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A306610 a(n) = (2*cos(Pi/15))^(-n) + (2*cos(7*Pi/15))^(-n) + (2*cos(11*Pi/15))^(-n) + (2*cos(13*Pi/15))^(-n), for n >= 1. 2
4, 24, 109, 524, 2504, 11979, 57299, 274084, 1311049, 6271254, 29997829, 143491199, 686373809, 3283190949, 15704770004, 75121978804, 359337430474, 1718849676159, 8221921677724, 39328626006254, 188124003629279, 899869747188249, 4304424455586134 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

-a(n) is the coefficient of x in the minimal polynomial for (2*cos(Pi/15))^n, for n >= 1. The coefficients of -x^3 are A306603(n), and those of x^2 are A306611(n).

a(n) is obtained from the Girard-Waring formula for the sum of powers of N = 4 indeterminates (see A324602), with the elementary symmetric functions e_1 = 4, e_2 = -4, e_3 = -1 and e_4 = 1. The arguments are e_j(1/x_1, 1/x_2, 1/x_3, 1/x_4), for j = 1..4, with the zeros {x_i}_{i=1..4} of the minimal polynomial of 2*cos(Pi/15), appearing under the negative powers of the formula given above. - Wolfdieter Lang, May 08 2019

LINKS

Table of n, a(n) for n=1..23.

Index entries for linear recurrences with constant coefficients, signature (4,4,-1,-1).

FORMULA

a(n) = 4a(n-1) + 4a(n-2) - a(n-3) - a(n-4).

G.f.: x*(-4x^3 -3x^2 +8x +4)/(x^4 +x^3 -4x^2 -4x +1).

a(n) = round((2*cos(7*Pi/15))^(-n)) for n >= 3.

MATHEMATICA

Table[Round[N[Sum[(2 Cos[k Pi/15])^(-n), {k, {1, 7, 11, 13}}], 50]], {n, 1, 30}]

CROSSREFS

Cf. A019887 (cos(Pi/15)), A019815 (cos(7*Pi/15)), A019851 (cos(11*Pi/15)), A019875 (cos(13*Pi/15)), A306603 (positive powers of these cosines), A306611, A324602.

Sequence in context: A120908 A145655 A265975 * A059153 A129032 A270686

Adjacent sequences:  A306607 A306608 A306609 * A306611 A306612 A306613

KEYWORD

nonn,easy

AUTHOR

Greg Dresden, Feb 28 2019

STATUS

approved

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Last modified August 21 18:38 EDT 2019. Contains 326168 sequences. (Running on oeis4.)