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A057029 Central column of arrays in A057027 and A057028.

%I A057029 #24 Sep 10 2024 20:22:27
%S A057029 1,6,12,27,39,64,82,117,141,186,216,271,307,372,414,489,537,622,676,
%T A057029 771,831,936,1002,1117,1189,1314,1392,1527,1611,1756,1846,2001,2097,
%U A057029 2262,2364,2539,2647,2832,2946,3141,3261,3466,3592,3807,3939,4164,4302,4537
%N A057029 Central column of arrays in A057027 and A057028.
%H A057029 Colin Barker, <a href="/A057029/b057029.txt">Table of n, a(n) for n = 1..1000</a>
%H A057029 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A057029 a(n) = C(2n-1, 2)+(n+1)/2 if n is odd, else a(n) = C(2n, 2)-(n-2)/2.
%F A057029 From _Colin Barker_, Jul 02 2016: (Start)
%F A057029 a(n) = (5-(-1)^n+2*(-4+(-1)^n)*n+8*n^2)/4.
%F A057029 a(n) = (4*n^2-3*n+2)/2 for n even, a(n) = (4*n^2-5*n+3)/2 for n odd.
%F A057029 a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5) for n>5.
%F A057029 G.f.: x*(1+5*x+4*x^2+5*x^3+x^4) / ((1-x)^3*(1+x)^2). (End)
%F A057029 E.g.f.: ((2 - x + 4*x^2)*cosh(x) + (3 + x + 4*x^2)*sinh(x) - 2)/2. - _Stefano Spezia_, Sep 10 2024
%p A057029 A057029:=n->(5-(-1)^n+2*(-4+(-1)^n)*n+8*n^2)/4: seq(A057029(n), n=1..80); # _Wesley Ivan Hurt_, Jul 03 2016
%t A057029 Table[(5 - (-1)^n + 2 (-4 + (-1)^n) n + 8 n^2)/4, {n, 49}] (* or *)
%t A057029 Table[If[OddQ@ n, Binomial[2 n - 1, 2] + (n + 1)/2 , Binomial[2 n, 2] - (n - 2)/2], {n, 49}] (* or *)
%t A057029 Rest@ CoefficientList[Series[x (1 + 5 x + 4 x^2 + 5 x^3 + x^4)/((1 - x)^3 (1 + x)^2), {x, 0, 49}], x] (* _Michael De Vlieger_, Jul 03 2016 *)
%o A057029 (PARI) Vec(x*(1+5*x+4*x^2+5*x^3+x^4)/((1-x)^3*(1+x)^2) + O(x^100)) \\ _Colin Barker_, Jul 02 2016
%o A057029 (Magma) [(5-(-1)^n+2*(-4+(-1)^n)*n+8*n^2)/4 : n in [1..80]]; // _Wesley Ivan Hurt_, Jul 03 2016
%Y A057029 Cf. A057027, A057028.
%K A057029 nonn,easy
%O A057029 1,2
%A A057029 _Clark Kimberling_, Jul 28 2000