cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

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

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Oct 01 2024

Keywords

Comments

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).

Crossrefs

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

Programs

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

Extensions

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(2*n*Pi) 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

Views

Author

Clark Kimberling, Oct 01 2024

Keywords

Examples

			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.

Crossrefs

Cf. A374987, A375057, A375053, A375054.

Programs

  • Mathematica
    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 *)

Formula

|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

Views

Author

Clark Kimberling, Oct 01 2024

Keywords

Comments

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).

Crossrefs

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

Programs

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

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

Views

Author

Clark Kimberling, Oct 01 2024

Keywords

Examples

			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.

Crossrefs

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

Programs

  • Mathematica
    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}]]

A376322 (1/4) times obverse convolution (2)**(2^n + 1); see Comments.

Original entry on oeis.org

1, 5, 35, 385, 7315, 256025, 17153675, 2247131425, 582007039075, 299733625123625, 307826433001962875, 631352014087025856625, 2587911905742718986305875, 21207938067561582092776645625, 347534481113131645754330891856875, 11389052480558437163015177657041650625
Offset: 0

Views

Author

Clark Kimberling, Sep 20 2024

Keywords

Comments

See A374848 for the definition of obverse convolution and a guide to related sequences.

Crossrefs

Cf. A000051, A000079, A007395, A010701, A374848, A375054, A139030.

Programs

  • Mathematica
    s[n_] := 2; t[n_] := 2^n + 1;
u[n_] := (1/4) Product[s[k] + t[n - k], {k, 0, n}];
Table[u[n], {n, 0, 20}]
(* or *)
Table[2^(n*(n+1)/2 - 2) * QPochhammer[-3, 1/2, n+1], {n, 0, 15}] (* Vaclav Kotesovec, Sep 20 2024 *)

Formula

a(n) = a(n-1)*A062709(n) for n>=1.
a(n) = (1/4)((3)**(2^n)) = (1/4)(A010701(n)**A000079(n)) for n>=0.

A376324 (1/6) times obverse convolution (4)**(2^n + 1); see Comments.

Original entry on oeis.org

1, 7, 63, 819, 17199, 636363, 43909047, 5839903251, 1524214748511, 788019024980187, 810871576704612423, 1664719346974569304419, 6827014041942708717422319, 55961034101804383356710748843, 917145387894472038833132462787927
Offset: 0

Views

Author

Clark Kimberling, Sep 20 2024

Keywords

References

  • See A374848 for the definition of obverse convolution and a guide to related sequences.

Crossrefs

Cf. A000051, A010709, A168614, A374848, A375053, A375054.

Programs

  • Mathematica
    s[n_] := 4; t[n_] := 2^n + 1;
u[n_] := (1/6) Product[s[k] + t[n - k], {k, 0, n}];
Table[u[n], {n, 0, 20}]
(* or *)
Table[2^(n*(n+1)/2 - 1) * QPochhammer[-5, 1/2, n+1]/3, {n, 0, 15}] (* Vaclav Kotesovec, Sep 20 2024 *)

Formula

a(n) = a(n-1)*A168614(n) for n>=1.
Showing 1-6 of 6 results.

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