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A192447
a(n) = n*(n-1)/2 if this is even, otherwise (n*(n-1)/2) + 1.
2
0, 2, 4, 6, 10, 16, 22, 28, 36, 46, 56, 66, 78, 92, 106, 120, 136, 154, 172, 190, 210, 232, 254, 276, 300, 326, 352, 378, 406, 436, 466, 496, 528, 562, 596, 630, 666, 704, 742, 780, 820, 862, 904, 946, 990, 1036, 1082, 1128, 1176, 1226, 1276, 1326, 1378, 1432
OFFSET
1,2
COMMENTS
Least number of swaps of passports of n persons so that each two have swapped at least once and finally each one gets his own passport (JBMO 2011 Shortlist).
FORMULA
a(n) = n*(n-1)/2 if this is even and a(n) = (n*(n-1)/2) + 1 otherwise.
a(n) = 2*A054925(n+1).
G.f.: 2*x*(x^2 - x + 1)/((1 - x)^3*(1 + x^2)).
a(n) = (n^2 - n + 1 - (-1)^(n*(n-1)/2))/2. - Guenther Schrack, Jun 04 2019
EXAMPLE
a(3)=4: Let the initial state be Aa, Bb, Cc. Swap(AB) to get Ab, Ba, Cc. Swap(AC) to get Ac, Ba, Cb. Swap(BC) to get Ac, Bb, Ca. Swap(AC) to get Aa, Bb, Cc, done.
MATHEMATICA
Table[(n^2 - n + 1 - (-1)^(n (n - 1)/2))/2, {n, 1, 60}] (* Bruno Berselli, Jun 07 2019 *)
LinearRecurrence[{3, -4, 4, -3, 1}, {0, 2, 4, 6, 10}, 54] (* Georg Fischer, Oct 26 2020 *)
PROG
(PARI) a(n) = my(m=n*(n-1)/2); if (m % 2, m+1, m); \\ Michel Marcus, Jun 07 2019
CROSSREFS
Equals the corresponding term of A000217 if it is even or is 1 more otherwise.
Cf. A054925.
Sequence in context: A339574 A258599 A101176 * A131882 A073150 A076529
KEYWORD
nonn,easy
AUTHOR
Ivaylo Kortezov, Jul 01 2011
STATUS
approved