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%I #23 Jun 24 2020 04:15:22
%S 0,1,3,6,10,16,25,38,56,81,116,165,233,327,457,637,886,1230,1705,2361,
%T 3267,4518,6245,8629,11920,16463,22734,31390,43338,59830,82594,114015,
%U 157385,217248,299876,413926,571347,788632,1088546,1502511,2073898,2862571
%N Expansion of x/((1 - x - x^4)*(1 - x)^2).
%C The coefficients of the recursion for a(n) are given by the 3rd row of A145152.
%C Comment from - _Enrique Navarrete_, May 25 2020: (Start)
%C a(n-4) is the number of subsets of {1,2,...,n} such that the difference of successive elements is at least 4. For example, for n = 9, a(5) = 16 and the subsets are: {1,5}, {1,6}, {1,7}, {1,8}, {1,9}, {2,6}, {2,7}, {2,8}, {2,9}, {3,7}, {3,8}, {3,9}, {4,8}, {4,9}, {5,9}, {1,5,9}.
%C For n >=0 the sequence contains the triangular numbers; for n >= 4 have to add the tetrahedral numbers; for n >= 8 have to add the numbers binomial(n,4) (starting with 0,1,5,..); for n >= 12 have to add the numbers binomial(n,5) (starting with 0,1,6,..); in general, for n >= 4*k have to add to the sequence the numbers binomial(n, k+2), k >= 0.
%C For example, a(15) = 120+286+210+21, where 120 is a triangular number, 286 is a tetrahedral number, 210 is a number binomial(n,4) and 21 is a number binomial(m,5) (with the proper n, m due to shifts in the names of the sequences).
%C First difference is A098578.
%C (End)
%H Alois P. Heinz, <a href="/A145131/b145131.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1,1,-2,1).
%F a(n) = 3a(n-1) -3a(n-2) +a(n-3) +a(n-4) -2a(n-5) +a(n-6).
%e a(7) = 38 = 3*25 -3*16 +10 +6 -2*3 +1.
%p col:= proc(k) local l, j, M, n; l:= `if` (k=0, [1, 0, 0, 1], [seq(coeff( -(1-x-x^4) *(1-x)^(k-1), x, j), j=1..k+3)]); M:= Matrix(nops(l), (i,j)-> if i=j-1 then 1 elif j=1 then l[i] else 0 fi); `if`(k=0, n->(M^n)[2,3], n->(M^n)[1,2]) end: a:= col(3): seq(a(n), n=0..40);
%t Series[x/((1-x-x^4)*(1-x)^2), {x, 0, 50}] // CoefficientList[#, x]& (* _Jean-François Alcover_, Feb 13 2014 *)
%t LinearRecurrence[{3,-3,1,1,-2,1},{0,1,3,6,10,16},50] (* _Harvey P. Dale_, Aug 08 2015 *)
%o (PARI) concat(0,Vec(1/(1-x-x^4)/(1-x)^2+O(x^99))) \\ _Charles R Greathouse IV_, Sep 25 2012
%Y 3rd column of A145153. Cf. A145152.
%K nonn,easy
%O 0,3
%A _Alois P. Heinz_, Oct 03 2008
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