A080882 a(n)*a(n+3) - a(n+1)*a(n+2) = 2^n, given a(0)=1, a(1)=3, a(2)=7.

1, 3, 7, 22, 52, 164, 388, 1224, 2896, 9136, 21616, 68192, 161344, 508992, 1204288, 3799168, 8988928, 28357376, 67094272, 211662336, 500798464, 1579869184, 3738010624, 11792304128, 27900891136, 88018956288, 208255086592

Offset: 0

Author

Paul D. Hanna, Feb 22 2003

Keywords

• nonn

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0, 8, 0, -4).

Crossrefs

Cf. A080876, A080877, A080878, A080879, A080880, A080881.

Programs

Maple

a:= n-> (Matrix([[22,7,3,1]]). Matrix(4, (i,j)-> if (i=j-1) then 1 elif j=1 then [0,8,0,-4][i] else 0 fi)^(n))[1,4]: seq(a(n), n=0..26); # Alois P. Heinz, Aug 23 2008

Mathematica

a[0]=1; a[1]=3; a[2]=7; a[3]=22; a[n_] := a[n] = 8*a[n-2] - 4*a[n-4]; Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Jun 15 2015, after Richard Choulet *)
LinearRecurrence[{0,8,0,-4},{1,3,7,22},30] (* Harvey P. Dale, Mar 23 2025 *)

Formula

a(2n)=A080879(2n+1)=A080876(2n+4)/4, a(2n+1)=A080879(2n+2)/2=A080876(2n+5)/4.

G.f.: (-2*x^3 - x^2 + 3*x + 1)/(4*x^4 - 8*x^2 + 1).

a(n + 4) = 8*a(n + 2) - 4*a(n). - Richard Choulet, Dec 06 2008

a(n) = (7/24*3^(1/2) + 1/2)*((1 + sqrt(3)))^n + ( - 7/24*3^(1/2) + 1/2)*((1 - sqrt(3)))^n + ( - 1/24*3^(1/2))*( - (1 + sqrt(3)))^n + (1/24*3^(1/2))*( - ((1 - sqrt(3))))^n. - Richard Choulet, Dec 06 2008