

A369304


Numbers k for which the polynomial (x1)^3*(x+1)^k has more than one zero coefficient.


1



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
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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 (x1)^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(n1) = 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



FORMULA

a(n) = ((m(n) + 3)^2  7)/3 where m(n) = A001651(n) is the nth natural number not divisible by 3.
G.f.: (x*(1+x+x^2)*(3x^2))/((1x)^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 (x1)^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 (x1)^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((x1)^3*(x+1)^k)) > 1; \\ Michel Marcus, Jan 19 2024
(Python)


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



