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A112421 Number of 6 element subsets of {1,2,3,...,n} for which the sum-set has 12 elements.

%I A112421 #29 Sep 07 2024 08:55:06
%S A112421 2,4,6,8,10,12,16,20,24,28,32,36,42,48,54,60,66,72,80,88,96,104,112,
%T A112421 120,130,140,150,160,170,180,192,204,216,228,240,252,266,280,294,308,
%U A112421 322,336,352,368,384,400,416,432,450,468,486,504,522,540,560,580,600
%N A112421 Number of 6 element subsets of {1,2,3,...,n} for which the sum-set has 12 elements.
%H A112421 Vincenzo Librandi, <a href="/A112421/b112421.txt">Table of n, a(n) for n = 7..1000</a>
%H A112421 <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 A112421 G.f.: 2*x^7/((1-x)^2*(1-x^6)).
%F A112421 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 A112421 a(n) = 2*j*(n-3*j-3), where j=floor(n/6). - _Jon E. Schoenfield_, Dec 20 2013
%e A112421 a(7)=2 because the two sets {1 2 3 4 5 7} and {1 3 4 5 6 7} have sum-sets
%e A112421 {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 A112421 CoefficientList[Series[2/((1 - x)^2 (1 - x^6)), {x, 0, 100}], x] (* _Vincenzo Librandi_, Dec 21 2013 *)
%o A112421 (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 A112421 (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 A112421 Cf. A008724.
%K A112421 nonn,easy
%O A112421 7,1
%A A112421 _David S. Newman_, Dec 10 2005
%E A112421 More terms from _Jon E. Schoenfield_, Dec 20 2013