OFFSET
1,1
COMMENTS
a(n)=a(n+30) are the coefficients of S(x) := T(x)-T(x/2)-T(x/3)-T(x/5)+T(x/30) with Chebyshev's function T(x) := sum(log(n),n=1..floor(x)), expanded in terms of psi(x/n) with psi(x) := log(A003418(floor(x))) (logarithm of least common multiple of {1,2,...,floor(x)}): S(x)=sum(a(n)*psi(x/n),n=1..infinity).
REFERENCES
M. I. Bashmakov: On Bertrand's Conjecture, pp. 21-26 in: Kvant Selecta: Algebra and Analysis, I, ed. S. Tabachnikov, Am. Math. Soc., 1999
L. G. Limanov: On n! and the Number e (Several Approaches to a Certain Problem), pp. 57-64 in: Kvant Selecta: Algebra and Analysis, I, ed. S. Tabachnikov, Am. Math. Soc., 1999
LINKS
Eric Weisstein's World of Mathematics, Chebyshev Functions.
Index entries for linear recurrences with constant coefficients, signature (-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1).
FORMULA
a(n)= 1-floor(2^(floor(n/2)-n/2)) - floor(2^(floor(n/3)-n/3)) - floor(2^(floor(n/5)-n/5)) + floor(2^(floor(n/30)-n/30)), from eq.(4), p. 64 of the Limanov reference.
From Chai Wah Wu, Dec 28 2021: (Start)
a(n) = - a(n-1) - a(n-2) - a(n-3) - a(n-4) - a(n-5) - a(n-6) - a(n-7) - a(n-8) - a(n-9) - a(n-10) - a(n-11) - a(n-12) - a(n-13) - a(n-14) - a(n-15) - a(n-16) - a(n-17) - a(n-18) - a(n-19) - a(n-20) - a(n-21) - a(n-22) - a(n-23) - a(n-24) - a(n-25) - a(n-26) - a(n-27) - a(n-28) - a(n-29) for n > 29.
G.f.: x*(x^28 + x^22 + x^18 + x^16 + x^13 + x^12 + x^10 + x^8 + x^7 + x^6 + x^4 + x^3 + x^2 + x + 1)/((x + 1)*(x^2 - x + 1)*(x^2 + x + 1)*(x^4 - x^3 + x^2 - x + 1)*(x^4 + x^3 + x^2 + x + 1)*(x^8 - x^7 + x^5 - x^4 + x^3 - x + 1)*(x^8 + x^7 - x^5 - x^4 - x^3 + x + 1)).
(End)
MATHEMATICA
PadRight[{}, 120, {1, 0, 0, 0, 0, -1, 1, 0, 0, -1, 1, -1, 1, 0, -1, 0, 1, -1, 1, -1, 0, 0, 1, -1, 0, 0, 0, 0, 1, -1}] (* Harvey P. Dale, Jan 16 2019 *)
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Wolfdieter Lang, Dec 01 2003
STATUS
approved