Displaying 51-60 of 202 results found.
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Number of (w,x,y,z) with all terms in {1,...,n} and w <= harmonic mean of {x,y,z}.
+10
3
0, 1, 9, 36, 106, 252, 528, 964, 1617, 2559, 3880, 5631, 7950, 10900, 14595, 19161, 24727, 31419, 39399, 48790, 59799, 72570, 87277, 104124, 123342, 145075, 169575, 197061, 227779, 261915, 299778, 341599, 387624, 438171, 493486
COMMENTS
A 4-tuple (w,x,y,z) is counted if 3/w<=1/x+1/y+1/z.
For a guide to related sequences, see A211795.
MATHEMATICA
t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w*(y*z + z*x + x*y) <= 3 x*y*z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 50]] (* A212104 *) FindLinearRecurrence[%]
Number of (w,x,y,z) with all terms in {1,...,n} and w > harmonic mean of {x,y,z}.
+10
3
0, 0, 7, 45, 150, 373, 768, 1437, 2479, 4002, 6120, 9010, 12786, 17661, 23821, 31464, 40809, 52102, 65577, 81531, 100201, 121911, 146979, 175717, 208434, 245550, 287401, 334380, 386877, 445366, 510222, 581922, 660952, 747750, 842850
COMMENTS
A 4-tuple (w,x,y,z) is counted if 3/w>1/x+1/y+1/z.
For a guide to related sequences, see A211795.
MATHEMATICA
t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w*(y*z + z*x + x*y) > 3 x*y*z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]] (* A212105 *)
Number of (w,x,y,z) with all terms in {1,...,n} and w < harmonic mean of {x,y,z}.
+10
3
0, 0, 7, 33, 96, 241, 498, 933, 1579, 2520, 3828, 5578, 7866, 10815, 14509, 19044, 24603, 31294, 39255, 48645, 59599, 72345, 87051, 103897, 123060, 144792, 169291, 196776, 227445, 261580, 299358, 341178, 387196, 437736, 493050
COMMENTS
A 4-tuple (w,x,y,z) is counted if 3/w<1/x+1/y+1/z.
For a guide to related sequences, see A211795.
MATHEMATICA
t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w*(y*z + z*x + x*y) < 3 x*y*z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]] (* A212106 *)
Number of (w,x,y,z) with all terms in {1,...,n} and w >= harmonic mean of {x,y,z}.
+10
3
0, 1, 9, 48, 160, 384, 798, 1468, 2517, 4041, 6172, 9063, 12870, 17746, 23907, 31581, 40933, 52227, 65721, 81676, 100401, 122136, 147205, 175944, 208716, 245833, 287685, 334665, 387211, 445701, 510642, 582343, 661380, 748185, 843286
COMMENTS
A 4-tuple (w,x,y,z) is counted if 3/w>=1/x+1/y+1/z.
For a guide to related sequences, see A211795.
MATHEMATICA
t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w*(y*z + z*x + x*y) >= 3 x*y*z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]] (* A212107 *)
Number of (w,x,y,z) with all terms in {1,...,n} and median<=mean.
+10
3
0, 1, 12, 57, 172, 405, 816, 1477, 2472, 3897, 5860, 8481, 11892, 16237, 21672, 28365, 36496, 46257, 57852, 71497, 87420, 105861, 127072, 151317, 178872, 210025, 245076, 284337, 328132, 376797, 430680, 490141, 555552, 627297, 705772, 791385, 884556, 985717
COMMENTS
Also, the number of (w,x,y,z) with all terms in {1,...,n} and median>=mean.
For a guide to related sequences, see A211795.
FORMULA
a(n) = n*(n^3 + 2*n^2 - 3*n + 2)/2.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: x*(1 + 7*x + 7*x^2 - 3*x^3) /(1 - x)^5. - Colin Barker, Dec 02 2017
MATHEMATICA
t = Compile[{{n, _Integer}}, Module[{s = 0}, (Do[If[Apply[Plus, Rest[Most[Sort[{w, x, y, z}]]]]/2 <= (w + x + y + z)/4, s = s + 1], {w, 1, #}, {x, 1, #}, {y, 1, #},
{z, 1, #}] &[n]; s)]];
Flatten[Map[{t[#]} &, Range[0, 50]]] (* A212134 *)
PROG
(PARI) concat(0, Vec(x*(1 + 7*x + 7*x^2 - 3*x^3) /(1 - x)^5 + O(x^40))) \\ Colin Barker, Dec 02 2017
Number of (w,x,y,z) with all terms in {1,...,n} and median<mean.
+10
3
0, 0, 4, 24, 84, 220, 480, 924, 1624, 2664, 4140, 6160, 8844, 12324, 16744, 22260, 29040, 37264, 47124, 58824, 72580, 88620, 107184, 128524, 152904, 180600, 211900, 247104, 286524, 330484, 379320, 433380, 493024, 558624, 630564, 709240, 795060, 888444
COMMENTS
Also, the number of (w,x,y,z) with all terms in {1,...,n} and median>mean.
For a guide to related sequences, see A211795.
FORMULA
a(n) = n*(n - 1)*(n^2 - n + 2)/2.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: 4*x*(1 + x + x^2) / (1 - x)^5. - Colin Barker, Dec 02 2017
MATHEMATICA
t = Compile[{{n, _Integer}}, Module[{s = 0}, (Do[If[
Apply[Plus, Rest[Most[Sort[{w, x, y, z}]]]]/2 > (w + x + y + z)/4, s = s + 1], {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Flatten[Map[{t[#]} &, Range[0, 20]]] (* A212135 *)
PROG
(PARI) concat(vector(2), Vec(4*x*(1 + x + x^2) / (1 - x)^5 + O(x^40))) \\ Colin Barker, Dec 02 2017
Number of (w,x,y,z) with all terms in {1,...,n} and 3w=x+y+z+n.
+10
3
0, 1, 4, 13, 29, 56, 95, 150, 222, 315, 430, 571, 739, 938, 1169, 1436, 1740, 2085, 2472, 2905, 3385, 3916, 4499, 5138, 5834, 6591, 7410, 8295, 9247, 10270, 11365, 12536, 13784, 15113, 16524, 18021, 19605, 21280, 23047, 24910, 26870, 28931
COMMENTS
For a guide to related sequences, see A211795.
FORMULA
a(n) = 3*a(n-1)-3*a(n-2)+2*a(n-3)-3*a(n-4)+3*a(n-5)-a(n-6).
a(n) = (2*n*(10*n^2+3*n+2)-9(-1)^n+9)/48. [ Bruno Berselli, May 30 2012]
MATHEMATICA
t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[2 w == x + y + z - n, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 60]] (* A212246 *) FindLinearRecurrence[%]
LinearRecurrence[{3, -2, -2, 3, -1}, {0, 1, 4, 13, 29}, 42] (* Ray Chandler, Aug 02 2015 *) CoefficientList[Series[x (1+x+3x^2)/((1+x)(1-x)^4), {x, 0, 50}], x] (* Harvey P. Dale, Jul 06 2021 *)
Number of (w,x,y,z) with all terms in {1,...,n} and 3w<x+y+z+n.
+10
3
0, 1, 12, 63, 202, 496, 1034, 1923, 3289, 5280, 8062, 11820, 16761, 23110, 31111, 41030, 53151, 67777, 85233, 105862, 130026, 158109, 190513, 227659, 269990, 317967, 372070, 432801, 500680, 576246, 660060, 752701, 854767, 966878
COMMENTS
For a guide to related sequences, see A211795.
FORMULA
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*(1+8*x+21*x^2+17*x^3+11*x^4+x^5)/((1+x+x^2)*(1-x)^5). [ Bruno Berselli, Jun 05 2012] a(n) = (59*n^4 -10*n^3 +5*n^2 -6*n -8*((((n+1) mod 3) +(-1)^((n+1) mod 3))*(-1)^(n mod 3)))/72. [ Bruno Berselli, Jun 05 2012]
MATHEMATICA
t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[3 w < x + y + z + n, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]] (* A212249 *)
Number of (w,x,y,z) with all terms in {1,...,n} and 3w>=x+y+z+n.
+10
3
0, 0, 4, 18, 54, 129, 262, 478, 807, 1281, 1938, 2821, 3975, 5451, 7305, 9595, 12385, 15744, 19743, 24459, 29974, 36372, 43743, 52182, 61786, 72658, 84906, 98640, 113976, 131035, 149940, 170820, 193809, 219043, 246664, 276819, 309657
COMMENTS
For a guide to related sequences, see A211795.
FORMULA
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*(4+2*x+6*x^2+x^3)/((1+x+x^2)*(1-x)^5). [ Bruno Berselli, Jun 05 2012] a(n) = (13*n^4+10*n^3-5*n^2+6*n+8*b)/72, where b = 0,-3,1,0,-3,1,... (repeated). [ Bruno Berselli, Jun 05 2012]
MATHEMATICA
t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[3 w >= x + y + z + n, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]] (* A212250 *)
Number of (w,x,y,z) with all terms in {1,...,n} and 3w = x + y + z + n + 1.
+10
3
0, 0, 1, 6, 15, 30, 54, 87, 131, 189, 261, 349, 456, 582, 729, 900, 1095, 1316, 1566, 1845, 2155, 2499, 2877, 3291, 3744, 4236, 4769, 5346, 5967, 6634, 7350, 8115, 8931, 9801, 10725, 11705, 12744, 13842, 15001, 16224, 17511, 18864, 20286, 21777, 23339, 24975
COMMENTS
Also, the number of (w,x,y,z) with all terms in {1,...,n} and 3w = x + y + z - n - 1.
For a guide to related sequences, see A211795.
FORMULA
a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3) - 3*a(n-4) + 3*a(n-5) - a(n-6).
G.f.: x^2*(1 + 3*x + x^3) / ((1 - x)^4*(1 + x + x^2)). - Colin Barker, Dec 02 2017
MATHEMATICA
t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[3 w == x + y + z + n + 1, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]] (* A212251 *)
PROG
(PARI) concat(vector(2), Vec(x^2*(1 + 3*x + x^3) / ((1 - x)^4*(1 + x + x^2)) + O(x^40))) \\ Colin Barker, Dec 02 2017
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