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Let v = (1,4,9,...,n^2), x = (0,1,2,4,6,...) [first n terms of A002620]; a(n) = v.v * x.x - (v.x)^2.
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%I #14 Aug 10 2021 16:47:34

%S 0,1,6,38,107,350,728,1752,3090,6215,9878,17654,26117,42924,60256,

%T 93024,125460,184509,241110,341110,434511,595562,742808,991640,

%U 1215110,1586403,1914822,2452646,2922185,3681560,4337024,5385600,6281704,7701561,8904294,10793862,12381939,14858038,16924440,20124440,22778042,26862143,30229430,35383062,39609933,46046276,51299936,59262560,65733500,75499125

%N Let v = (1,4,9,...,n^2), x = (0,1,2,4,6,...) [first n terms of A002620]; a(n) = v.v * x.x - (v.x)^2.

%H N. J. A. Sloane, Vinay A. Vaishampayan and Sueli I. R. Costa, <a href="http://neilsloane.com/doc/Exists.pdf">Fat Struts: Constructions and a Bound</a>, Proceedings Information Theory Workshop, Taormino, Italy, 2009. [<a href="/A047896/a047896.pdf">Cached copy</a>]

%H N. J. A. Sloane, Vinay A. Vaishampayan and Sueli I. R. Costa, <a href="http://neilsloane.com/doc/FATS.pdf">A Note on Projecting the Cubic Lattice</a>, Discrete and Computational Geometry, Vol. 46 (No. 3, 2011), 472-478.

%H N. J. A. Sloane, Vinay A. Vaishampayan and Sueli I. R. Costa, <a href="http://neilsloane.com/doc/main_fat_strut.pdf">The Lifting Construction: A General Solution to the Fat Strut Problem</a>, Proceedings International Symposium on Information Theory (ISIT), 2010, IEEE Press. [<a href="/A047896/a047896_1.pdf">Cached copy</a>]

%H <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (1,6,-6,-15,15,20,-20,-15,15,6,-6,-1,1).

%F G.f. -x^2*(1+5*x+26*x^2+39*x^3+66*x^4+39*x^5+26*x^6+5*x^7+x^8) / ( (1+x)^6*(x-1)^7 ). - R. J. Mathar, Apr 04 2012

%p fv := n->1/30*n*(1+n)*(2*n+1)*(3*n^2+3*n-1); # this is A000538

%p f1 := n->1/160*(n-1)*(1+n)*(2*n^3+5*n^2+2*n-5);

%p f2 := n->1/160*n*(n+2)*(2*n^3+n^2-2*n+4);

%p f7 := n->if n mod 2 = 0 then f2(n) else f1(n) end if; # this is A059859

%p f3 := n->1/20*n^5+1/8*n^4+1/24*n^3-11/120*n-1/8*n^2;

%p f4 := n->1/20*n^5+1/8*n^4+1/24*n^3+1/30*n;

%p f5:-n-> if `mod`(n,2) = 0 then f4(n) else f3(n) end if; # this is A060453

%p A060452 := n->f7(n)*fv(n)-f5(n)^2;

%t Table[Module[{nn=n,v,x},v=Range[nn]^2;x=Floor[v/4];v.v x.x-(v.x)^2],{n,50}] (* or *) LinearRecurrence[{1,6,-6,-15,15,20,-20,-15,15,6,-6,-1,1},{0,1,6,38,107,350,728,1752,3090,6215,9878,17654,26117},50] (* _Harvey P. Dale_, Aug 10 2021 *)

%Y Cf. A002620, A000538, A059859. Agrees with A060453 for first 37 terms.

%K nonn

%O 1,3

%A _N. J. A. Sloane_ and Vinay Vaishampayan, Apr 09 2001