A297928 - OEIS

%I #9 Apr 22 2018 08:26:18

%S 4,13,43,151,559,2143,8383,33151,131839,525823,2100223,8394751,

%T 33566719,134242303,536920063,2147581951,8590131199,34360131583,

%U 137439739903,549757386751,2199026401279,8796099313663,35184384671743,140737513521151,562950003752959,2251799914348543

%N a(n) = 2*4^n + 3*2^n - 1.

%C For n > 0, in binary, this is a 1 followed by n-1 0's followed by 10 followed by n 1's.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-14,8)

%F G.f.: (4 - 15*x + 8*x^2)/((1 - x)*(1 - 2*x)*(1 - 4*x)).

%F E.g.f.: 2*e^(4*x) + 3*e^(2*x) - e^x.

%F a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3), n > 2.

%F a(n) = A000918(n) + A085601(n).

%e a(0) = 2*4^0 + 3*2^0 - 1 = 4;   in binary, 100.

%e a(1) = 2*4^1 + 3*2^1 - 1 = 13;  in binary, 1101.

%e a(2) = 2*4^2 + 3*2^2 - 1 = 43;  in binary, 101011.

%e a(3) = 2*4^3 + 3*2^3 - 1 = 151; in binary, 10010111.

%e a(4) = 2*4^4 + 3*2^4 - 1 = 559; in binary, 1000101111.

%e ...

%t Table[2 4^n+3 2^n-1,{n,0,30}] (* or *) LinearRecurrence[{7,-14,8},{4,13,43},30] (* _Harvey P. Dale_, Apr 22 2018 *)

%o (PARI) a(n) = 2*4^n + 3*2^n - 1

%o (PARI) first(n) = Vec((4 - 15*x + 8*x^2)/((1 - x)*(1 - 2*x)*(1 - 4*x)) + O(x^n))

%Y A lower bound for A296807.

%Y Cf. A000918, A085601.

%K nonn,easy

%O 0,1

%A _Iain Fox_, Jan 08 2018