A329723 Coefficients of expansion of (1-2x^3)/(1-x-x^2) in powers of x.

1, 1, 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349, 15127, 24476, 39603, 64079, 103682, 167761, 271443, 439204, 710647, 1149851, 1860498, 3010349, 4870847, 7881196, 12752043, 20633239, 33385282, 54018521, 87403803, 141422324, 228826127

Offset

0,3

Comments

Two terms 1, 1 followed by the Lucas sequence (A000032), i.e., A000032(n) = a(n+2). The run length transform is given by Sum_{k=0..n} ((binomial(n+2k,2n-k)*binomial(n,k)) mod 2) (A329722).

Links

Chai Wah Wu, Table of n, a(n) for n = 0..4786

Narad Rampersad and Max Wiebe, Sums of products of binomial coefficients mod 2 and 2-regular sequences, arXiv:2309.04012 [math.NT], 2023.

Chai Wah Wu, Sums of products of binomial coefficients mod 2 and run length transforms of sequences, arXiv:1610.06166 [math.CO], 2016.

Index entries for linear recurrences with constant coefficients, signature (1,1).

Formula

a(n) = A000032(n-2) for n > 1.

a(n) = a(n-1) + a(n-2) for n > 3. - Chai Wah Wu, Feb 04 2022

Mathematica

CoefficientList[Series[(1 - 2 x^3)/(1 - x - x^2), {x, 0, 42}], x] (* Michael De Vlieger, Feb 04 2022 *)

Prog

(Python)
from sympy import lucas
def A329723(n): return 1 if n <= 1 else lucas(n-2) # Chai Wah Wu, Feb 04 2022

Crossrefs

Cf. A000032, A329722.

Sequence in context: A268615 A061084 A000032 * A267551 A055391 A177940

Adjacent sequences: A329720 A329721 A329722 * A329724 A329725 A329726

Keyword

nonn,easy

Author

Chai Wah Wu, Nov 19 2019

Status

approved