A047896 -id:A047896 - 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

A060453 Dot product of the squares and the quarter-squares: a(n) = sum(i=1..n, i^2 * floor(i^2/4)).

Original entry on oeis.org

0, 4, 22, 86, 236, 560, 1148, 2172, 3792, 6292, 9922, 15106, 22204, 31808, 44408, 60792, 81600, 107844, 140334, 180334, 228844, 287408, 357236, 440180, 537680, 651924, 784602, 938266, 1114876, 1317376, 1548016, 1810160, 2106368, 2440452

Offset: 1

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

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 (3,0,-8,6,6,-8,0,3,-1).

Cf. A002620.

Maple
```
See A060452.
```

A060453[n_]:=Sum[i^2*Floor[i^2/4],{i,1,n}]; Array[A060453,100] (* Enrique Pérez Herrero, Mar 19 2012 *)

a(n)=sum(i=1,n,i^2\4*i^2) \\ Charles R Greathouse IV, Mar 20 2012

G.f.: 2*x^2*(2+5*x+10*x^2+5*x^3+2*x^4) / ( (1+x)^3*(x-1)^6 ). - R. J. Mathar, Apr 04 2012

A124255 Forest-and-trees problem: square of distance to most distant visible tree.

Original entry on oeis.org

2, 5, 13, 17, 34, 41, 61, 74, 97, 113, 137, 157, 194, 221, 250, 281, 317, 353, 397, 433, 482, 521, 569, 617, 674, 725, 778, 829, 898, 953, 1021, 1082, 1154, 1217, 1289, 1361, 1433, 1517, 1597, 1669, 1762, 1825, 1933, 2018, 2113, 2197, 2297, 2393, 2498, 2594

Offset: 2

Jon E. Schoenfield, Oct 22 2006

nonn

In an arbitrarily large pine plantation, a tree with a trunk of radius 1/n is located at each of the lattice points of a square lattice (whose rows are spaced one unit apart), except for one empty lattice point near the center of the plantation. For an observer located at the empty lattice point, how far away is the most distant visible tree trunk? The sequence a(n) is defined as the square of the distance from the observer to the most distant lattice point at which a visible tree trunk is located. (Each tree trunk is assumed to be a vertical cylinder, centered at its respective lattice point. A tree trunk is considered "visible" unless it is completely obscured from view by one or more other tree trunks.)

It is known that, for any coprime x and y, the closest point to the line from (0,0) to (x,y) is 1/sqrt(x^2 + y^2) units away from it (see e.g. the first linked paper in A047896). Since tree trunks intersect lines that are closer than 1/n units, we must have that a(n) < n^2. In addition, a(n) cannot be divisible by the square of any prime p not congruent to 1 modulo 4, since this forces x and y to have common factor p. Combining this with the criteria for a(n) to be a sum of two squares, we have that a(n) is the largest number < n^2 that is either a product of primes congruent to 1 modulo 4 or twice such a product. - Charlie Neder, Jan 15 2019

			Example: at n = 5, there are 40 visible tree trunks; defining the origin as the location of the observer, they are the ones located at (1,0), (4,1), (3,1), (2,1), (3,2), (1,1) and all the additional locations that result from using every possible reflection of them across the x-axis, the y-axis, or the diagonal, y=x. (The tree trunk at (4,3) is considered completely obscured by ones at (3,2) and (1,1), each of which is tangent to the line 4y = 3x.)
The most distant visible tree trunks are the ones located at the lattice point (4,1) and its symmetrical locations; the square of their distance from the origin is 17, so a(5) = 17.

Jon E. Schoenfield, Table of n, a(n) for n = 2..3000
A different but related problem is addressed at Forests.

Cf. A124254, A124256.

A059774 Consider the line segment in R^n from the origin to the point P=(1,2,3,...,n); let d = squared distance to this line from the closest point of Z^n (excluding the endpoints). Sequence gives d times P.P.

Original entry on oeis.org

1, 3, 9, 21, 40, 75, 120, 189, 285, 385, 506, 650, 819, 1015, 1240, 1496, 1785, 2109, 2470, 2870, 3311, 3795, 4324, 4900, 5525, 6201, 6930, 7714, 8555, 9455, 10416, 11440, 12529, 13685, 14910, 16206, 17575, 19019, 20540, 22140, 23821, 25585, 27434, 29370

Offset: 2

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

P.P is given by A000330(n). For n >= 10, a(n) = A000330(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. A000330, A059804, A047896.

A060454 Consider the line segment in R^n from the origin to the point v = (1,4,9,...,n^2); let d = squared distance to this line from the closest point of Z^n (excluding the endpoints). Sequence gives d times v.v.

Original entry on oeis.org

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, 14822755, 16907891, 19221332, 21781332, 24607093, 27718789, 31137590

Offset: 0

N. J. A. Sloane, Apr 09 2001

nonn

v.v is given by A000538(n).

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. A059804.

For n<=37, a(n) = A060452(n); for n >= 38, a(n) = A000538(n-1).

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.

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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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A060453 Dot product of the squares and the quarter-squares: a(n) = sum(i=1..n, i^2 * floor(i^2/4)).

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Maple

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A124255 Forest-and-trees problem: square of distance to most distant visible tree.

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A059774 Consider the line segment in R^n from the origin to the point P=(1,2,3,...,n); let d = squared distance to this line from the closest point of Z^n (excluding the endpoints). Sequence gives d times P.P.

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Author

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Crossrefs

A060454 Consider the line segment in R^n from the origin to the point v = (1,4,9,...,n^2); 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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Author

Keywords

Comments

Links

Crossrefs

Formula