A167167 A001045 with a(0) replaced by -1.

-1, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, 1365, 2731, 5461, 10923, 21845, 43691, 87381, 174763, 349525, 699051, 1398101, 2796203, 5592405, 11184811, 22369621, 44739243, 89478485, 178956971, 357913941, 715827883, 1431655765, 2863311531

Offset: 0

Author

Paul Curtz, Oct 29 2009

Keywords

• sign
• less

Comments

Essentially the same as A001045, and perhaps also A152046.

Also the binomial transform of the sequence with terms (-1)^(n+1)*A128209(n).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,2).

Programs

GAP

Concatenation([-1], List([1..35], n-> (2^n -(-1)^n)/3) ); # G. C. Greubel, Dec 01 2019

Magma

[-1] cat [(2^n -(-1)^n)/3 : n in [1..35]]; // G. C. Greubel, Dec 01 2019

Maple

seq( `if`(n=0, -1, (2^n -(-1)^n)/3), n=0..35); # G. C. Greubel, Dec 01 2019

Mathematica

CoefficientList[Series[(2*x-1+2*x^2)/((1+x)*(1-2*x)), {x, 0, 35}], x] (* G. C. Greubel, Jun 04 2016 *)
Table[If[n==0, -1, (2^n -(-1)^n)/3], {n,0,35}] (* G. C. Greubel, Dec 01 2019 *)
LinearRecurrence[{1,2},{-1,1,1},40] (* Harvey P. Dale, Jul 23 2025 *)

PARI

vector(36, n, if(n==1, -1, (2^(n-1) +(-1)^n)/3 ) ) \\ G. C. Greubel, Dec 01 2019

Sage

[-1]+[lucas_number1(n, 1, -2) for n in (1..35)] # G. C. Greubel, Dec 01 2019

Formula

a(n) = A001045(n), n>0.

a(n) + a(n+1) = 2*A001782(n) = 2*A131577(n) = A155559(n) = A090129(n+2), n>0.

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

E.g.f.: (exp(2*x) - exp(-x) - 3)/3. - G. C. Greubel, Dec 01 2019

Extensions

Edited and extended by R. J. Mathar, Nov 01 2009