A248229 -id:A248229 - cp's OEIS frontend

A248669 Triangular array of coefficients of polynomials q(n,k) defined in Comments.

1, 2, 1, 5, 4, 1, 16, 17, 7, 1, 65, 84, 45, 11, 1, 326, 485, 309, 100, 16, 1, 1957, 3236, 2339, 909, 196, 22, 1, 13700, 24609, 19609, 8702, 2281, 350, 29, 1, 109601, 210572, 181481, 89225, 26950, 5081, 582, 37, 1, 986410, 2004749, 1843901, 984506, 331775

Offset: 1

Clark Kimberling, Oct 11 2014

q(n,x) = 1 + k+x + (k+x)(k-1+x) + (k+x)(k-1+x)(k-2+x) + ... + (k+x)(k-1+x)(k-2+x)...(1+x). The arrays at A248229 and A248664 have the same first column, given by A000522(n) for n >= 0. The alternating row sums of the array at A248669 are also given by A000522; viz., q(n,-1) = q(n-1,0) = A000522(n-2) for n >= 2. Column 2 of A248669 is given by A093344(n) for n >= 1.

			The first six polynomials:
p(1,x) = 1
p(2,x) = 2 + x
p(3,x) = 5 + 4 x + x^2
p(4,x) = 16 + 17 x + 7 x^2 + x^3
p(5,x) = 65 + 8 x + 45 x^2 + 11 x^3 + x^4
p(6,x) = 326 + 485 x + 309 x^2 + 100 x^3 + 16 x^4 + x^5
First six rows of the triangle:
1
2     1
5     4     1
16    17    7    1
65    84    45   11    1
326   485  309   100   16   1

Clark Kimberling, Table of n, a(n) for n = 1..5000

Cf. A248665, A248666, A248667, A248668, A248670.

t[x_, n_, k_] := t[x, n, k] = Product[x + n - i, {i, 1, k}];
q[x_, n_] := Sum[t[x, n, k], {k, 0, n - 1}];
TableForm[Table[q[x, n], {n, 1, 6}]];
TableForm[Table[Factor[q[x, n]], {n, 1, 6}]];
c[n_] := c[n] = CoefficientList[q[x, n], x];
TableForm[Table[c[n], {n, 1, 12}]] (* A248669 array *)
Flatten[Table[c[n], {n, 1, 12}]]   (* A248669 sequence *)

q(n,x) = (x + n - 1)*q(n-1,x) + 1, with q(1,x) = 1.

A248227 Least k such that zeta(4) - sum{1/h^4, h = 1..k} < 1/n^3.

Original entry on oeis.org

1, 1, 2, 3, 3, 4, 5, 6, 6, 7, 8, 8, 9, 10, 10, 11, 12, 12, 13, 14, 15, 15, 16, 17, 17, 18, 19, 19, 20, 21, 21, 22, 23, 24, 24, 25, 26, 26, 27, 28, 28, 29, 30, 31, 31, 32, 33, 33, 34, 35, 35, 36, 37, 37, 38, 39, 40, 40, 41, 42, 42, 43, 44, 44, 45, 46, 46, 47

Offset: 1

Clark Kimberling, Oct 05 2014

This sequence and A248230 provide insight into the manner of convergence of sum{1/h^4, h = 0..k}. Since a(n+1) - a(n) is in {0,1} for n >= 1, A248228 and A248229 are complementary.

			Let s(n) = sum{1/h^4, h = 1..n}.  Approximations are shown here:
n ... zeta(4) - s(n) ... 1/n^3
1 ... 0.0823232 .... 1
2 ... 0.0198232 .... 0.125
3 ... 0.0074775 .... 0.037
4 ... 0.0035713 .... 0.015
5 ... 0.0019713 .... 0.008
6 ... 0.0011997 .... 0.005
a(6) = 4 because zeta(4) - s(4) < 1/216 < zeta(4) - s(3).

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

Cf. A248228, A248229, A248230, A013662.

$MaxExtraPrecision = Infinity; z = 400; p[k_] := p[k] = Sum[1/h^4, {h, 1, k}];
N[Table[Zeta[4] - p[n], {n, 1, z/10}]]
f[n_] := f[n] = Select[Range[z], Zeta[4] - p[#] < 1/n^3 &, 1];
u = Flatten[Table[f[n], {n, 1, z}]]   (* A248227 *)
Flatten[Position[Differences[u], 0]]  (* A248228 *)
Flatten[Position[Differences[u], 1]]  (* A248229 *)
f = Table[Floor[1/(Zeta[4] - p[n])], {n, 1, z}]  (* A248230 *)

a(n) ~ 3^(-1/3) * n. - Vaclav Kotesovec, Oct 09 2014

A248228 Numbers k such that A248227(k+1) = A248227(k).

Original entry on oeis.org

1, 4, 8, 11, 14, 17, 21, 24, 27, 30, 34, 37, 40, 44, 47, 50, 53, 57, 60, 63, 66, 70, 73, 76, 79, 83, 86, 89, 92, 96, 99, 102, 105, 109, 112, 115, 119, 122, 125, 128, 132, 135, 138, 141, 145, 148, 151, 154, 158, 161, 164, 167, 171, 174, 177, 180, 184, 187

Offset: 1

Clark Kimberling, Oct 05 2014

Since A248227(k+1) - A248227(k) is in {0,1} for k >= 1, A248228 and A248229 are complementary.

			The difference sequence of A248227 is (0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, ...), so that A248228 = (1, 4, 8, 11, 14, 17, 2,...) and A248229 = (2, 3, 5, 6, 7, 9, 10, 12, 13, 15, 16, 18,...), the complement of A248228.

Clark Kimberling, Table of n, a(n) for n = 1..300

Cf. A248227, A248229, A248230.

$MaxExtraPrecision = Infinity; z = 400; p[k_] := p[k] = Sum[1/h^4, {h, 1, k}];
N[Table[Zeta[4] - p[n], {n, 1, z/10}]]
f[n_] := f[n] = Select[Range[z], Zeta[4] - p[#] < 1/n^3 &, 1];
u = Flatten[Table[f[n], {n, 1, z}]]   (* A248227 *)
Flatten[Position[Differences[u], 0]]  (* A248228 *)
Flatten[Position[Differences[u], 1]]  (* A248229 *)
f = Table[Floor[1/(Zeta[4] - p[n])], {n, 1, z}]  (* A248230 *)

A248230 a(n) = floor(1/(zeta(4) - Sum_{h=1..n} 1/h^4)).

Original entry on oeis.org

12, 50, 133, 280, 507, 833, 1276, 1855, 2586, 3488, 4579, 5878, 7401, 9167, 11194, 13501, 16104, 19022, 22273, 25876, 29847, 34205, 38968, 44155, 49782, 55868, 62431, 69490, 77061, 85163, 93814, 103033, 112836, 123242, 134269, 145936, 158259, 171257, 184948

Offset: 1

Clark Kimberling, Oct 05 2014

This sequence provides insight into the manner of convergence of Sum_{h=1..n} 1/h^4.

Clark Kimberling, Table of n, a(n) for n = 1..1000
Soumyadip Sahu, On Certain Reciprocal Sums, arXiv:1807.05454 [math.NT], 2018.

Cf. A248227, A248228, A248229, A013662.

$MaxExtraPrecision = Infinity; z = 400; p[k_] := p[k] = Sum[1/h^4, {h, 1, k}];
N[Table[Zeta[4] - p[n], {n, 1, z/10}]]
f[n_] := f[n] = Select[Range[z], Zeta[4] - p[#] < 1/n^3 &, 1];
u = Flatten[Table[f[n], {n, 1, z}]]   (* A248227 *)
Flatten[Position[Differences[u], 0]]  (* A248228 *)
Flatten[Position[Differences[u], 1]]  (* A248229 *)
f = Table[Floor[1/(Zeta[4] - p[n])], {n, 1, z}]  (* A248230 *)

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

Conjecture: a(n) = 1 + 7*n/2 + 9*n^2/2 + 3*n^3 + floor(n/4), holds for all n <= 10000. - Vaclav Kotesovec, Oct 09 2014

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A248669 Triangular array of coefficients of polynomials q(n,k) defined in Comments.

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A248227 Least k such that zeta(4) - sum{1/h^4, h = 1..k} < 1/n^3.

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A248228 Numbers k such that A248227(k+1) = A248227(k).

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A248230 a(n) = floor(1/(zeta(4) - Sum_{h=1..n} 1/h^4)).

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