A227012 -id:A227012 - cp's OEIS frontend

A227017 Floor(M(g(n-1)+1,..,g(n))), where M = harmonic mean and g(n) = n(3n-1)/2 = A000326(n).

1, 3, 8, 17, 28, 43, 60, 81, 104, 131, 160, 193, 228, 267, 308, 353, 400, 451, 504, 561, 620, 683, 748, 817, 888, 963, 1040, 1121, 1204, 1291, 1380, 1473, 1568, 1667, 1768, 1873, 1980, 2091, 2204, 2321, 2440, 2563, 2688, 2817, 2948, 3083, 3220, 3361, 3504

Offset: 1

Clark Kimberling, Jul 01 2013

Also a(n) = floor(G(g(n-1)+1,g(n))), where G = geometric mean. See A227012.

			a(1) = floor(1/(1/1)); a(2) = floor(4/(1/2 + 1/3 + 1/4 + 1/5)) = 3.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A227012.

z = 100; f[x_] := f[x] = 1/x; g[n_] := g[n] = n (3 n - 1)/2; s[n_] := s[n] = Sum[f[k], {k, g[n - 1] + 1, g[n]}]; v[n_] := v[n] = (g[n] - g[n - 1])/s[n]; Table[g[n], {n, 1, z}]; Table[Floor[v[n]], {n, 1, z}]

a(n) = (1/4)*(1 - (-1)^n + 4*n + 6*n^2) (conjectured).
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n > 1 (conjectured).
G.f.: (-1 - x - 2*x^2 - 3*x^3 + x^4)/((-1 + x)^3 (1 + x)). (conjectured)

A227013 a(n) = floor(M(g(n-1)+1,..,g(n))), where M is the harmonic mean and g(n) = n^4.

Original entry on oeis.org

1, 6, 40, 152, 413, 920, 1792, 3173, 5232, 8160, 12173, 17512, 24440, 33245, 44240, 57760, 74165, 93840, 117192, 144653, 176680, 213752, 256373, 305072, 360400, 422933, 493272, 572040, 659885, 757480, 865520, 984725, 1115840, 1259632, 1416893, 1588440, 1775112

Offset: 1

Clark Kimberling, Jul 01 2013

See A227012.

			a(1) = floor(1/(1/1)) = 1.
a(2) = floor(15/(1/2 + 1/3 + ... + 1/16)) = 6.

Cf. A227012.

z = 30; f[x_] := f[x] = 1/x; g[n_] := g[n] = n^4; s[n_] := s[n] = Sum[f[k], {k, g[n - 1] + 1, g[n]}]; v[n_] := v[n] = (g[n] - g[n - 1])/s[n]; Table[Floor[v[n]], {n, 1, z}]

a(n) = 47/9 + 14*n + (41*n^2)/3 + 6*n^3 + n^4 - (2/9)Cos(2*n*pi/3) (conjectured).
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) for n > 2 (conjectured).
G.f.: (-1 - 2*x - 22*x^2 - 23*x^3 - 20*x^4 - 4*x^5 + 2*x^6 - 3*x^7 + x^8)/((-1 + x)^5 (1 + x + x^2)) (conjectured).

Extended by Ray Chandler, Jul 15 2015

A227016 Floor(M(g(n-1)+1,..,g(n))), where M = harmonic mean and g(n) = n(n + 1)(n + 2)/6.

Original entry on oeis.org

1, 2, 7, 14, 27, 45, 69, 101, 141, 191, 252, 323, 408, 506, 618, 746, 890, 1052, 1233, 1432, 1653, 1895, 2159, 2447, 2759, 3097, 3462, 3853, 4274, 4724, 5204, 5716, 6260, 6838, 7451, 8098, 8783, 9505, 10265, 11065, 11905, 12787, 13712, 14679, 15692, 16750

Offset: 1

Clark Kimberling, Jul 01 2013

See A227012.

			a(1) = floor(1/(1/1)) = 1; a(2) = floor(3/(1/2 + 1/3 + 1/4)) = 2; a(3) = floor(6/(1/5 + 1/6 + ... + 1/10)) = 7.

Clark Kimberling, Table of n, a(n) for n = 1..200

Cf. A227012, A227013.

z = 200; f[x_] := f[x] = 1/x; g[n_] := g[n] = n (n + 1) (n + 2)/6; s[n_] := s[n] = Sum[f[k], {k, g[n - 1] + 1, g[n]}]; v[n_] := v[n] = (g[n] - g[n - 1])/s[n]; Table[g[n], {n, 1, z}];
Table[Floor[v[n]], {n, 1, z}]

a(n) + 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-8) - 3*a(n-9) + 3*a(n-10) - a(n-11) for n > 1 (conjectured).

G.f.: (1 - x + 4*x^2 - 2*x^3 + 4*x^4 - x^5 + x^6 + 2*x^7 - x^8 + 3*x^9 - 3*x^10 + x^11)/((x - 1)^4 (1+x) (1+x^2) (1+x^4)) (conjectured). (G.f. found by Peter J. C. Moses, Jul 01 2013)

A227014 a(n) = floor(M(g(n-1)+1,..,g(n))), where M = harmonic mean and g(n) = n^5.

Original entry on oeis.org

1, 10, 104, 543, 1883, 5102, 11717, 23906, 44626, 77735, 128110, 201769, 305989, 449428, 642243, 896212, 1224852, 1643541, 2169636, 2822595, 3624095, 4598154, 5771249, 7172438, 8833478, 10788947, 13076362, 15736301, 18812521, 22352080

Offset: 1

Clark Kimberling, Jul 01 2013

See A227012. It is conjectured that A227014 is a linear recurrence sequence with signature (5,-10,10,-5,1,...Z...,1,-5,-10,-10,-1,0,0), where ...Z... represents a string of 138 zeros; has been confirmed for a(1), a(2),..., a(150000).

			a(1) = floor(1/(1/1)) = 1.
a(2) = floor(31/(1/2 + 1/3 + ... + 1/32)) = 10.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A227012, A227013.

Clear[g]; g[n_] := N[n^5, 100]; a = {1}; Do[AppendTo[a, Floor[(#2 - #1 + 1)/(HarmonicNumber[#2]-HarmonicNumber[#1 - 1])] &[g[k - 1] + 1, g[k]]], {k, 2, 200}]; a (* Peter J. C. Moses, Jul 05 2012 *)
(* confirm generating function *)
p = {1, -4, 5, 9, 54, 117, 117, 122, 118, 122, 118, 122, 118, 122,
   118, 122, 118, 122, 118, 122, 118, 122, 118, 122, 118, 122, 118,
   122, 118, 122, 118, 122, 118, 122, 118, 122, 118, 122, 118, 122,
   118, 122, 118, 122, 118, 122, 118, 122, 118, 122, 118, 122, 118,
   122, 118, 122, 118, 122, 118, 122, 118, 122, 118, 122, 118, 122,
   119, 117, 129, 107, 134, 106, 134, 106, 134, 106, 134, 106, 134,
   106, 134, 106, 134, 107, 129, 117, 119, 122, 118, 122, 118, 122,
   118, 122, 118, 122, 118, 122, 118, 122, 118, 122, 118, 122, 118,
   122, 118, 122, 118, 122, 118, 122, 118, 122, 118, 122, 118, 122,
   118, 122, 118, 122, 118, 122, 118, 122, 118, 122, 118, 122, 118,
   122, 118, 122, 118, 122, 118, 122, 118, 122, 118, 122, 118, 122,
   117, 126, 113, 113, 64, 5, 1};
q = {0, 0, 1, -5, 10, -10, 5, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
   0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
   0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
   0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
   0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
   0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
   0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 5, -10, 10, -5,
    1}; gf = Fold[x #1 + #2 &, 0, p]/Fold[x #1 + #2 &, 0, q]; CoefficientList[Series[Factor[gf], {x, 0, 100}], x] (* Peter J. C. Moses, Jul 08 2012 *)

A227015 a(n) = floor(M(g(n-1)+1, ..., g(n))), where M = harmonic mean and g(n) = n^3 + n^2 + n + 1.

Original entry on oeis.org

2, 8, 26, 60, 117, 203, 324, 487, 696, 958, 1279, 1666, 2123, 2657, 3274, 3981, 4782, 5684, 6693, 7816, 9057, 10423, 11920, 13555, 15332, 17258, 19339, 21582, 23991, 26573, 29334, 32281, 35418, 38752, 42289, 46036, 49997, 54179, 58588, 63231, 68112, 73238

Offset: 1

Clark Kimberling, Jul 01 2013

nonn

See A227012.

Clark Kimberling, Table of n, a(n) for n = 1..100

Cf. A227012, A227013.

z = 100; f[x_] := f[x] = 1/x; g[n_] := g[n] = n^3 + n^2 + n + 1; s[n_] := s[n] = Sum[f[k], {k, g[n - 1] + 1, g[n]}]; v[n_] := v[n] = (g[n] - g[n - 1])/s[n]; Table[g[n], {n, 1, z}];
Table[Floor[v[n]], {n, 1, z}]

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-4) - 3*a(n-5) + 3*a(n-6) - a(n-7) for n > 2 (conjectured).
G.f.: x*(2 + 2*x + 8*x^2 + 4*x^3 + 5*x^4 + 4*x^5 - 2*x^6 + 3*x^7 - 3*x^8 + x^9)/((x - 1)^4*(1 + x + x^2 + x^3)) (conjectured).
From Franck Maminirina Ramaharo, Apr 16 2018: (Start)
a(n) = (1/2)*((-1)^(n - 1)! + 2*n^3 - n^2 + n + 3 + 2*floor(max(0, n - 4)/4)) (conjectured).
E.g.f.: (1/24)*exp(-x)*(exp(x)*(6*sin(x) + 6*cos(x) + 4*x^3 - 24) + exp(2*x)*(24*x^3 + 60*x^2 + 30*x + 15) + 3) (conjectured).
(End)

A227018 a(n) = floor(M(g(n-1)+1,..,g(n))), where M = harmonic mean and g(n) = n(n + 1)(n + 2)*(n + 3)/24.

Original entry on oeis.org

1, 3, 9, 24, 51, 95, 164, 266, 407, 598, 850, 1174, 1582, 2087, 2706, 3452, 4342, 5395, 6628, 8060, 9714, 11609, 13768, 16215, 18975, 22072, 25534, 29388, 33662, 38387, 43591, 49307, 55568, 62407, 69858, 77957, 86740, 96245, 106511, 117577, 129482, 142270

Offset: 1

Clark Kimberling, Jul 06 2013

See A227012.

			a(1) = [1/(1/1)] = 1;
a(2) = [4/(1/2 + 1/3 + 1/4 + 1/5)] = 3;
a(3) = [10/(1/6 + 1/7 + ... + 1/15)] = 9.

Clark Kimberling, Table of n, a(n) for n = 1..97

Cf. A227012, A227016.

Clear[g]; g[n_] := N[Binomial[n + # - 1, #] &[4], 100]; a = {1}; Do[
AppendTo[a, Floor[(#2 - #1 + 1)/(HarmonicNumber[#2] - HarmonicNumber[#1 - 1])] &[g[k - 1] + 1, g[k]]], {k, 2, 100}]; a

Conjectured g.f.: (-1 + x - 3 x^2 - 2 x^3 + 2 x^4 - 2 x^5 - 3 x^6 + 2 x^8 - 5 x^9 - x^12 - x^13 - 4 x^14 + 4 x^15 - 4 x^16 - 2 x^17 + 2 x^18 - 2 x^19 - 4 x^20 + 4 x^21 - 4 x^22 - x^23 - x^24 + x^25 - 4 x^26 + x^27 - x^28 - 3 x^29 + 3 x^30 - 5 x^31 - x^34 - 2 x^35 + x^36 - 3 x^37 + 2 x^38 + 2 x^39 - 2 x^40 - 3 x^41 + 6 x^42 - 3 x^43 - 3 x^44 + 6 x^45 - 4 x^46 + x^47)/((-1 + x)^5 (1 + x) (1 + x^2) (1 - x + x^2) (1 + x + x^2) (1 - x^2 + x^4) (1 - x^3 + x^6) (1 + x^3 + x^6) (1 - x^6 + x^12)).

cp's OEIS Frontend

A227017 Floor(M(g(n-1)+1,..,g(n))), where M = harmonic mean and g(n) = n(3n-1)/2 = A000326(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A227013 a(n) = floor(M(g(n-1)+1,..,g(n))), where M is the harmonic mean and g(n) = n^4.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions

A227016 Floor(M(g(n-1)+1,..,g(n))), where M = harmonic mean and g(n) = n(n + 1)(n + 2)/6.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A227014 a(n) = floor(M(g(n-1)+1,..,g(n))), where M = harmonic mean and g(n) = n^5.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A227015 a(n) = floor(M(g(n-1)+1, ..., g(n))), where M = harmonic mean and g(n) = n^3 + n^2 + n + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A227018 a(n) = floor(M(g(n-1)+1,..,g(n))), where M = harmonic mean and g(n) = n*(n + 1)*(n + 2)*(n + 3)/24.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A227018 a(n) = floor(M(g(n-1)+1,..,g(n))), where M = harmonic mean and g(n) = n(n + 1)(n + 2)*(n + 3)/24.