A374987 -id:A374987 - cp's OEIS frontend

A375057 a(n) = least k such that (n*pi)^(2k)/(2 k)! < 1.

1, 4, 8, 12, 16, 20, 25, 29, 33, 37, 42, 46, 50, 54, 59, 63, 67, 71, 76, 80, 84, 88, 93, 97, 101, 105, 110, 114, 118, 122, 127, 131, 135, 140, 144, 148, 152, 157, 161, 165, 169, 174, 178, 182, 186, 191, 195, 199, 203, 208, 212, 216, 221, 225, 229, 233, 238

Offset: 0

Clark Kimberling, Oct 01 2024

nonn

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

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

Cf. A374987, A376456, A376457, A375053, A375054.

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

Edited by Clark Kimberling, Oct 10 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; *)

cp's OEIS Frontend

A375057 a(n) = least k such that (n*pi)^(2k)/(2 k)! < 1.

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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.

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

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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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cp's OEIS Frontend

A375057 a(n) = least k such that (n*pi)^(2k)/(2 k)! < 1.

Views

Author

Keywords

Comments

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Programs

Mathematica

Extensions

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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Formula

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

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Author

Keywords

Comments

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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Programs

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.