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%I #48 Jun 14 2023 17:18:51
%S 0,6,26,68,140,250,406,616,888,1230,1650,2156,2756,3458,4270,5200,
%T 6256,7446,8778,10260,11900,13706,15686,17848,20200,22750,25506,28476,
%U 31668,35090,38750,42656,46816,51238,55930,60900,66156,71706,77558,83720
%N Sums of the products of n consecutive pairs of numbers.
%C Number of integer solutions to 1-n <= x <= y <= z <= n-1 where x - 2*y + z is nonzero. - _Michael Somos_, Dec 27 2011
%C This sequence is related to A001105 by the transform a(n) = (n-1)*A001105(n)-Sum_{i=0..n-1} A001105(i). - _Bruno Berselli_, Mar 12 2012
%C a(n) is the maximum value obtainable by partitioning the set {x in the natural numbers | 0 <= x <= 2n+1} into pairs, taking the product of all such pairs, and taking the sum of all such products. - _Thomas Anton_, Oct 20 2020
%H Vincenzo Librandi, <a href="/A135036/b135036.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = 0*1 + 2*3 + 4*5 + ... + 2*n*(2*n + 1).
%F a(n) = (4*n^3 - 3*n^2 - n)/3 = (n - 1)*n*(4*n + 1)/3.
%F From _R. J. Mathar_, Feb 14 2008: (Start)
%F O.g.f.: 2*x^2*(3 + x)/(1 - x)^4.
%F a(n) = 2*A016061(n-1). (End)
%F a(0)=0, a(1)=6, a(2)=26, a(3)=68, a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - _Harvey P. Dale_, Jun 20 2013
%F From _Amiram Eldar_, Apr 30 2023: (Start)
%F Sum_{n>=2} 1/a(n) = 6*Pi/5 + 36*log(2)/5 - 213/25.
%F Sum_{n>=2} (-1)^n/a(n) = 6*sqrt(2)*Pi/5 + 6*(sqrt(2)+3)*log(2)/5 - 12*sqrt(2)*log(2-sqrt(2))/5 - 267/25. (End)
%F E.g.f.: exp(x)*x^2*(9 + 4*x)/3. - _Stefano Spezia_, Jun 14 2023
%e For n = 3, the sum of the first 3 pairs is 0*1+2*3+4*5 = 26, the 3rd entry in the sequence.
%e G.f.: 6*x^2 + 26*x^3 + 68*x^4 + 140*x^5 + 250*x^6 + 406*x^7 + 616*x^8 + 888*x^9 + ...
%t Accumulate[Times@@@Partition[Range[0,81],2]] (* or *) LinearRecurrence[ {4,-6,4,-1},{0,6,26,68},40] (* _Harvey P. Dale_, Jun 20 2013 *)
%t a[ n_] := n (n - 1) (4 n + 1)/3; (* _Michael Somos_, Oct 15 2015 *)
%t a[ n_] := If[ n >= 0, Length @ FindInstance[ 1 - n <= x <= y <= z <= n - 1 && x - 2 y + z != 0, {x, y, z}, Integers, 10^9], -(Length @ FindInstance[ n <= x < y <= z <= -n && x - 2 y + z != 0, {x, y, z}, Integers, 10^9] + n)]; (* _Michael Somos_, Oct 15 2015 *)
%o (PARI) sumprod(n) = { local(x,s=0); forstep(x=0,n,2, s+=x*(x+1); print1(s",") ) }
%o (PARI) {a(n) = n * (n - 1) * (4*n + 1) / 3}; /* _Michael Somos_, Dec 27 2011 */
%o (Magma) [(n-1)*n*(4*n+1)/3: n in [1..40]]; // _Bruno Berselli_, Mar 12 2012
%Y Cf. A016061, A001105.
%K nonn,easy
%O 1,2
%A _Cino Hilliard_, Feb 10 2008
%E First formula corrected by _Harvey P. Dale_, Jun 20 2013
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