A348405 a(0) = 1, a(n) + a(n+1) = round(2^n/9), n >= 0.

1, -1, 1, -1, 2, 0, 4, 3, 11, 17, 40, 74, 154, 301, 609, 1211, 2430, 4852, 9712, 19415, 38839, 77669, 155348, 310686, 621382, 1242753, 2485517, 4971023, 9942058, 19884104, 39768220, 79536427, 159072867, 318145721, 636291456, 1272582898, 2545165810

Offset: 0

Author

Paul Curtz, Oct 17 2021

Keywords

• sign

Links

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

Crossrefs

Cf. A139797 (a(n) + a(n+1) = round(2^n/9) too, but a(0) = 0).

Cf. A001045, A104581, A113405, A131666.

Cf. A136298, A172481, A175656, A348407.

Programs

Mathematica

CoefficientList[ Series[(x^4-x^3+2x-1)/((2*x^3-3*x^2+3*x-1)*(x+1)^2), {x, 0, 40}], x] (* Thomas Scheuerle, Oct 17 2021 *)
nxt[{n_,a_}]:={n+1,Round[(2^n)/9]-a}; NestList[nxt,{0,1},40][[All,2]] (* or *) LinearRecurrence[{1,2,-1,1,2},{1,-1,1,-1,2},40] (* Harvey P. Dale, Apr 28 2022 *)

Formula

a(n+1) = 2*a(n) - A104581(n+6).

a(n) + a(n+1) = A113405(n).

a(n) + a(n+3) = A001045(n).

a(n+2) = a(n) + A131666(n).

From Thomas Scheuerle, Oct 18 2021: (Start)

G.f.: (x^4-x^3+2x-1)/((2*x^3-3*x^2+3*x-1)*(x+1)^2).

A172481(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*a(2*n-k). With negative sign for ...*a(1+2*n-k) and ...*a(3+2*n-k) too.

A175656(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*a(2+2*n-k).

A136298(n+1) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*a(4+2*n-k).

A348407(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*(a(2+2*n-k) - 2*a(1+2*n-k) - a(2*n-k)).

(End)

Extensions

a(22)-a(36) from Thomas Scheuerle, Oct 17 2021