A211795 -id:A211795 - cp's OEIS frontend

A212067 Number of (w,x,y,z) with all terms in {1,...,n} and w^3 = xyz.

0, 1, 2, 3, 10, 11, 12, 13, 26, 45, 46, 47, 60, 61, 62, 63, 88, 89, 120, 121, 128, 129, 130, 131, 162, 199, 200, 255, 262, 263, 264, 265, 332, 333, 334, 335, 402, 403, 404, 405, 436, 437, 438, 439, 446, 477, 478, 479, 540, 601, 674, 675, 682, 683, 786

Offset: 0

Clark Kimberling, Apr 30 2012

nonn

For a guide to related sequences, see A211795.

			G.f. = x + 2*x^2 + 3*x^3 + 10*x^4 + 11*x^5 + 12*x^6 + 13*x^7 + 26*x^8 + ...
a(4) counts these ten 4-tuples:
(1,1,1,1), (2,2,2,2), (3,3,3,3), (4,4,4,4),
(2,1,2,4), (2,1,4,2), (2,2,1,4), (2,2,4,1),
(2,4,1,2), (2,4,2,1).

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Cf. A211795, A212068.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w^3 == x*y*z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 60]] (* A212067 *)
(* Peter J. C. Moses, Apr 13 2012 *)
a[ n_] := Length@FindInstance[ w^3 == x y z && 0 < w <= n && 0 < x <= n && 0 < y <= n && 0 < z <= n, {w, x, y, z}, Integers, 10^9]; (* Michael Somos, Nov 26 2016 *)

A212068 Number of (w,x,y,z) with all terms in {1,...,n} and 2w=x+y+z.

Original entry on oeis.org

0, 0, 3, 10, 25, 49, 86, 137, 206, 294, 405, 540, 703, 895, 1120, 1379, 1676, 2012, 2391, 2814, 3285, 3805, 4378, 5005, 5690, 6434, 7241, 8112, 9051, 10059, 11140, 12295, 13528, 14840, 16235, 17714, 19281, 20937, 22686, 24529, 26470, 28510, 30653, 32900

Offset: 0

Clark Kimberling, May 01 2012

For a guide to related sequences, see A211795.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Cf. A211795.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[2 w == x + y + z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 50]] (* A212068 *)
FindLinearRecurrence[%]
(* Peter J. C. Moses, Apr 13 2012 *)
LinearRecurrence[{3, -2, -2, 3, -1},{0, 0, 3, 10, 25},42] (* Ray Chandler, Aug 02 2015 *)

concat(vector(2), Vec(x^2*(3 + x + x^2) / ((1 - x)^4*(1 + x)) + O(x^40))) \\ Colin Barker, Dec 02 2017

a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5).
From Colin Barker, Dec 02 2017: (Start)
G.f.: x^2*(3 + x + x^2) / ((1 - x)^4*(1 + x)).
a(n) = n*(10*n^2 - 3*n + 2)/24 for n even.
a(n) = (n - 1)*(10*n^2 + 7*n + 9)/24 for n odd.
(End)

A212088 Number of (w,x,y,z) with all terms in {1,...,n} and w

Original entry on oeis.org

0, 0, 7, 36, 117, 292, 612, 1143, 1963, 3159, 4833, 7099, 10080, 13914, 18751, 24750, 32085, 40942, 51516, 64017, 78667, 95697, 115353, 137893, 163584, 192708, 225559, 262440, 303669, 349576, 400500, 456795, 518827, 586971, 661617

Offset: 0

Clark Kimberling, May 01 2012

Also, number of (w,x,y,z) with all terms in {1,...,n} and w>average{x,y,z}.
a(n) + A212089(n) = n^4.
For a guide to related sequences, see A211795.

Index entries for linear recurrences with constant coefficients, signature (4, -6, 5, -5, 6, -4, 1).

Cf. A211795, A212069, A212089.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[3 w < x + y + z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 50]] (* A212088 *)
FindLinearRecurrence[%]
(* Peter J. C. Moses, Apr 13 2012 *)
LinearRecurrence[{4, -6, 5, -5, 6, -4, 1},{0, 0, 7, 36, 117, 292, 612},35] (* Ray Chandler, Aug 02 2015 *)

a(n) = 4*a(n-1)-6*a(n-2)+5*a(n-3)-5*a(n-4)+6*a(n-5)-4*a(n-6)+a(n-7).

G.f.: x^2*(x^4+5*x^3+15*x^2+8*x+7) / ((x^2+x+1)*(1-x)^5). - Alois P. Heinz, May 18 2012

A212091 Number of (w,x,y,z) with all terms in {1,...,n} and w^2=x^2+y^2+z^2.

Original entry on oeis.org

0, 0, 0, 3, 3, 3, 6, 12, 12, 24, 24, 33, 36, 42, 48, 63, 63, 72, 84, 99, 99, 132, 141, 159, 162, 174, 180, 219, 225, 243, 258, 282, 282, 330, 339, 369, 381, 405, 420, 465, 465, 492, 525, 558, 567, 627, 645, 681, 684, 732, 744, 804, 810, 846, 885, 930

Offset: 0

Clark Kimberling, May 02 2012

nonn

Every term is divisible by 3. For a guide to related sequences, see A211795.

Cf. A211795.

Partial sums of A181787.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w^2 == x^2 + y^2 + z^2, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 50]] (* A212091 *)
%/3  (* integers *)
(* Peter J. C. Moses, Apr 13 2012 *)

A212092 Number of (w,x,y,z) with all terms in {1,...,n} and w^2

Original entry on oeis.org

0, 1, 15, 73, 231, 562, 1155, 2133, 3632, 5801, 8830, 12907, 18264, 25131, 33766, 44461, 57524, 73274, 92043, 114213, 140173, 170289, 205057, 244884, 290251, 341658, 399594, 464582, 537200, 618046, 707665, 806727, 915828, 1035607

Offset: 0

Clark Kimberling, May 02 2012

nonn

a(n)+A212095(n)=n^4.

For a guide to related sequences, see A211795.

Cf. A211795.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w^2 < x^2 + y^2 + z^2, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 50]] (* A212092 *)
(* Peter J. C. Moses, Apr 13 2012 *)

A212093 Number of (w,x,y,z) with all terms in {1,...,n} and w^2<=x^2+y^2+z^2.

Original entry on oeis.org

0, 1, 15, 76, 234, 565, 1161, 2145, 3644, 5825, 8854, 12940, 18300, 25173, 33814, 44524, 57587, 73346, 92127, 114312, 140272, 170421, 205198, 245043, 290413, 341832, 399774, 464801, 537425, 618289, 707923, 807009, 916110, 1035937

Offset: 0

Clark Kimberling, May 02 2012

nonn

a(n)+A212094(n)=n^4. For a guide to related sequences, see A211795.

Cf. A211795.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w^2 <= x^2 + y^2 + z^2, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 50]] (* A212093 *)
(* Peter J. C. Moses, Apr 13 2012 *)

A212094 Number of (w,x,y,z) with all terms in {1,...,n} and w^2>x^2+y^2+z^2.

Original entry on oeis.org

0, 0, 1, 5, 22, 60, 135, 256, 452, 736, 1146, 1701, 2436, 3388, 4602, 6101, 7949, 10175, 12849, 16009, 19728, 24060, 29058, 34798, 41363, 48793, 57202, 66640, 77231, 88992, 102077, 116512, 132466, 149984, 169206, 190205, 213109, 238015

Offset: 0

Clark Kimberling, May 02 2012

nonn

a(n)+A212093(n)=n^4. For a guide to related sequences, see A211795.

Cf. A211795.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w^2 > x^2 + y^2 + z^2, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 50]] (* A212094 *)
(* Peter J. C. Moses, Apr 13 2012 *)

A212095 Number of (w,x,y,z) with all terms in {1,...,n} and w^2>=x^2+y^2+z^2.

Original entry on oeis.org

0, 0, 1, 8, 25, 63, 141, 268, 464, 760, 1170, 1734, 2472, 3430, 4650, 6164, 8012, 10247, 12933, 16108, 19827, 24192, 29199, 34957, 41525, 48967, 57382, 66859, 77456, 89235, 102335, 116794, 132748, 150314, 169545, 190574, 213490, 238420

Offset: 0

Clark Kimberling, May 02 2012

nonn

a(n)+A212092(n)=n^4. For a guide to related sequences, see A211795.

Cf. A211795.

Partial sums of A253663.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w^2 >= x^2 + y^2 + z^2, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 50]] (* A212095 *)
(* Peter J. C. Moses, Apr 13 2012 *)

A212097 Number of (w,x,y,z) with all terms in {1,...,n} and w^3

Original entry on oeis.org

0, 1, 15, 72, 221, 536, 1098, 2028, 3445, 5502, 8368, 12234, 17304, 23794, 31963, 42089, 54439, 69332, 87070, 108013, 132549, 161043, 193924, 231550, 274463, 323023, 377776, 439254, 507882, 584329, 669046, 762686, 865804, 979052

Offset: 0

Clark Kimberling, May 03 2012

nonn

a(n)+A212100(n)=4^n. For a guide to related sequences, see A211795.

Cf. A211795.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w^3 < x^3 + y^3 + z^3, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]] (* A212097 *)
(* Peter J. C. Moses, Apr 13 2012 *)

A212098 Number of (w,x,y,z) with all terms in {1,...,n} and w^3<=x^3+y^3+z^3.

Original entry on oeis.org

0, 1, 15, 72, 221, 536, 1104, 2034, 3451, 5514, 8380, 12246, 17322, 23812, 31981, 42107, 54457, 69350, 87100, 108049, 132591, 161085, 193966, 231592, 274511, 323077, 377830, 439314, 507948, 584401, 669124, 762764, 865882, 979130

Offset: 0

Clark Kimberling, May 03 2012

nonn

For a guide to related sequences, see A211795.

Giovanni Resta, Table of n, a(n) for n = 0..1000

Cf. A211795, A212099.

f:= proc(n) local x,y,z, r, t;
   r:= 0:
   for x from 1 to n do
     for y from x to n do
       for z from y to n do
          t:= min(n, floor((x^3 + y^3 + z^3)^(1/3)));
          if x = z then r:= r+t
          elif x=y or y=z then r:= r+3*t
          else r:= r+6*t
          fi
   od od od;
   r
end proc:
map(f, [$0..40]); # Robert Israel, May 08 2017

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w^3 <= x^3 + y^3 + z^3, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]] (* A212098 *)
(* Peter J. C. Moses, Apr 13 2012 *)

A212098(n)={my(s=0,c=[6,3,1]);forvec(v=vector(4,i,if(i>1,[1,n],[-n,-1])),sum(i=1,4,v[i]^3)>=0&s+=c[1+(v[2]==v[3])+(v[3]==v[4])],1);s} /* not very efficient */ \\ M. F. Hasler, May 20 2012

a(n) + A212099(n) = n^4.

cp's OEIS Frontend

A212067 Number of (w,x,y,z) with all terms in {1,...,n} and w^3 = x*y*z.

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A212068 Number of (w,x,y,z) with all terms in {1,...,n} and 2w=x+y+z.

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A212088 Number of (w,x,y,z) with all terms in {1,...,n} and w

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A212091 Number of (w,x,y,z) with all terms in {1,...,n} and w^2=x^2+y^2+z^2.

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A212092 Number of (w,x,y,z) with all terms in {1,...,n} and w^2

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A212093 Number of (w,x,y,z) with all terms in {1,...,n} and w^2<=x^2+y^2+z^2.

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A212094 Number of (w,x,y,z) with all terms in {1,...,n} and w^2>x^2+y^2+z^2.

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A212095 Number of (w,x,y,z) with all terms in {1,...,n} and w^2>=x^2+y^2+z^2.

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A212097 Number of (w,x,y,z) with all terms in {1,...,n} and w^3

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A212067 Number of (w,x,y,z) with all terms in {1,...,n} and w^3 = xyz.