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a(n) = 9*a(n-1) - 9*a(n-2) + a(n-3); given a(1) = 1, a(2) = 10, a(3) = 81.
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%I #27 Jun 07 2021 01:09:26

%S 1,10,81,640,5041,39690,312481,2460160,19368801,152490250,1200553201,

%T 9451935360,74414929681,585867502090,4612525087041,36314333194240,

%U 285902140466881,2250902790540810,17721320183859601,139519658680336000,1098435949258828401,8647967935390291210

%N a(n) = 9*a(n-1) - 9*a(n-2) + a(n-3); given a(1) = 1, a(2) = 10, a(3) = 81.

%C A sequence derived from A076765, with a(n)/a(n-1) tending to 4 + sqrt(15).

%C a(n)/a(n-1) tends to C = 4 + sqrt(15) = 7.87298334... (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.

%C This is the r=10 member of the r-family of sequences S_r(n), n>=1, defined in A092184, where more information can be found.

%H Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, <a href="https://www.emis.de/journals/INTEGERS/papers/p38/p38.Abstract.html">Polynomial sequences on quadratic curves</a>, Integers, Vol. 15, 2015, #A38.

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (9,-9,1).

%F a(n) = A076765(n-1) + A076765(n-2).

%F Let M be the 3 X 3 matrix [1 1 1 / 1 2 3 / 1 3 6]; then M^n * [1 0 0] = [A095002(n) A095003(n) a(n)].

%F a(n)= (T(n, 4)-1)/3 with Chebyshev's polynomials of the first kind evaluated at x=4: T(n, 4)=A001091(n). a(0):=0. - _Wolfdieter Lang_, Oct 18 2004

%F G.f.: x*(1+x)/((1-x)*(1-8*x+x^2)) = x*(1+x)/(1-9*x+9*x^2-x^3).

%e a(4) = 640 = 568 + 72 = A076765(3) + A076765(2).

%e a(4) = 640 = 9*81 - 9*10 + 1.

%e a(4) = 640, rightmost term in M^4 * [1 0 0]: [145 352 640] = [A095002(4) A095003(4) A095004(4)].

%p a:= n-> (<<1|1|1>, <1|2|3>, <1|3|6>>^n)[1, 3]:

%p seq(a(n), n=1..23); # _Alois P. Heinz_, Jun 06 2021

%t a[n_] := (MatrixPower[{{1, 1, 1}, {1, 2, 3}, {1, 3, 6}}, n].{{1}, {0}, {0}})[[3, 1]]; Table[ a[n], {n, 20}]; (* _Robert G. Wilson v_, May 29 2004 *)

%Y Cf. A095002, A095003, A076765.

%K nonn,easy

%O 1,2

%A _Gary W. Adamson_, May 27 2004

%E Edited and extended by _Robert G. Wilson v_, May 29 2004

%E Definition aligned with A095002, A095003 by _Georg Fischer_, Jun 06 2021