A375057 -id:A375057 - cp's OEIS frontend

A376952 a(n) = least k such that (n*Pi/2)^(2k)/(2 k)! < 1.

1, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 23, 25, 27, 29, 31, 33, 35, 37, 39, 42, 44, 46, 48, 50, 52, 54, 56, 59, 61, 63, 65, 67, 69, 71, 73, 76, 78, 80, 82, 84, 86, 88, 91, 93, 95, 97, 99, 101, 103, 105, 108, 110, 112, 114, 116, 118, 120, 122, 125, 127, 129

Offset: 0

Clark Kimberling, Oct 12 2024

nonn

The numbers (n*Pi/2)^(2k)/(2 k)! are the coefficients in the Maclaurin series for cos x when x = n*Pi/2. If m>a(n), then (m*Pi/2)^(2k)/(2 k)! < 1. A375057 is a bisection of this sequence.

Cf. A370507, A376953, A376954, A376955, A376956, A376957, A376958, A376959, A376960.

a[n_] := Select[Range[300], (n Pi/2)^(2 #)/(2 #)! < 1 &, 1];
Flatten[Table[a[n], {n, 0, 200}]]

a(n) ~ Pi*exp(1)*n/4 - log(n)/4. - Vaclav Kotesovec, Oct 13 2024

A376953 a(n) = least k such that (n*Pi/3)^(2k)/(2 k)! < 1.

Original entry on oeis.org

1, 1, 2, 4, 5, 6, 8, 9, 11, 12, 13, 15, 16, 18, 19, 20, 22, 23, 25, 26, 27, 29, 30, 32, 33, 35, 36, 37, 39, 40, 42, 43, 44, 46, 47, 49, 50, 52, 53, 54, 56, 57, 59, 60, 61, 63, 64, 66, 67, 69, 70, 71, 73, 74, 76, 77, 78, 80, 81, 83, 84, 86, 87, 88, 90, 91, 93

Offset: 0

Clark Kimberling, Oct 12 2024

nonn

The numbers (n*Pi/3)^(2k)/(2 k)! are the coefficients in the Maclaurin series for cos x when x = n*Pi/3. If m>a(n), then (m*Pi/3)^(2k)/(2 k)! < 1. A375057 is a trisection of this sequence.

Cf. A370507, A376952, A376954, A376955, A376956, A376957, A376958, A376959, A376960.

a[n_] := Select[Range[300], (n Pi/3)^(2 #)/(2 #)! < 1 &, 1];
Flatten[Table[a[n], {n, 0, 200}]]

a(n) ~ Pi*exp(1)*n/6 - log(n)/4. - Vaclav Kotesovec, Oct 13 2024

A376954 a(n) = least k such that (2n*Pi/3)^(2k)/(2 k)! < 1.

Original entry on oeis.org

1, 2, 5, 8, 11, 13, 16, 19, 22, 25, 27, 30, 33, 36, 39, 42, 44, 47, 50, 53, 56, 59, 61, 64, 67, 70, 73, 76, 78, 81, 84, 87, 90, 93, 95, 98, 101, 104, 107, 110, 113, 115, 118, 121, 124, 127, 130, 132, 135, 138, 141, 144, 147, 149, 152, 155, 158, 161, 164, 167

Offset: 0

Clark Kimberling, Oct 12 2024

nonn

The numbers (2n*Pi/3)^(2k)/(2 k)! are the coefficients in the Maclaurin series for cos x when x = 2n*Pi/3. If m>a(n), then (2m*Pi/3)^(2k)/(2 k)! < 1. A375057 is a trisection of this sequence.

Cf. A370507, A376952, A376953, A376955, A376956, A376957, A376958, A376959, A376960.

a[n_] := Select[Range[200], (2n Pi/3)^(2 #)/(2 #)! < 1 &, 1];
Flatten[Table[a[n], {n, 0, 200}]]

a(n) ~ Pi*exp(1)*n/3 - log(n)/4. - Vaclav Kotesovec, Oct 13 2024

A376457 Let s(x) be the Maclaurin series for cos(x); then a(n) is the index k for which (k+1)-st partial sum of s(2nPi) is least among all partial sums.

Original entry on oeis.org

3, 5, 9, 11, 15, 19, 21, 25, 27, 31, 33, 37, 41, 43, 47, 49, 53, 55, 59, 63, 65, 69, 71, 75, 77, 81, 85, 87, 91, 93, 97, 99, 103, 107, 109, 113, 115, 119, 121, 125, 129, 131, 135, 137, 141, 143, 147, 151, 153, 157, 159, 163, 165, 169, 173, 175, 179, 181, 185

Offset: 1

Clark Kimberling, Oct 01 2024

nonn

			For n = 2 the partial sums (of which the 1st is for k=0) are approximately 1, -18.7, 46.2, -39.2, 20.9, -5.4, ..., where the least, -39.2..., is the 4th, so that a(2) = 3.

Cf. A374987, A375057, A375053, A375054.

z = 200; r = Pi;
f[n_, m_] := f[n, m] = N[Sum[(-1)^k  (2 n  r)^(2 k)/(2 k)!, {k, 0, m}], 10]
t[n_] := Table[f[n, m], {m, 1, z}]
g[n_] := Select[Range[z], f[n, #] == Max[t[n]] &]
h[n_] := Select[Range[z], f[n, #] == Min[t[n]] &]
Flatten[Table[g[n], {n, 1, 60}]]  (* A376456 *)
Flatten[Table[h[n], {n, 1, 60}]]  (* this sequence *)

|a(n)-A376457(n)| = 1 for n>=1.

A375053 a(n) = least k such that (n Pi)^(2 k + 1)/(2 k + 1)! < 1.

Original entry on oeis.org

1, 3, 7, 11, 16, 20, 24, 28, 33, 37, 41, 45, 50, 54, 58, 62, 67, 71, 75, 79, 84, 88, 92, 96, 101, 105, 109, 113, 118, 122, 126, 131, 135, 139, 143, 148, 152, 156, 160, 165, 169, 173, 177, 182, 186, 190, 194, 199, 203, 207, 212, 216, 220, 224, 229, 233, 237

Offset: 0

Clark Kimberling, Oct 01 2024

nonn

The numbers (n Pi)^(2 k + 1)/(2 k + 1)! are the coefficients in the Maclaurin series for sin x when x = n*Pi.

(n Pi)^(2 k + 1)/(2 k + 1)! < 1 for every k >= a(n).

Cf. A374987, A376456, A376457, A375054, A375057.

z = 300; r = Pi;
a[n_] := Select[Range[z], (n  r)^(2 # + 1)/(2 # + 1)! < 1 &, 1]
Flatten[Table[a[n], {n, 0, 100}]]

A375054 Let M(n,x) denote the Maclaurin polynomial of degree 2n for cos x. Let u(n) be the number of nonreal zeros of M(n,x) and v(n) the number of real zeros of M(n,x). Then a(n) = u(n) - v(n).

Original entry on oeis.org

-2, -4, 2, 0, 6, 4, 2, 8, 6, 4, 10, 8, 14, 12, 10, 16, 14, 12, 18, 16, 22, 20, 18, 24, 22, 28, 26, 24, 30, 28, 26, 32, 30, 36, 34, 32, 38, 36, 34, 40, 38, 44, 42, 40, 46, 44, 50, 48, 46, 52, 50, 48, 54, 52, 58, 56, 54, 60, 58, 64, 62, 60, 66, 64, 62, 68, 66

Offset: 1

Clark Kimberling, Oct 01 2024

sign

1

			The 6 zeros of the Maclaurin polynomial x^2/2! - x^4/4! - x^6/6! are approximately {-3.92 - 1.28 i, -3.92 + 1.2 i, -1.56, 1.56, 3.92 - 1.28 i, 3.92 + 1.28 i}; there are 4 nonreal zero and 2 real zeros, so that a(3) = 4 - 2 = 2.

Cf. A374987, A376456, A376457, A375053, A375057.

z = 100;
a[n_] := CountRoots[Sum[(-1)^k*x^k/(2 k)!, {k, 0, n}], {x, 0, Infinity}];
t = 2 Table[a[n], {n, 1, z}] ; (* # real zeros of M(n,x) *)
2 Range[z] - t (* # nonreal zeros *)
2 Range[z] - 2 t (* # nonreal zeros minus # real zeros; *)

A374987 Let s(x) be the Maclaurin series for cos(x); then a(n) is the least index k for which all partial sums of cos(2m*Pi) are positive.

Original entry on oeis.org

6, 14, 24, 32, 40, 48, 58, 66, 74, 82, 92, 100, 108, 116, 126, 134, 142, 150, 160, 168, 176, 184, 194, 202, 210, 218, 228, 236, 244, 254, 262, 270, 278, 288, 296, 304, 312, 322, 330, 338, 346, 356, 364, 372, 382, 390, 398, 406, 416, 424, 432, 440, 450, 458

Offset: 0

Clark Kimberling, Oct 01 2024

nonn

			For n=1, the partial sums (for k = 0,1,2,3,4,5,6,7) are approximately 1, -18.7, 46.2, -39.2, 20.9, -5.4, 2.4, 0.7; beginning with k=6, the partials sums are all positive, so a(1)=6.

Cf. A376456, A376457, A375057, A375053, A375054.

z = 800; r = Pi;
f[m_, n_] := f[m, n] = N[Sum[(-1)^k  (2 m  r)^(2 k)/(2 k)!, {k, 0, n}], 10]
g[m_] := Select[Range[z], f[m, #] > 0 && f[m, # + 1] > 0 &, 1]
Flatten[Table[g[m], {m, 1, 80}]]

cp's OEIS Frontend

A376952 a(n) = least k such that (n*Pi/2)^(2k)/(2 k)! < 1.

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A376953 a(n) = least k such that (n*Pi/3)^(2k)/(2 k)! < 1.

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Mathematica

Formula

A376954 a(n) = least k such that (2n*Pi/3)^(2k)/(2 k)! < 1.

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Mathematica

Formula

A376457 Let s(x) be the Maclaurin series for cos(x); then a(n) is the index k for which (k+1)-st partial sum of s(2*n*Pi) is least among all partial sums.

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A375053 a(n) = least k such that (n Pi)^(2 k + 1)/(2 k + 1)! < 1.

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Crossrefs

Programs

Mathematica

A375054 Let M(n,x) denote the Maclaurin polynomial of degree 2n for cos x. Let u(n) be the number of nonreal zeros of M(n,x) and v(n) the number of real zeros of M(n,x). Then a(n) = u(n) - v(n).

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A374987 Let s(x) be the Maclaurin series for cos(x); then a(n) is the least index k for which all partial sums of cos(2m*Pi) are positive.

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Mathematica

A376457 Let s(x) be the Maclaurin series for cos(x); then a(n) is the index k for which (k+1)-st partial sum of s(2nPi) is least among all partial sums.