A096978 Sum of the areas of the first n Jacobsthal rectangles.

0, 1, 4, 19, 74, 305, 1208, 4863, 19398, 77709, 310612, 1242907, 4970722, 19884713, 79535216, 318148151, 1272578046, 5090341317, 20361307020, 81445344595, 325781145370, 1303125047521, 5212499258024, 20849998896239, 83399991856694

Offset: 0

Author

Paul Barry, Jul 17 2004

Keywords

• easy
• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (4,3,-14,8).

Crossrefs

Cf. A096977, A064831.

Programs

Magma

[8*4^n/27-2*(-2)^n/27-(n+2)/9: n in [0..30]]; // Vincenzo Librandi, May 31 2011

Mathematica

LinearRecurrence[{4,3,-14,8},{0,1,4,19},30] (* or *) Table[(2^(2n+1)-3n - 3+(-2)^n)/27,{n,30}] (* Harvey P. Dale, Aug 08 2011 *)

Formula

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

a(n) = 8*4^n/27 - 2*(-2)^n/27 - (n+2)/9;

a(n) = Sum_{k=0..n} (2*4^k/3 + (-2)^k/3)*(n-k).

a(n) = 4*a(n-1) + 3*a(n-2) - 14*a(n-3) + 8*a(n-4).

a(n) = Sum_{k=0..n} A001045(k)*A001045(k+1).

a(n-1) = Sum_{k=0..n} (-1)^(k+1)*A001045(k)*A001045(2*(n-k)). - Paul Barry, Aug 11 2009