A086953 Binomial transform of (-1)^mod(n,3) (A257075).

1, 0, 0, 2, 6, 12, 22, 42, 84, 170, 342, 684, 1366, 2730, 5460, 10922, 21846, 43692, 87382, 174762, 349524, 699050, 1398102, 2796204, 5592406, 11184810, 22369620, 44739242, 89478486, 178956972, 357913942, 715827882, 1431655764, 2863311530, 5726623062

Offset: 0

Author

Paul Barry, Jul 25 2003

Keywords

• nonn
• easy

Links

• Index entries for linear recurrences with constant coefficients, signature (3,-3,2).

Crossrefs

Cf. A024493, A024495, A131370, A131708, A257075.

Programs

Mathematica

Join[{1, a = 0, b = 0}, Table[c = 2^n - a + b; a = b; b = c, {n, 1, 100}]] (* Vladimir Joseph Stephan Orlovsky, Jun 28 2011 *)
LinearRecurrence[{3,-3,2},{1,0,0},40] (* Harvey P. Dale, Aug 02 2017 *)

Formula

a(n+3)/2 = A024495(n+2). - corrected by Vladimir Shevelev, Aug 08 2017

a(n) = 0^n + Sum{k=0..floor((n-1)/3)} C(n-1, 3*k+2).

a(n) = Sum{k=0..n} C(n, k)(-1)^mod(k, 3).

G.f.: (1 - 3*x + 3*x^2)/((1 - 2*x)*(1 - x + x^2)). - Paul Barry, Dec 14 2004

From Vladimir Shevelev, Aug 02 2017: (Start)

a(n) = A024493(n) - A131708(n) + A024495(n);

a(n) = A024495(n) if and only if n == 1 (mod 3);

a(n) = A024495(n) - 1 if and only if n == 2 or 3 (mod 6);

a(n) = A024495(n) + 1 if and only if n == 0 or 5 (mod 6);

a(3*k+1) = 2*A024495(3*k). (End)

a(n) = A131370(n+1)/2. - Rick L. Shepherd, Aug 02 2017

3*a(n) = 2^n + 2*A057079(n+2). - R. J. Mathar, Aug 04 2017