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A219844 T(n,k)=Number of -k..k arrays of length n whose end-around centered second difference is a constant times that array. 1
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified May 29 21:28 EDT 2017. Contains 287257 sequences.