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Let M = 2 X 2 matrix [0, 1; 1, Pi]; then a(n) = 0 if n=0, 1 if n=1; and otherwise a(n) = floor(M^(n-1)[2,2]).
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%I #11 Oct 27 2019 02:08:49

%S 0,1,3,10,37,128,439,1508,5179,17779,61033,209522,719266,2469165,

%T 8476377,29098491,99891986,342918422,1177201983,4041207525,

%U 13873029857,47624616207,163490174266,561244146614

%N Let M = 2 X 2 matrix [0, 1; 1, Pi]; then a(n) = 0 if n=0, 1 if n=1; and otherwise a(n) = floor(M^(n-1)[2,2]).

%e a(2) = 3 since M^1 = matrix [0, 1; 1, Pi], and the (2,2) entry is Pi, with floor(Pi)=3.

%K nonn

%O 0,3

%A _Gary W. Adamson_, Mar 30 2008

%E Corrected by _N. J. A. Sloane_, Oct 27 2019 at the suggestion of _Harvey P. Dale_.