A280186 Number of 3-element subsets of S = {1..n} whose sum is odd.

0, 0, 0, 0, 2, 4, 10, 16, 28, 40, 60, 80, 110, 140, 182, 224, 280, 336, 408, 480, 570, 660, 770, 880, 1012, 1144, 1300, 1456, 1638, 1820, 2030, 2240, 2480, 2720, 2992, 3264, 3570, 3876, 4218, 4560, 4940, 5320, 5740, 6160, 6622, 7084, 7590, 8096, 8648, 9200

Offset: 0

Author

Necip Fazil Patat, Dec 28 2016

Keywords

• nonn
• easy

Comments

The same as A006584 (apart from the offset). - R. J. Mathar, Jan 15 2017

There are two cases: n is odd and n is even.

Let n be an odd integer and n > 3, the sum of 3 integers is odd when all of them are odd or one is odd and the others are even. Number of ways to choose 3 odd numbers: C((n+1)/2, 3). Number of ways to choose 2 even numbers and 1 odd: C((n-1)/2, 2)*C((n+1)/2, 1). Total number of ways: C((n+1)/2, 3) + C((n-1)/2, 2)*C((n+1)/2,1).

Let n be an even integer and n > 3. Number of ways to choose 3 odd numbers: C(n/2, 3). Number of ways to choose 2 even numbers and 1 odd: C(n/2, 2)*C(n/2, 1). Total number of ways: C(n/2, 3) + C(n/2, 2)*C(n/2, 1).

Take a chessboard of n X n unit squares in which the a1 square is black. a(n) is the number of composite squares having white unit squares on their vertices. For the number of composite squares having black unit squares on their vertices see A005993. - Ivan N. Ianakiev, Aug 19 2018

Examples

			For n = 5 then a(5) = 4. The subsets are: {1, 2, 4}, {1, 3, 5}, {2, 3, 4}, {2, 4, 5}.
		

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Crossrefs

Essentially 2*A006918.

Same as A006584.

Programs

Mathematica

Table[Binomial[(n + #)/2, 3] + Binomial[(n - #)/2, 2] Binomial[(n + #)/2, 1] &@ Boole@ OddQ@ n, {n, 0, 49}] (* or *)
CoefficientList[Series[2 x^4/((1 - x)^4*(1 + x)^2), {x, 0, 49}], x] (* Michael De Vlieger, Jan 07 2017 *)

PARI

concat(vector(4), Vec(2*x^4 / ((1-x)^4*(1+x)^2) + O(x^60))) \\ Colin Barker, Dec 28 2016

Formula

a(n) = C((n+1)/2, 3) + C((n-1)/2, 2)*C((n+1)/2,1) when n is odd.

a(n) = C(n/2, 3) + C(n/2, 2)*C(n/2, 1) when n is even.

From Colin Barker, Dec 28 2016: (Start)

a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) for n>5.

a(n) = n*(n - 1)*(n - 2)/12 for n even.

a(n) = (n - 1)*(n + 1)*(n - 3)/12 for n odd.

G.f.: 2*x^4 / ((1-x)^4*(1+x)^2). (End)

a(n) = ((-1)^n)*(-1+n)*(3 - 3*(-1)^n - 4*((-1)^n)*n + 2*((-1)^n)*n^2)/24. - Ivan N. Ianakiev, Aug 19 2018

Extensions

More terms from Colin Barker, Dec 28 2016