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A095002 a(n) = 9*a(n-1) - 9*a(n-2) + a(n-3); given a(1) = 1, a(2) = 3, a(3) = 19.

%I A095002 #26 Sep 02 2022 13:24:12
%S A095002 1,3,19,145,1137,8947,70435,554529,4365793,34371811,270608691,
%T A095002 2130497713,16773373009,132056486355,1039678517827,8185371656257,
%U A095002 64443294732225,507360986201539,3994444594880083,31448195772839121,247591121587832881,1949280776929823923
%N A095002 a(n) = 9*a(n-1) - 9*a(n-2) + a(n-3); given a(1) = 1, a(2) = 3, a(3) = 19.
%C A095002 A companion to A095003, A005004; a(n)/a(n-1) tending to 4 + sqrt(15).
%C A095002 a(n)/a(n-1) tends to C = 4 + sqrt(15); C having the property that C + 1/C = 8. Eigenvalues of M (1, C, 1/C) are roots to x^3 - 9x^2 + 9x - 1.
%H A095002 Harvey P. Dale, <a href="/A095002/b095002.txt">Table of n, a(n) for n = 1..1000</a>
%H A095002 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (9,-9,1).
%F A095002 Let M be the 3 X 3 matrix [1 1 1 / 1 2 3 / 1 3 6]. M^n * [1 0 0] = [a(n) A095003(n) A095004(n)].
%F A095002 From _R. J. Mathar_, Aug 22 2008: (Start)
%F A095002 O.g.f.: x*(1-6x+x^2)/((1-x)*(1-8x+x^2)).
%F A095002 a(n) = (2 + A001090(n+1) - 7*A001090(n))/3. (End)
%e A095002 a(4) = 145 = 9*19 - 9*3 + 1.
%e A095002 a(4) = 145, leftmost term in M^4 * [1 0 0] = [145 352 640].
%p A095002 a:= n-> (<<1|1|1>, <1|2|3>, <1|3|6>>^n)[1$2]:
%p A095002 seq(a(n), n=1..23);  # _Alois P. Heinz_, Jun 06 2021
%t A095002 a[n_] := (MatrixPower[{{1, 1, 1}, {1, 2, 3}, {1, 3, 6}}, n].{{1}, {0},
%t A095002 {0}})[[1, 1]]; Table[ a[n], {n, 20}]; (* _Robert G. Wilson v_, May 29 2004 *)
%t A095002 nxt[{a_,b_,c_}]:={b,c,9c-9b+a}; NestList[nxt,{1,3,19},30][[All,1]] (* _Harvey P. Dale_, Sep 02 2022 *)
%o A095002 (PARI) Vec(x*(1-6*x+x^2)/((1-x)*(1-8*x+x^2)) + O(x^20)) \\ _Michel Marcus_, Mar 21 2015
%Y A095002 Cf. A095003, A095004, A076765.
%K A095002 nonn,easy
%O A095002 1,2
%A A095002 _Gary W. Adamson_, May 27 2004
%E A095002 Edited and extended by _Robert G. Wilson v_, May 29 2004
%E A095002 Edited by _Georg Fischer_, Jun 06 2021