A369304 Numbers k for which the polynomial (x-1)^3*(x+1)^k has more than one zero coefficient.

3, 6, 14, 19, 31, 38, 54, 63, 83, 94, 118, 131, 159, 174, 206, 223, 259, 278, 318, 339, 383, 406, 454, 479, 531, 558, 614, 643, 703, 734, 798, 831, 899, 934, 1006, 1043, 1119, 1158, 1238, 1279, 1363, 1406, 1494, 1539, 1631, 1678, 1774, 1823, 1923, 1974, 2078, 2131, 2239, 2294, 2406, 2463

Offset

1,1

Comments

In this sequence, pairs of consecutive even numbers (excluding the leading term) alternate with pairs of consecutive odd numbers. When in the sequence a(n) is even (resp. when a(n) is odd), the polynomial (x-1)^3*(x+1)^a(n) has two (resp. three) vanishing coefficients.

These are the coefficients of x^j with j = (m(n) +- 2)*(m(n) +- 1)/6, where m(n) = (6*n - 3 - (-1)^n)/4, and for odd a(n), also with j = (a(n) + 3)/2.

The first differences are a(n) - a(n-1) = n+1 if n even, or 2*(n+1) if n odd, for n >= 2 (A022998).

a(n) = A001082(n+2)-2. Indeed, this formula is valid for n=1,...,20 and the even and odd terms of both sequences A001082 and A369304 are the values of quadratic polynomials in n.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Vladimir Petrov Kostov, On universal sign patterns, arXiv:2405.18895 [math.CA], 2024. See p. 5.

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Formula

a(n) = ((m(n) + 3)^2 - 7)/3 where m(n) = A001651(n) is the n-th natural number not divisible by 3.

G.f.: (x*(1+x+x^2)*(3-x^2))/((1-x)^3*(1+x)^2). - Joerg Arndt, Jan 19 2024

E.g.f.: (4 + (3*x^2 + 13*x - 4)*cosh(x) + (3*x^2 + 11*x - 1)*sinh(x))/4. - Stefano Spezia, Feb 13 2024

Sum_{n>=1} 1/a(n) = 3/2 + (tan((1+2*sqrt(7))*Pi/6) - cot((1+sqrt(7))*Pi/3)) * Pi/(2*sqrt(7)). - Amiram Eldar, Mar 07 2024

Example

For n=1, a(1)=3 and the polynomial (x-1)^3*(x+1)^3 = x^6 - 3*x^4 + 3*x^2 - 1 has three vanishing coefficients, those of x^5, x^3 and x.
For n=2, a(2)=6 and the polynomial (x-1)^3*(x+1)^6 = x^9 + 3*x^8 - 8*x^6 - 6*x^5 + 6*x^4 + 8*x^3 - 3*x - 1 has two vanishing coefficients, those of x^7 and x^2.

Mathematica

LinearRecurrence[{1, 2, -2, -1, 1}, {3, 6, 14, 19, 31}, 56] (* Hugo Pfoertner, Feb 12 2024 *)

Prog

(PARI) isok(k) = #select(x->(x==0), Vec((x-1)^3*(x+1)^k)) > 1; \\ Michel Marcus, Jan 19 2024
(Python)
def A369304(n): return ((n+1<<1)-(n>>1))**2//3-2 # Chai Wah Wu, Mar 05 2024

Crossrefs

Cf. A001651, A022998.

Sequence in context: A118523 A097633 A263620 * A083356 A096337 A175318

Adjacent sequences: A369301 A369302 A369303 * A369305 A369306 A369307

Keyword

nonn,easy

Author

Vladimir Petrov Kostov, Jan 19 2024

Status

approved