%I #29 Sep 07 2024 08:55:06

%S 2,4,6,8,10,12,16,20,24,28,32,36,42,48,54,60,66,72,80,88,96,104,112,

%T 120,130,140,150,160,170,180,192,204,216,228,240,252,266,280,294,308,

%U 322,336,352,368,384,400,416,432,450,468,486,504,522,540,560,580,600

%N Number of 6 element subsets of {1,2,3,...,n} for which the sum-set has 12 elements.

%H Vincenzo Librandi, <a href="/A112421/b112421.txt">Table of n, a(n) for n = 7..1000</a>

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,0,0,1,-2,1).

%F G.f.: 2*x^7/((1-x)^2*(1-x^6)).

%F a(n) = 2*A008724(n-3). a(n) = 2*a(n-1) - a(n-2) + a(n-6) - 2*a(n-7) + a(n-8). - _R. J. Mathar_, Jul 26 2010

%F a(n) = 2*j*(n-3*j-3), where j=floor(n/6). - _Jon E. Schoenfield_, Dec 20 2013

%e a(7)=2 because the two sets {1 2 3 4 5 7} and {1 3 4 5 6 7} have sum-sets

%e {2 3 4 5 6 7 8 9 10 11 12 14} and {2 4 5 6 7 8 9 10 11 12 13 14}, respectively and each of these sum-sets has 12 elements.

%t CoefficientList[Series[2/((1 - x)^2 (1 - x^6)), {x, 0, 100}], x] (* _Vincenzo Librandi_, Dec 21 2013 *)

%o (PARI) lista(n) = {my(x = xx + O(xx^n)); gf = 2*x^7/((1-x)^2*(1-x^6)); for (i=7, n-1, print1(polcoeff(gf, i, xx), ", "));} \\ - _Michel Marcus_, Dec 20 2013

%o (Magma) I:=[2,4,6,8,10,12,16,20]; [n le 8 select I[n] else 2*Self(n-1)-Self(n-2)+Self(n-6)-2*Self(n-7)+Self(n-8): n in [1..70]]; // _Vincenzo Librandi_, Dec 21 2013

%Y Cf. A008724.

%K nonn,easy

%O 7,1

%A _David S. Newman_, Dec 10 2005

%E More terms from _Jon E. Schoenfield_, Dec 20 2013