A192447 a(n) = n*(n-1)/2 if this is even, otherwise (n*(n-1)/2) + 1.

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).

Links

Table of n, a(n) for n=1..54.

Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).

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).

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

Sum_{n>=2} 1/a(n) = 2 - Pi/2 + Pi*sinh(sqrt(7)*Pi/4)/(sqrt(7)*(1/sqrt(2) + cosh(sqrt(7)*Pi/4))). - Amiram Eldar, Dec 14 2024

a(n) = 2*(A213484(n+1) - 1)/3 = (A373584(n) - 1)/3. - Hugo Pfoertner, Dec 15 2024

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, A213484, A373584.

Sequence in context: A339574 A258599 A101176 * A131882 A073150 A076529

Adjacent sequences: A192444 A192445 A192446 * A192448 A192449 A192450

Keyword

nonn,easy

Author

Ivaylo Kortezov, Jul 01 2011

Status

approved