A219844 T(n,k)=Number of -k..k arrays of length n whose end-around centered second difference is a constant times that array.

3, 5, 5, 7, 9, 9, 9, 13, 23, 13, 11, 17, 43, 33, 3, 13, 21, 69, 61, 5, 17, 15, 25, 101, 97, 7, 45, 3, 17, 29, 139, 141, 9, 85, 5, 13, 19, 33, 183, 193, 11, 137, 7, 33, 9, 21, 37, 233, 253, 13, 201, 9, 61, 23, 5, 23, 41, 289, 321, 15, 277, 11, 97, 43, 9, 3, 25, 45, 351, 397, 17, 365, 13

Offset

1,1

Comments

The empirical formula below depends on eigenvectors of the system existing only with the discovered periods of 1, 2, 3, 4 and 6

The eigenvectors are sampled sines and cosines, and so the formula depends on no sampling x(j) of sin(j*2Pi/n+offset) existing with exclusively rational ratios of the x(j) except for those already found for n=1,2,3,4 or 6

The empirical formula dependency on k then describes how those existing cycles can be populated with new values

Links

R. H. Hardin, Table of n, a(n) for n = 1..476

Wikipedia, Crystallographic restriction theorem.

Formula

.Empirical: t(n,k) { #pseudocode

. if (n modulo 12 = 0) return 10*k^2+14*k+1;

. if (n modulo 12 = 1 || n modulo 12 = 5 || n modulo 12 = 7 || n modulo 12 = 11) return 2*k+1;

. if (n modulo 12 = 2 || n modulo 12 = 10) return 4*k+1;

. if (n modulo 12 = 3 || n modulo 12 = 9) return 3*k^2+5*k+1;

. if (n modulo 12 = 4 || n modulo 12 = 8) return 4*k^2+8*k+1;

. if (n modulo 12 = 6) return 6*k^2+10*k+1;

.}

Example

Table starts
..3..5...7...9..11..13..15..17..19...21...23...25...27...29...31...33...35...37
..5..9..13..17..21..25..29..33..37...41...45...49...53...57...61...65...69...73
..9.23..43..69.101.139.183.233.289..351..419..493..573..659..751..849..953.1063
.13.33..61..97.141.193.253.321.397..481..573..673..781..897.1021.1153.1293.1441
..3..5...7...9..11..13..15..17..19...21...23...25...27...29...31...33...35...37
.17.45..85.137.201.277.365.465.577..701..837..985.1145.1317.1501.1697.1905.2125
..3..5...7...9..11..13..15..17..19...21...23...25...27...29...31...33...35...37
.13.33..61..97.141.193.253.321.397..481..573..673..781..897.1021.1153.1293.1441
..9.23..43..69.101.139.183.233.289..351..419..493..573..659..751..849..953.1063
..5..9..13..17..21..25..29..33..37...41...45...49...53...57...61...65...69...73
..3..5...7...9..11..13..15..17..19...21...23...25...27...29...31...33...35...37
.25.69.133.217.321.445.589.753.937.1141.1365.1609.1873.2157.2461.2785.3129.3493
..3..5...7...9..11..13..15..17..19...21...23...25...27...29...31...33...35...37
Some solutions for n=6 k=5, alternating array and its second difference
..2...-2....3...-3....1...-1....0....0....3...-9....3..-12....1...-3....2...-6
..2...-2...-1....1....0....0....4...-4....2...-6...-3...12...-3....9...-4...12
..0....0...-4....4...-1....1....4...-4...-5...15....3..-12....2...-6....2...-6
.-2....2...-3....3...-1....1....0....0....3...-9...-3...12....1...-3....2...-6
.-2....2....1...-1....0....0...-4....4....2...-6....3..-12...-3....9...-4...12
..0....0....4...-4....1...-1...-4....4...-5...15...-3...12....2...-6....2...-6
Some solutions for n=12 k=5, alternating array and its second difference
.-3....3...-3....9...-5...10...-4...12....4...-4...-3....3....2...-8....1...-3
..0....0....0....0...-2....4....2...-6...-1....1...-2....2...-2....8....2...-6
..3...-3....3...-9....5..-10....2...-6...-5....5....1...-1....2...-8...-3....9
..3...-3...-3....9....2...-4...-4...12...-4....4....3...-3...-2....8....1...-3
..0....0....0....0...-5...10....2...-6....1...-1....2...-2....2...-8....2...-6
.-3....3....3...-9...-2....4....2...-6....5...-5...-1....1...-2....8...-3....9
.-3....3...-3....9....5..-10...-4...12....4...-4...-3....3....2...-8....1...-3
..0....0....0....0....2...-4....2...-6...-1....1...-2....2...-2....8....2...-6
..3...-3....3...-9...-5...10....2...-6...-5....5....1...-1....2...-8...-3....9
..3...-3...-3....9...-2....4...-4...12...-4....4....3...-3...-2....8....1...-3
..0....0....0....0....5..-10....2...-6....1...-1....2...-2....2...-8....2...-6
.-3....3....3...-9....2...-4....2...-6....5...-5...-1....1...-2....8...-3....9

Prog

(PARI) t(n, k)={1+[[10, 14], [0, 2], [0, 4], [3, 5], [4, 8], [0, 2], [6, 10]][7-abs(n%12-6)]*[k^2, k]~} \\ Using the pseudocode given as formula. - M. F. Hasler, Feb 20 2013

Crossrefs

Sequence in context: A276501 A131421 A088743 * A168056 A122800 A227950

Adjacent sequences: A219841 A219842 A219843 * A219845 A219846 A219847

Keyword

nonn,tabl

Author

R. H. Hardin Feb 19 2013

Status

approved