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a(n) = 6*binomial(n+1, 6).
7

%I #71 Sep 08 2022 08:46:11

%S 6,42,168,504,1260,2772,5544,10296,18018,30030,48048,74256,111384,

%T 162792,232560,325584,447678,605682,807576,1062600,1381380,1776060,

%U 2260440,2850120,3562650,4417686,5437152,6645408,8069424,9738960,11686752,13948704,16564086

%N a(n) = 6*binomial(n+1, 6).

%C For a set of integers {1, 2, ..., n}, a(n) is the sum of the 3 smallest elements of each subset with 5 elements, which is 6*C(n+1, 6) (for n >= 5), hence a(n) = 6*C(n+1, 6) = 6 * A000579(n+1).

%H Colin Barker, <a href="/A253946/b253946.txt">Table of n, a(n) for n = 5..1000</a>

%H Serhat Bulut and Oktay Erkan Temizkan, <a href="https://web.archive.org/web/20160708101054/http://matematikproje.com/dosyalar/7e1cdSubset_smallest_elements_Sum.pdf">Subset Sum Problem</a>, 2015.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).

%F a(n) = 6*C(n+1,6) = 6*A000579(n+1).

%F G.f.: 6*x^5 / (1-x)^7. - _Colin Barker_, Apr 03 2015

%F From _Amiram Eldar_, Jan 09 2022: (Start)

%F Sum_{n>=5} 1/a(n) = 1/5.

%F Sum_{n>=5} (-1)^(n+1)/a(n) = 32*log(2) - 661/30. (End)

%e For A = {1, 2, 3, 4, 5, 6} the subsets with 5 elements are {1, 2, 3, 4, 5}, {1, 2, 3, 4, 6}, {1, 2, 3, 5, 6}, {1, 2, 4, 5, 6}, {1, 3, 4, 5, 6}, {2, 3, 4, 5, 6}.

%e The sum of 3 smallest elements of each subset: a(6) = (1 + 2 + 3) + (1 + 2 + 3) + (1 + 2 + 3) + (1 + 2 + 4) + (1 + 3 + 4) + (2 + 3 + 4) = 42 = 6*C(6 + 1, 6) = 6*A000579(6+1).

%p A253946:=n->6*binomial(n+1,6): seq(A253946(n), n=5..50); # _Wesley Ivan Hurt_, Feb 13 2015

%t Drop[Plus @@ Flatten[Part[#, 1 ;; 3] & /@ Subsets[Range@ #, {5}]] & /@

%t Range@ 30, 4] (* _Michael De Vlieger_, Jan 20 2015 *)

%t 6Binomial[Range[6, 29], 6] (* _Alonso del Arte_, Feb 05 2015 *)

%t LinearRecurrence[{7,-21,35,-35,21,-7,1},{6,42,168,504,1260,2772,5544},40] (* _Harvey P. Dale_, May 14 2019 *)

%o (Magma) [6*Binomial(n+1, 6): n in [5..40]]; // _Vincenzo Librandi_, Feb 13 2015

%o (PARI) Vec(6*x^5/(1-x)^7 + O(x^100)) \\ _Colin Barker_, Apr 03 2015

%Y Cf. A000579.

%Y Sixth column of A003506.

%K nonn,easy

%O 5,1

%A _Serhat Bulut_, Jan 20 2015

%E More terms from _Vincenzo Librandi_, Feb 13 2015