A060453 -id:A060453 - cp's OEIS frontend

A059804 Consider the line segment in R^n from the origin to the point v=(2,3,5,7,11,...) with prime coordinates; let d = squared distance to this line from the closest point of Z^n (excluding the endpoints). Sequence gives d times v.v.

1, 3, 9, 39, 87, 215, 391, 711, 1326, 1975, 2925, 4256, 5696, 7537, 9774, 12488, 16322, 20477, 24966, 30007, 35336, 41577, 48466, 56387, 65796, 75997, 86606, 98055, 109936, 122705, 138834, 155995, 174764, 194085, 216286, 239087, 263736, 290305

Offset: 2

N. J. A. Sloane and Vinay Vaishampayan, Feb 21 2001

v.v is given by A024450(n). For n >= 19, a(n) = A024450(n-1).

Officially these are just conjectures so far.

N. J. A. Sloane, Vinay A. Vaishampayan and Sueli I. R. Costa, Fat Struts: Constructions and a Bound, Proceedings Information Theory Workshop, Taormino, Italy, 2009. [Cached copy]
N. J. A. Sloane, Vinay A. Vaishampayan and Sueli I. R. Costa, A Note on Projecting the Cubic Lattice, Discrete and Computational Geometry, Vol. 46 (No. 3, 2011), 472-478.
N. J. A. Sloane, Vinay A. Vaishampayan and Sueli I. R. Costa, The Lifting Construction: A General Solution to the Fat Strut Problem, Proceedings International Symposium on Information Theory (ISIT), 2010, IEEE Press. [Cached copy]

Cf. A059774, A024450, A047896, A060453.

Cf. A137609 (where the minimum distance occurs along the line segment).

A060452 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.

Original entry on oeis.org

0, 1, 6, 38, 107, 350, 728, 1752, 3090, 6215, 9878, 17654, 26117, 42924, 60256, 93024, 125460, 184509, 241110, 341110, 434511, 595562, 742808, 991640, 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

Offset: 1

N. J. A. Sloane and Vinay Vaishampayan, Apr 09 2001

nonn

N. J. A. Sloane, Vinay A. Vaishampayan and Sueli I. R. Costa, Fat Struts: Constructions and a Bound, Proceedings Information Theory Workshop, Taormino, Italy, 2009. [Cached copy]
N. J. A. Sloane, Vinay A. Vaishampayan and Sueli I. R. Costa, A Note on Projecting the Cubic Lattice, Discrete and Computational Geometry, Vol. 46 (No. 3, 2011), 472-478.
N. J. A. Sloane, Vinay A. Vaishampayan and Sueli I. R. Costa, The Lifting Construction: A General Solution to the Fat Strut Problem, Proceedings International Symposium on Information Theory (ISIT), 2010, IEEE Press. [Cached copy]
Index entries for linear recurrences with constant coefficients, signature (1,6,-6,-15,15,20,-20,-15,15,6,-6,-1,1).

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

fv := n->1/30*n*(1+n)*(2*n+1)*(3*n^2+3*n-1); # this is A000538
f1 := n->1/160*(n-1)*(1+n)*(2*n^3+5*n^2+2*n-5);
f2 := n->1/160*n*(n+2)*(2*n^3+n^2-2*n+4);
f7 := n->if n mod 2 = 0 then f2(n) else f1(n) end if; # this is A059859
f3 := n->1/20*n^5+1/8*n^4+1/24*n^3-11/120*n-1/8*n^2;
f4 := n->1/20*n^5+1/8*n^4+1/24*n^3+1/30*n;
f5:-n-> if `mod`(n,2) = 0 then f4(n) else f3(n) end if; # this is A060453
A060452 := n->f7(n)*fv(n)-f5(n)^2;

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 *)

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

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A059804 Consider the line segment in R^n from the origin to the point v=(2,3,5,7,11,...) with prime coordinates; let d = squared distance to this line from the closest point of Z^n (excluding the endpoints). Sequence gives d times v.v.

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A060452 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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