A084174 a(n) = 3*a(n-1) - a(n-2) - 3*a(n-3) + 2*a(n-4).

1, 1, 3, 6, 14, 29, 61, 124, 252, 507, 1019, 2042, 4090, 8185, 16377, 32760, 65528, 131063, 262135, 524278, 1048566, 2097141, 4194293, 8388596, 16777204, 33554419, 67108851, 134217714, 268435442, 536870897, 1073741809, 2147483632

Offset: 0

Author

Paul Barry, May 18 2003

Keywords

• nonn
• easy

Comments

Original name was: Generalized Jacobsthal numbers.

Row sums of triangle A131087. - Gary W. Adamson, Jun 14 2007

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,2).

Crossrefs

Cf. A131087.

Programs

Magma

[2^n+(-1)^n/4-(2*n+1)/4: n in [0..35]]; // Vincenzo Librandi, Jul 05 2011

Maple

seq(floor(2^n-n(n+1)/2),n=0..31); # Zerinvary Lajos, Apr 29 2007

Mathematica

LinearRecurrence[{3,-1,-3,2},{1,1,3,6},40] (* Harvey P. Dale, Feb 17 2021 *)

Formula

a(n) = 2^n + (-1)^n/4 - (2*n+1)/4.

a(n+2) = a(n+1) + 2*a(n) + n, a(0)=1, a(1)=1.

a(n) = A084172(n) - A001045(n).

a(n) = floor(2^n - n*(n+1)/2). - Zerinvary Lajos, Apr 29 2007

G.f.: (1 - 2*x + x^2 + x^3)/(1 - 3*x + x^2 + 3*x^3 - 2*x^4). - Colin Barker, Jan 16 2012