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 A369304 Numbers k for which the polynomial (x-1)^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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified September 10 06:17 EDT 2024. Contains 375773 sequences. (Running on oeis4.)