A147678 Double, add 0, double, add 1, double, add 2, double, add 3, etc.

1, 2, 2, 4, 5, 10, 12, 24, 27, 54, 58, 116, 121, 242, 248, 496, 503, 1006, 1014, 2028, 2037, 4074, 4084, 8168, 8179, 16358, 16370, 32740, 32753, 65506, 65520, 131040, 131055, 262110, 262126, 524252, 524269, 1048538, 1048556, 2097112, 2097131

Offset

1,2

Comments

A147675 to A147677, and this sequence, are from a quiz that someone asked me to help them with.

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (0,4,0,-5,0,2).

Formula

From R. J. Mathar, Apr 22 2009: (Start)

a(n) = 4*a(n-2) - 5*a(n-4) + 2*a(n-6).

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

a(n) = (1 + (-1)^n)/2 + 2*A016116(n) - A105811(n+3)/4 - 3*(n+1)/4. (End)

a(n) = 2*a(n-1) - (n mod 2)*(a(n-1) - (n-3)/2). - Reinhard Zumkeller, Apr 22 2009

E.g.f.: -2 + 2*cosh(sqrt(2)*x) + sqrt(2)*sinh(sqrt(2)*x) - (1/2)*(x*cosh(x) + (1 + 2*x)*sinh(x)). - G. C. Greubel, Dec 01 2025

Mathematica

Transpose[NestList[Flatten[{Rest[#], 2First[#]-5#[[3]]+ 4#[[5]]}]&, {1, 2, 2, 4, 5, 10}, 40]][[1]] (* Harvey P. Dale, Mar 24 2011 *)
dp[a_, n_]:=Flatten[{{x=a}, Table[{2x, x=2x+m}, {m, 0, n}]}]; A147678=dp[1, 20] (* Zak Seidov, Mar 24 2011 *)
LinearRecurrence[{0, 4, 0, -5, 0, 2}, {1, 2, 2, 4, 5, 10}, 20] (* T. D. Noe, Mar 25 2011 *)

Prog

(Magma)
A147678:= func< n | -(n+1)/2 + ((n+1) mod 2)*(2^Floor(2+n/2) -n+1)/2 + (n mod 2)*2^Floor((n+1)/2) >;
[A147678(n): n in [1..70]]; // G. C. Greubel, Dec 01 2025
(SageMath)
def A147678(n): return (-(n+1) + ((n+1)%2)*(4*2**int(n/2) -n+1) + 2*(n%2)*2**int((n+1)/2))//2
print([A147678(n) for n in range(71)]) # G. C. Greubel, Dec 01 2025

Crossrefs

Cf. A016116, A105811, A147675, A147676, A147677.

Sequence in context: A378831 A306318 A091188 * A195865 A222006 A127712

Adjacent sequences: A147675 A147676 A147677 * A147679 A147680 A147681

Keyword

nonn,easy

Author

N. J. A. Sloane, Apr 21 2009

Extensions

More terms from R. J. Mathar, Apr 22 2009

Status

approved