%I #16 May 06 2024 12:37:44

%S 1,4,17,74,323,1400,6005,25478,107015,445556,1841273,7561922,30897227,

%T 125714672,509767421,2061390206,8317305359,33498803948,134727010049,

%U 541232563130,2172291241811,8712410196584,34922863258757,139921580805494,560408087592983

%N a(n) = 2^(2n-1) - (n+2)*3^(n-2).

%C Number of n X n nonsingular real matrices with entries {0,1} in which the top left n-1 X n-1 submatrix is the identity matrix. See A125587 for proof.

%C The number of singular matrices is given by A006234.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (10,-33,36).

%F G.f.: -x*(10*x^2-6*x+1) / ((3*x-1)^2*(4*x-1)). - _Colin Barker_, Feb 26 2014

%e a(2) = 4:

%e 10 10 11 11

%e 01 11 01 10

%p A125586:=n->2^(2n-1)-(n+2)*3^(n-2); seq(A125586(n), n=1..30); # _Wesley Ivan Hurt_, Feb 26 2014

%t Table[2^(2n-1)-(n+2)*3^(n-2), {n, 30}] (* _Wesley Ivan Hurt_, Feb 26 2014 *)

%t LinearRecurrence[{10,-33,36},{1,4,17},50] (* _Harvey P. Dale_, Sep 15 2019 *)

%o (PARI) Vec(-x*(10*x^2-6*x+1)/((3*x-1)^2*(4*x-1)) + O(x^100)) \\ _Colin Barker_, Feb 26 2014

%Y Cf. A125587, A006234.

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_ and _Vinay Vaishampayan_, Jan 05 2007