A329723 - OEIS

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

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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)