A164356 Expansion of (1 - x^2)^4 / ((1 - x)^4 * (1 - x^4)) in powers of x.

1, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Michael Somos, Rational Function Multiplicative Coefficients

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

Formula

Euler transform of length 4 sequence [4, -4, 0, 1].

Moebius transform is length 4 sequence [4, 2, 0, -4].

a(n) = 4 * b(n) unless n=0 and b(n) is multiplicative with b(2) = 3/2, b(2^e) = 1/2 if e>1, b(p^e) = 1 if p>2.

a(n) = a(-n) for all n in Z. a(n+4) = a(n) unless n=0 or n=-4. a(2*n + 1) = 4. a(4*n) = 2 unless n=0. a(4*n + 2) = 6.

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

a(n) = 2 * A068073(n) unless n=0. - Michael Somos, Apr 17 2015

Example

G.f. = 1 + 4*x + 6*x^2 + 4*x^3 + 2*x^4 + 4*x^5 + 6*x^6 + 4*x^7 + 2*x^8 + ...

Mathematica

a[ n_] := -Boole[n == 0] + 4 - If[ EvenQ[n], (-1)^(n/2) 2, 0]; (* Michael Somos, Apr 17 2015 *)
a[ n_] := SeriesCoefficient[ -1 + 4/(1 - x) - 2/(1 + x^2), {x, 0, Abs@n}]; (* Michael Somos, Jan 07 2019 *)
LinearRecurrence[{1, -1, 1}, {1, 4, 6, 4}, 120] (* or *) PadRight[{1}, 120, {2, 4, 6, 4}] (* Harvey P. Dale, Aug 30 2024 *)

Prog

(PARI) {a(n) = -(n==0) + 4 - if( n%2 == 0, (-1)^(n/2) * 2, 0)};

Crossrefs

Cf. A068073.

Sequence in context: A131890 A062751 A135911 * A181774 A291379 A001138

Adjacent sequences: A164353 A164354 A164355 * A164357 A164358 A164359

Keyword

nonn,easy

Author

Michael Somos, Aug 13 2009

Status

approved