A253942 a(n) = 3*binomial(n+1, 5).

0, 0, 0, 3, 18, 63, 168, 378, 756, 1386, 2376, 3861, 6006, 9009, 13104, 18564, 25704, 34884, 46512, 61047, 79002, 100947, 127512, 159390, 197340, 242190, 294840, 356265, 427518, 509733, 604128, 712008, 834768, 973896, 1130976, 1307691, 1505826, 1727271, 1974024, 2248194

Offset

1,4

Comments

For a set of integers {1,2,...,n}, a(n) is the sum of the 2 smallest elements of each subset with 4 elements, which is 3*binomial(n+1, 5) (for n >= 4), hence a(n) = 3*binomial(n+1, 5) = 3*A000389(n+1).

a(n) is also the number of tetrahedra in the n-triangular honeycomb queen graph. - Eric W. Weisstein, Jan 08 2026

Links

Colin Barker, Table of n, a(n) for n = 1..1000 [a(1)-a(3) prepended by Georg Fischer, Feb 18 2026]

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

Eric Weisstein's World of Mathematics, Graph Tetrahedron.

Eric Weisstein's World of Mathematics, Triangular Honeycomb Queen Graph.

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Formula

a(n) = 3*A000389(n+1).

a(n) = (n-3)*(n-2)*(n-1)*n*(1+n)/40. - Colin Barker, Jan 20 2015

G.f.: 3*x^4 / (x-1)^6. - Colin Barker, Jan 20 2015

E.g.f.: x^4*(x+5)*exp(x)/40. - G. C. Greubel, Nov 25 2017

a(n) = Sum_{k=3..n-1} A050534(k). - Ivan N. Ianakiev, Oct 08 2023

From Amiram Eldar, Sep 29 2025: (Start)

Sum_{n>=4} 1/a(n) = 5/12.

Sum_{n>=4} (-1)^n/a(n) = 80*log(2)/3 - 655/36. (End)

Example

For A = {1,2,3,4}, the only subset with 4 elements is {1,2,3,4}; sum of 2 minimum elements of this subset: a(4) = 1+2 = 3 = 3*binomial(4+1, 5).
For A = {1,2,3,4,5}, the subsets with 4 elements are {1,2,3,4}, {1,2,3,5}, {1,2,4,5}, {1,3,4,5}, {2,3,4,5}; sum of 2 smallest elements of each subset: a(5) = (1+2)+(1+2)+(1+2)+(1+3)+(2+3) = 18 = 3*binomial(5+1, 5).

Mathematica

a253942[n_] := Drop[Plus @@ Flatten[Part[#, 1 ;; 2] & /@ Subsets[Range @ #, {4}]] & /@ Range @ n, 3]; a253942[28] (* Michael De Vlieger, Jan 20 2015 *)
Table[3 Binomial[n + 1, 5], {n, 35}] (* Vincenzo Librandi, Feb 14 2015 *)
3 Binomial[Range[20] + 1, 5] (* Eric W. Weisstein, Jan 08 2026 *)
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 0, 0, 3, 18, 63}, 20] (* Eric W. Weisstein, Jan 08 2026 *)
CoefficientList[Series[3 x^3/(x - 1)^6, {x, 0, 20}], x] (* Eric W. Weisstein, Jan 08 2026 *)

Prog

(PARI) a(n) = 3*binomial(n+1, 5); \\ Michel Marcus, Jan 20 2015
(PARI) Vec(3*x^4/(x-1)^6 + O(x^100)) \\ Colin Barker, Jan 20 2015
(Magma) [3*Binomial(n+1, 5): n in [4..40]]; // Vincenzo Librandi, Feb 14 2015

Crossrefs

Cf. A000389, A050534.

Sequence in context: A210366 A000648 A235988 * A317404 A110689 A027333

Adjacent sequences: A253939 A253940 A253941 * A253943 A253944 A253945

Keyword

nonn,easy

Author

Serhat Bulut, Jan 20 2015

Extensions

a(1)-a(3) prepended and offset adjusted by Eric W. Weisstein, Jan 08 2026

Status

approved