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 A240328 Inverse of 37th cyclotomic polynomial. 95

%I

%S 1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,

%T 0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,

%U 0,0,0,0,0,0,1,-1,0,0,0,0,0,0

%N Inverse of 37th cyclotomic polynomial.

%C Periodic with period length 37. - _Ray Chandler_, Apr 03 2017

%C In general the expansion of 1/Phi(N) is N-periodic, but also satisfies a linear recurrence of lower order given by degree(Phi(N)) = phi(N) = A000010(N) < N. The signature is given by the coefficients of (1-Phi(N)). - _M. F. Hasler_, Feb 18 2018

%H Vincenzo Librandi, <a href="/A240328/b240328.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_36">Index entries for linear recurrences with constant coefficients</a>, order 36, 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, -1, -1, -1, -1, -1, -1, -1).

%H <a href="/index/Pol#poly_cyclo_inv">Index to sequences related to inverse of cyclotomic polynomials</a>

%t CoefficientList[Series[1/Cyclotomic[37, x], {x, 0, 100}], x]

%t LinearRecurrence[{-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, -1, -1, -1, -1, -1, -1, -1},{1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},82] (* _Ray Chandler_, Sep 15 2015 *)

%o (PARI) Vec(1/polcyclo(37) + O(x^99));

%o (MAGMA) &cat[[n le 2 select -(-1)^n else 0: n in [1..37]]: m in [1..4]];

%o (MAGMA) t:=37; u:=3; m:=u*t+2; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/CyclotomicPolynomial(t))); // _Bruno Berselli_, Apr 04 2014

%Y Cf. A049347 (3), A056594 (4), A010891 (5), A010892 (6), A014016 (7) - A014045 (36), this sequence (37), A014047 (38) - A014049 (40), A240329 (41), A014051 (42), A240330 (43), A014053 (44) - A014055 (46), A240331 (47), A014057 (48), A240332 (49), A014059 (50) - A014061 (52), A240348 (53), A014063 (54) - A014067 (58), A240349 (59), A014069 (60), A240350 (61), A014071 (62), A014072 (63), A240351 (64), A014074 (65), A014075 (66), A240352 (67), A240353 (68), A014078 (69), A014079 (70), A240354 (71) - A240357 (74), A014084 (75).

%Y Cf. sequences listed in A240467 for 1/Phi(N) with N = 76 .. 253.

%K sign,easy

%O 0

%A _Vincenzo Librandi_, Apr 04 2014

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Last modified April 19 04:19 EDT 2019. Contains 322237 sequences. (Running on oeis4.)