A192447 - OEIS

%I #38 Dec 15 2024 07:25:21

%S 0,2,4,6,10,16,22,28,36,46,56,66,78,92,106,120,136,154,172,190,210,

%T 232,254,276,300,326,352,378,406,436,466,496,528,562,596,630,666,704,

%U 742,780,820,862,904,946,990,1036,1082,1128,1176,1226,1276,1326,1378,1432

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

%C 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).

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-4,4,-3,1).

%F a(n) = n*(n-1)/2 if this is even and a(n) = (n*(n-1)/2) + 1 otherwise.

%F a(n) = 2*A054925(n).

%F G.f.: 2*x*(x^2 - x + 1)/((1 - x)^3*(1 + x^2)).

%F a(n) = (n^2 - n + 1 - (-1)^(n*(n-1)/2))/2. - _Guenther Schrack_, Jun 04 2019

%F 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

%F a(n) = 2*(A213484(n+1) - 1)/3 = (A373584(n) - 1)/3. - _Hugo Pfoertner_, Dec 15 2024

%e 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.

%t Table[(n^2 - n + 1 - (-1)^(n (n - 1)/2))/2, {n, 1, 60}] (* _Bruno Berselli_, Jun 07 2019 *)

%t LinearRecurrence[{3,-4,4,-3,1}, {0,2,4,6,10}, 54] (* _Georg Fischer_, Oct 26 2020 *)

%o (PARI) a(n) = my(m=n*(n-1)/2); if (m % 2, m+1, m); \\ _Michel Marcus_, Jun 07 2019

%Y Equals the corresponding term of A000217 if it is even or is 1 more otherwise.

%Y Cf. A054925, A213484, A373584.

%K nonn,easy

%O 1,2

%A _Ivaylo Kortezov_, Jul 01 2011