A253944 a(n) = 3*binomial(n+1,7).

3, 24, 108, 360, 990, 2376, 5148, 10296, 19305, 34320, 58344, 95472, 151164, 232560, 348840, 511632, 735471, 1038312, 1442100, 1973400, 2664090, 3552120, 4682340, 6107400, 7888725, 10097568, 12816144, 16138848, 20173560, 25043040, 30886416, 37860768

Offset

6,1

Comments

For a set of integers {1,2,...,n}, a(n) is the sum of the 2 smallest elements of each subset with 6 elements, which is 3*C(n+1,7) (for n>=6), hence a(n) = 3*C(n+1,7) = 3*A000580(n+1).

Links

G. C. Greubel, Table of n, a(n) for n = 6..1000

Serhat Bulut and Oktay Erkan Temizkan, Subset Sum Problem, 2015. [Wayback Machine link]

Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Formula

a(n) = 3*C(n+1,7) = 3*A000580(n+1).

a(n) = n*(n^6 - 14*n^5 + 70*n^4 - 140*n^3 + 49*n^2 + 154*n - 120)/1680.

From G. C. Greubel, Apr 03 2025: (Start)

G.f.: 3*x^6/(1-x)^8.

E.g.f.: (3/7!)*x^6*(x+7)*exp(x). (End)

From Amiram Eldar, Sep 28 2025: (Start)

Sum_{n>=6} 1/a(n) = 7/18.

Sum_{n>=6} (-1)^n/a(n) = 448*log(2)/3 - 9289/90. (End)

Example

For A={1,2,3,4,5,6,7}, subsets with 6 elements are {1,2,3,4,5,6}, {1,2,3,4,5,7}, {1,2,3,4,6,7}, {1,2,3,5,6,7}, {1,2,4,5,6,7}, {1,3,4,5,6,7}, {2,3,4,5,6,7}.
Sum of 2 smallest elements of each subset:
a(7) = (1+2)+(1+2)+(1+2)+(1+2)+(1+2)+(1+3)+(2+3) = 24 = 3*C(7+1,7) = 3*A000580(7+1).

Mathematica

Drop[Plus @@ Flatten[Part[#, 1 ;; 2] & /@ Subsets[Range@ #, {6}]] & /@
  Range@ 28, 5] (* Michael De Vlieger, Jan 20 2015 *)
3 Binomial[Range[7, 29], 7] (* Michael De Vlieger, Feb 13 2015, after Alonso del Arte at A253946 *)

Prog

(PARI) a(n)=3*binomial(n+1, 7) \\ Charles R Greathouse IV, Feb 04 2015
(Magma) [3*Binomial(n+1, 7): n in [6..40]]; // Vincenzo Librandi, Feb 13 2015
(SageMath)
def A253944(n): return 3*binomial(n+1, 7)
print([A253944(n) for n in range(6, 51)]) # G. C. Greubel, Apr 03 2025

Crossrefs

Cf. A000580.

Sequence in context: A342112 A050545 A354676 * A139031 A065692 A127520

Adjacent sequences: A253941 A253942 A253943 * A253945 A253946 A253947

Keyword

nonn,easy

Author

Serhat Bulut, Jan 20 2015

Extensions

More terms from Vincenzo Librandi, Feb 13 2015

Status

approved