Displaying 31-40 of 202 results found.
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Number of (w,x,y,z) with all terms in {1,...,n} and w*x>y*z+1.
+10
3
0, 0, 3, 25, 98, 260, 571, 1089, 1898, 3084, 4755, 7017, 9994, 13836, 18691, 24705, 32066, 40964, 51579, 64145, 78850, 95956, 115723, 138385, 164170, 193436, 226467, 263521, 304930, 351076, 402195, 458777, 521082, 589532, 664547
MATHEMATICA
t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w*x > y*z + 1, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 50]] (* A212054 *)
Number of (w,x,y,z) with all terms in {1,...,n} and w*x>y*z+2.
+10
3
0, 0, 1, 19, 80, 230, 521, 1019, 1800, 2966, 4593, 6819, 9768, 13566, 18353, 24339, 31640, 40478, 51025, 63523, 78168, 95230, 114881, 137459, 163184, 192374, 225265, 262251, 303568, 349606, 400633, 457099, 519280, 587654, 662481
MATHEMATICA
t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w*x > y*z + 2, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 50]] (* A212056 *)
Number of (w,x,y,z) with all terms in {1,...,n} and w<x*y*z.
+10
3
0, 0, 11, 69, 231, 584, 1230, 2307, 3964, 6385, 9771, 14356, 20377, 28125, 37894, 50008, 64809, 82681, 104005, 129216, 158743, 193063, 232668, 278080, 329812, 388452, 454585, 528822, 611791, 704167, 806610, 919852, 1044607, 1181643
MATHEMATICA
t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w < x*y*z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 50]] (* A212057 *)
Number of (w,x,y,z) with all terms in {1,...,n} and w>=x*y*z.
+10
3
0, 1, 5, 12, 25, 41, 66, 94, 132, 176, 229, 285, 359, 436, 522, 617, 727, 840, 971, 1105, 1257, 1418, 1588, 1761, 1964, 2173, 2391, 2619, 2865, 3114, 3390, 3669, 3969, 4278, 4596, 4923, 5286, 5652, 6027, 6411, 6825, 7242, 7686, 8133, 8598
FORMULA
a(n) = Sum_{i=1..n+1} Sum_{j=1..n+1} tau(i)*floor((n+1-j)/i). - Ridouane Oudra, Oct 03 2020
MATHEMATICA
t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w >= x*y*z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 50]] (* A212058 *)
Number of (w,x,y,z) with all terms in {1,...,n} and w^2<x*y*z.
+10
3
0, 0, 8, 50, 169, 440, 943, 1796, 3118, 5090, 7877, 11683, 16708, 23253, 31552, 41892, 54589, 70030, 88524, 110484, 136289, 166434, 201327, 241465, 287278, 339444, 398407, 464742, 539068, 622021, 714192, 816319, 929007, 1053100
MATHEMATICA
t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w^2 < x*y*z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 50]] (* A212063 *)
Number of (w,x,y,z) with all terms in {1,...,n} and w^2>=x*y*z.
+10
3
0, 1, 8, 31, 87, 185, 353, 605, 978, 1471, 2123, 2958, 4028, 5308, 6864, 8733, 10947, 13491, 16452, 19837, 23711, 28047, 32929, 38376, 44498, 51181, 58569, 66699, 75588, 85260, 95808, 107202, 119569, 132821, 147102, 162427, 178898
MATHEMATICA
t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w^2 >= x*y*z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 50]] (* A212064 *)
Number of (w,x,y,z) with all terms in {1,...,n} and w^2<=x*y*z.
+10
3
0, 1, 12, 57, 185, 459, 974, 1830, 3168, 5158, 7957, 11766, 16830, 23378, 31689, 42041, 54774, 70218, 88757, 110720, 136558, 166715, 201620, 241761, 287646, 339854, 398829, 465198, 539557, 622513, 714762, 816892, 929637, 1053742
MATHEMATICA
t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w^2 <= x*y*z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 50]] (* A212065 *)
Number of (w,x,y,z) with all terms in {1,...,n} and w^2>x*y*z.
+10
3
0, 0, 4, 24, 71, 166, 322, 571, 928, 1403, 2043, 2875, 3906, 5183, 6727, 8584, 10762, 13303, 16219, 19601, 23442, 27766, 32636, 38080, 44130, 50771, 58147, 66243, 75099, 84768, 95238, 106629, 118939, 132179, 146448, 161761, 178106
MATHEMATICA
t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w^2 > x*y*z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 50]] (* A212066 *)
Number of (w,x,y,z) with all terms in {1,...,n} and w^3 = x*y*z.
+10
3
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
COMMENTS
For a guide to related sequences, see A211795.
EXAMPLE
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).
MATHEMATICA
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 *) 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 *)
Number of (w,x,y,z) with all terms in {1,...,n} and 2w=x+y+z.
+10
3
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
COMMENTS
For a guide to related sequences, see A211795.
FORMULA
a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5).
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)
MATHEMATICA
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[%]
LinearRecurrence[{3, -2, -2, 3, -1}, {0, 0, 3, 10, 25}, 42] (* Ray Chandler, Aug 02 2015 *)
PROG
(PARI) concat(vector(2), Vec(x^2*(3 + x + x^2) / ((1 - x)^4*(1 + x)) + O(x^40))) \\ Colin Barker, Dec 02 2017
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