206702,771421,2878981,10744502,
%T A101265 40099026,149651601,558507377,2084377906,7779004246,29031639077,
%U A101265 108347552061,404358569166,1509086724602,5631988329241,21018866592361
%N A101265 a(1) = 1, a(2) = 2, a(3) = 6; a(n) = 5*a(n-1) - 5*a(n-2) + a(n-3) for 
               n>3.
%C A101265 Let M = [ 1, 1, 0; 1, 3, 1; 0, 1, 1]; then [1,0,0]*M^n = [a(n), A001353(n), 
               A061278(n-1)] for n>1; Further a(n) = A061278(n) + 1; A001353(n) 
               is first difference of a(n) and first difference of A061278(n). Let 
               v(n) = [1,0,0]*M^n, then sum(v_i(n)) = A001075(n) for n>=0; and v_1(n) 
               + v_3(n) = A001835(n) for n>=0; Characteristic polynomial of M = 
               x^3 - 5x^2 + 5x - 1; a(n)/a(n-1) tends to 2 + sqrt(3) = 3.732.... 
               (see A019973) (a root of the polynomial and an eigenvalue of the 
               matrix).
%C A101265 a(n) = 1 + A061278(n). Also numbers n such that the RootMeanSquare(1,
               ...,6*n-5) is an integer. [From Ctibor O. Zizka (c.zizka(AT)email.cz), 
               Dec 17 2008]
%F A101265 a(n) = A005246(n)*A005246(n+1). a(n+1) = a(n)*(a(n)+1)/a(n-1). - Frank 
               Adams-Watters (FrankTAW(AT)Netscape.net), Apr 24 2006
%F A101265 a(n) = (A001835(n) + 1) / 2. - Ralf Stephan, May 16 2007
%F A101265 O.g.f.: x(1-3x+x^2)/((1-x)(1-4x+x^2)). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Aug 22 2008]
%o A101265 (PARI) M = [ 1, 1, 0; 1, 3, 1; 0, 1, 1]; for(i=1,30,print1(([1,0,0]*M^i)[1],
               ","))
%Y A101265 Cf. A001353, A061278, A001835, A001075, A019973.
%Y A101265 Cf. A005246.
%Y A101265 Sequence in context: A131792 A144904 A151287 this_sequence A101879 A063023 
               A150188
%Y A101265 Adjacent sequences: A101262 A101263 A101264 this_sequence A101266 A101267 
               A101268
%K A101265 nonn
%O A101265 1,2
%A A101265 Lambert Klasen (lambert.klasen(AT)gmx.net) and Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Jan 25 2005

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