A198808 -id:A198808

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A198800

Number of closed paths of length n whose steps are 20th roots of unity, U_20(n).

+10
2

1, 0, 20, 0, 1140, 480, 102800, 151200, 12310900, 38707200, 1812247920, 9574488000, 313983978000, 2391608419200, 62051403928800, 611744666332800, 13627749414064500, 160896284989440000, 3253345101771050000, 43527416858084016000, 829176006298475046640

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

U_20(n) (comment in article) : For each m >= 1, the sequence (U_m(N)), N >= 0 is P-recursive but is not algebraic when m > 2.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..200

Gilbert Labelle and Annie Lacasse, Closed paths whose steps are roots of unity, in FPSAC 2011, Reykjavik, Iceland DMTCS proc. AO, 2011, 599-610.

FORMULA

E.g.f.: g(x)^2 where g(x) is the e.g.f. of A070190. - Andrew Howroyd, Nov 01 2018

a(n) ~ 2^(2*n) * 5^(n+3) / (Pi^4 * n^4). - Vaclav Kotesovec, Apr 30 2024

PROG

(PARI) seq(n)={Vec(serlaplace(sum(k=0, n, if(k, 2, 1)*(x^k*besseli(k, 2*x + O(x^(n-k+1)))/k!)^5)^2))} \\ Andrew Howroyd, Nov 01 2018

CROSSREFS

Cf. A070190, A198808.

KEYWORD

nonn

AUTHOR

Simon Plouffe, Oct 30 2011

STATUS

approved

A321417

Number of n element multisets of the 12th roots of unity with zero sum.

+10
2

1, 0, 6, 4, 21, 24, 64, 84, 174, 236, 420, 576, 926, 1260, 1896, 2540, 3639, 4800, 6618, 8592, 11499, 14700, 19200, 24204, 30972, 38544, 48480, 59620, 73884, 89892, 109960, 132480, 160221, 191308, 229038, 271248, 321809, 378264, 445128, 519608, 606954, 704016

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Equivalently, the number of closed convex paths of length n whose steps are the 12th roots of unity up to translation. For even n, there will be 6 paths of zero area consisting of n/2 steps in one direction followed by n/2 steps in the opposite direction.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,3,-6,-6, 6,13,-2,-18, -2,13,6,-6, -6,3,2,-1)

FORMULA

G.f.: ((2/(1 - x^3) - 1)/(1 - x^2)^3)^2.

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

PROG

(PARI) Vec(((2/(1 - x^3) - 1)/(1 - x^2)^3)^2 + O(x^40))

CROSSREFS

Column k=6 of A321414.

Cf. A053090, A198808.

KEYWORD

nonn,easy

AUTHOR

Andrew Howroyd, Nov 09 2018

STATUS

approved

A198802

Number of closed paths of length n whose steps are 18th roots of unity, U_18(n).

+10
1

1, 0, 18, 36, 918, 5400, 82800, 801360, 10907190, 132053040, 1802041668, 24199809480, 340640607384, 4834708246368, 70229958125184, 1032223723667136, 15391538570569590, 231935110984687968, 3531542904056225916, 54244559313713885688, 839979883121036697468

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

U_18(n), comment in article: For each m >= 1, the sequence (U_m(N)), N >= 0 is P-recursive but is not algebraic when m > 2.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..200

Gilbert Labelle and Annie Lacasse, Closed paths whose steps are roots of unity, in FPSAC 2011, Reykjavik, Iceland DMTCS proc. AO, 2011, 599-610.

FORMULA

E.g.f.: g(x)^3 where g(x) is the e.g.f. of A002898.

a(n) ~ 2^(n-3) * 3^(2*n + 9/2) / (Pi^3 * n^3). - Vaclav Kotesovec, Apr 30 2024

PROG

(PARI) seq(n)={Vec(serlaplace(sum(k=0, n, if(k, 2, 1)*(x^k*besseli(k, 2*x + O(x^(n-k+1)))/k!)^3)^3))} \\ Andrew Howroyd, Nov 01 2018

CROSSREFS

Cf. A002898, A198808.

KEYWORD

nonn

AUTHOR

Simon Plouffe, Oct 30 2011

STATUS

approved

A198805

Number of closed paths of length n whose steps are 15th roots of unity, U_15(n).

+10
1

1, 0, 0, 30, 0, 360, 7650, 0, 302400, 4544400, 11226600, 324324000, 4310633250, 24324300000, 437404968000, 5634178329780, 45972927000000, 697866761592000, 8962716395833200, 88725951057744000, 1258898645656852200

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

U_15(n), (comment in article): For each m >= 1, the sequence (U_m(N)), N >= 0 is P-recursive but is not algebraic when m > 2.

LINKS

Table of n, a(n) for n=0..20.

Gilbert Labelle and Annie Lacasse, Closed paths whose steps are roots of unity, in FPSAC 2011, Reykjavik, Iceland DMTCS proc. AO, 2011, 599-610.

CROSSREFS

Cf. A198808, A321418.

KEYWORD

nonn

AUTHOR

Simon Plouffe, Oct 30 2011

STATUS

approved

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