A253942 - OEIS

%I #73 Oct 09 2023 12:57:55

%S 3,18,63,168,378,756,1386,2376,3861,6006,9009,13104,18564,25704,34884,

%T 46512,61047,79002,100947,127512,159390,197340,242190,294840,356265,

%U 427518,509733,604128,712008,834768,973896,1130976,1307691,1505826,1727271,1974024,2248194

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

%C 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).

%H Colin Barker, <a href="/A253942/b253942.txt">Table of n, a(n) for n = 4..1000</a>

%H Serhat Bulut and Oktay Erkan Temizkan, <a href="http://www.matematikproje.com/dosyalar/05cb6Subset_smallest_elements_Sum.pdf">Subset Sum Problem</a> [Dead link]

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).

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

%F a(n) = (n-3)*(n-2)*(n-1)*n*(1+n)/40. - _Colin Barker_, Jan 20 2015

%F G.f.: 3*x^4 / (x-1)^6. - _Colin Barker_, Jan 20 2015

%F E.g.f.: x^4*(x+5)*exp(x)/40. - _G. C. Greubel_, Nov 25 2017

%F a(n) = Sum_{k=3..n-1} A050534(k). - _Ivan N. Ianakiev_, Oct 08 2023

%e 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).

%e 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).

%t a253942[n_] := Drop[Plus @@ Flatten[Part[#, 1 ;; 2] & /@ Subsets[Range@ #, {4}]] & /@ Range@ n, 3]; a253942[28] (* _Michael De Vlieger_, Jan 20 2015 *)

%t Table[3 Binomial[n + 1, 5], {n, 4, 35}] (* _Vincenzo Librandi_, Feb 14 2015 *)

%o (PARI) a(n) = 3*binomial(n+1, 5); \\ _Michel Marcus_, Jan 20 2015

%o (PARI) Vec(3*x^4/(x-1)^6 + O(x^100)) \\ _Colin Barker_, Jan 20 2015

%o (Magma) [3*Binomial(n+1, 5): n in [4..40]]; _Vincenzo Librandi_, Feb 14 2015

%Y Cf. A000389.

%K nonn,easy

%O 4,1

%A _Serhat Bulut_, Jan 20 2015