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

Author

Chai Wah Wu, Nov 19 2019

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A000032, A329722.

Programs

Mathematica

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

Python

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

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