A285009 Subset sums (see Comments).

9, 17, 28, 42, 59, 79, 102, 128, 157, 189, 224, 262, 303, 347, 394, 444, 497, 553, 612, 674, 739, 807, 878, 952, 1029, 1109, 1192, 1278, 1367, 1459, 1554, 1652, 1753, 1857, 1964, 2074, 2187, 2303, 2422, 2544, 2669, 2797, 2928, 3062, 3199, 3339, 3482, 3628, 3777, 3929, 4084

Offset: 3

Author

Ivan N. Ianakiev, Apr 07 2017

Keywords

• easy
• nonn

Comments

For n > 2, take the set [3*(n-1)] and form three subsets all of which: a) have cardinality of n, b) have the same sum of elements, and c) share one element with the other subset and another element with the third subset. a(n) is the sum of the elements of each subset.

a(n) is the minimum value of the magic constant in a normal magic triangle of order n (see formula 5 in Trotter). - Stefano Spezia, Feb 18 2021

Examples

			For n = 3, the set is S = {1,2,3,4,5,6} and the subsets are S1 = {1,2,6}, S2 = {1,3,5} and S3 = {2,3,4}. Therefore, a(3) = 9.
		

References

• a(4) is mentioned in: Gary Gruber, "The World's 200 Hardest Brain Teasers", Sourcebooks, 2010, p. 55.

Links

• Colin Barker, Table of n, a(n) for n = 3..1000
• Terrel Trotter, Normal Magic Triangles of Order n, Journal of Recreational Mathematics Vol. 5, No. 1, 1972, pp. 28-32.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A005449.

Programs

Mathematica

Table[(8+(n-2)*(3 *n+1))/2,{n,3,53}]
Drop[CoefficientList[Series[x^3*(9 - 10*x + 4*x^2) / (1 - x)^3 , {x, 0, 60}], x], 3] (* Indranil Ghosh, Apr 08 2017 *)

PARI

Vec(x^3*(9 - 10*x + 4*x^2) / (1 - x)^3 + O(x^60)) \\ Colin Barker, Apr 08 2017

Formula

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

a(n) = (8 + (n-2)*(3*n+1))/2, for n > 2.

G.f.: x^3*(9 - 10*x + 4*x^2) / (1 - x)^3. - Colin Barker, Apr 08 2017

E.g.f.: (1/2)*exp(x)*(3*x^2 - 2*x + 6) - 2*x*(x + 1) - 3. - Indranil Ghosh, Apr 08 2017; corrected by Ilya Gutkovskiy, Apr 10 2017

a(n) = A005449(n-1) + 2. - Hugo Pfoertner, Feb 18 2021