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%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
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