A299754 -id:A299754

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A299807

Rectangular array read by antidiagonals: T(n,k) is the number of distinct sums of k complex n-th roots of 1, n >= 1, k >= 0.

+10
3

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 9, 10, 5, 1, 1, 6, 15, 16, 15, 6, 1, 1, 7, 19, 35, 25, 21, 7, 1, 1, 8, 28, 37, 70, 36, 28, 8, 1, 1, 9, 33, 84, 61, 126, 49, 36, 9, 1, 1, 10, 45, 96, 210, 91, 210, 64, 45, 10, 1, 1, 11, 51, 163, 225, 462, 127, 330, 81, 55, 11, 1, 1, 12, 66, 180, 477, 456, 924, 169, 495, 100, 66

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

OFFSET

1,5

LINKS

Max Alekseyev, Table of n, a(n) for n = 1..351

FORMULA

From Chai Wah Wu, May 28 2018: (Start)

The following are all conjectures.

For m >= 0, the 2^(m+1)-th row are the figurate numbers based on the 2^m-dimensional regular convex polytope with g.f.: (1+x)^(2^m-1)/(1-x)^(2^m+1).

For p prime, the n=p row corresponds to binomial(k+p-1,p-1) for k = 0,1,2,3,..., which is the maximum possible (i.e., the number of combinations with repetitions of k choices from p categories) with g.f.: 1/(1-x)^p.

(End)

EXAMPLE

Array starts:

n=1: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

n=2: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ...

n=3: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, ...

n=4: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, ...

n=5: 1, 5, 15, 35, 70, 126, 210, 330, 495, 715, 1001, ...

n=6: 1, 6, 19, 37, 61, 91, 127, 169, 217, 271, 331, ...

n=7: 1, 7, 28, 84, 210, 462, 924, 1716, 3003, 5005, 8008, ...

n=8: 1, 8, 33, 96, 225, 456, 833, 1408, 2241, 3400, 4961, ...

n=9: 1, 9, 45, 163, 477, 1197, 2674, 5454, 10341, 18469, 31383, ...

n=10: 1, 10, 51, 180, 501, 1131, 2221, 3951, 6531, 10201, 15231, ...

...

CROSSREFS

Rows: A000012 (n=1), A000027 (n=2), A000217 (n=3), A000290 (n=4), A000332 (n=5), A354343 (n=6), A000579 (n=7), A014820 (n=8).

Columns: A000012 (k=0), A000027 (k=1), A031940 (k=2).

Diagonal: A299754 (n=k).

Cf. A103314, A107754, A107861, A108380, A107848, A107753, A108381, A143008.

KEYWORD

nonn,tabl

AUTHOR

Max Alekseyev, Feb 24 2018

EXTENSIONS

Row 6 corrected by Max Alekseyev, Aug 14 2022

STATUS

approved

A354343

Number of distinct sums of n complex 6th power roots of unity.

+10
1

1, 6, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397, 469, 547, 631, 721, 817, 919, 1027, 1141, 1261, 1387, 1519, 1657, 1801, 1951, 2107, 2269, 2437, 2611, 2791, 2977, 3169, 3367, 3571, 3781, 3997, 4219, 4447, 4681, 4921, 5167, 5419, 5677, 5941, 6211, 6487, 6769, 7057, 7351, 7651, 7957

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

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..51.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

For n >= 2, a(n) = 3*n^2 + 3*n + 1 = A003215(n).

For n >= 5, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

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

MATHEMATICA

LinearRecurrence[{3, -3, 1}, {1, 6, 19, 37, 61}, 60] (* Harvey P. Dale, Nov 03 2024 *)