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.

User: Cheng-Jun Li

Cheng-Jun Li's wiki page.

Cheng-Jun Li has authored 3 sequences.

A376499 Array read by ascending antidiagonals: A(n,k) = A376484/(2*n+1).

Original entry on oeis.org

1, 1, 3, 1, 3, 9, 1, 3, 10, 27, 1, 3, 10, 35, 81, 1, 3, 10, 35, 125, 243, 1, 3, 10, 35, 126, 450, 729, 1, 3, 10, 35, 126, 462, 1625, 2187, 1, 3, 10, 35, 126, 462, 1715, 5875, 6561, 1, 3, 10, 35, 126, 462, 1716, 6419, 21250, 19683, 1, 3, 10, 35, 126, 462, 1716, 6435, 24157, 76875, 59049
Offset: 1

Views

Author

Cheng-Jun Li, Sep 25 2024

Keywords

Comments

It is only a conjecture that the A(n,k) are always integers.
Values repeated as a staircase for all A(n+x,2*n) (x > 0 and are equal to A(n,2*n)).

Examples

			First ten rows start as follows:
  1 3  9 27  81 243  729 2187  6561 19683  59049  177147  531441  1594323  4782969
  1 3 10 35 125 450 1625 5875 21250 76875 278125 1006250 3640625 13171875 47656250
  1 3 10 35 126 462 1715 6419 24157 91238 345401 1309574 4970070 18874261 71705865
  1 3 10 35 126 462 1716 6435 24309 92358 352485 1350054 5185350 19960020 76964985
  1 3 10 35 126 462 1716 6435 24310 92378 352715 1352054 5199975 20055024 77531355
  1 3 10 35 126 462 1716 6435 24310 92378 352716 1352078 5200299 20058272 77558325
  1 3 10 35 126 462 1716 6435 24310 92378 352716 1352078 5200300 20058300 77558759
  1 3 10 35 126 462 1716 6435 24310 92378 352716 1352078 5200300 20058300 77558760
  1 3 10 35 126 462 1716 6435 24310 92378 352716 1352078 5200300 20058300 77558760
  1 3 10 35 126 462 1716 6435 24310 92378 352716 1352078 5200300 20058300 77558760

Crossrefs

All of these are conjectures. Rows: A000244, A081567, A122068. Columns: A000012, A000012 * 3, A095049 for n >= 20. A(1,k) = A000244, A(2,k) = A081567, A(3,k) = A122068 (First 3 rows of the array).A(n,1) = A(n,2) / 3 = A000012, A(n,3) = A095049 for n >= 20 (First 3 columns of the array). When k increases, the row of A(n,k) gets closer to A001700.

Formula

A(n,k) = A376484(n,k)/(2*n+1)

A376498 Array read by ascending antidiagonals: A(n, k) = 2^k*Sum_{j=1..n} cos((2*j - 1)*Pi/(2*n + 1))^k.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 1, 1, 0, 4, 1, 3, 1, 0, 5, 1, 5, 4, 1, 0, 6, 1, 7, 4, 7, 1, 0, 7, 1, 9, 4, 13, 11, 1, 0, 8, 1, 11, 4, 19, 16, 18, 1, 0, 9, 1, 13, 4, 25, 16, 38, 29, 1, 0, 10, 1, 15, 4, 31, 16, 58, 57, 47, 1, 0, 11, 1, 17, 4, 37, 16, 78, 64, 117, 76, 1, 0
Offset: 0

Views

Author

Cheng-Jun Li, Sep 25 2024

Keywords

Comments

It is only a conjecture that the A(n, k) are always integers.

Examples

			Array starts:
[0] 0, 0,  0, 0,  0,  0,   0,  0,   0,   0,    0,    0, ...  [A000004]
[1] 1, 1,  1, 1,  1,  1,   1,  1,   1,   1,    1,    1, ...  [A000012]
[2] 2, 1,  3, 4,  7, 11,  18, 29,  47,  76,  123,  199, ...  [A000032]
[3] 3, 1,  5, 4, 13, 16,  38, 57, 117, 193,  370,  639, ...  [A096975]
[4] 4, 1,  7, 4, 19, 16,  58, 64, 187, 247,  622,  925, ...  [A094649]
[5] 5, 1,  9, 4, 25, 16,  78, 64, 257, 256,  874, 1013, ...  [A189234]
[6] 6, 1, 11, 4, 31, 16,  98, 64, 327, 256, 1126, 1024, ...  [A216605]
[7] 7, 1, 13, 4, 37, 16, 118, 64, 397, 256, 1378, 1024, ...
[8] 8, 1, 15, 4, 43, 16, 138, 64, 467, 256, 1630, 1024, ...
[9] 9, 1, 17, 4, 49, 16, 158, 64, 537, 256, 1882, 1024, ...

Crossrefs

Rows: A000004 (n=0), A000012 (n=1), A000032 (n=2), A096975 (n=3), A094649 (n=4), A189234 (n=5), A216605 (n=6, with alternate signs).
Columns: A001477 (k=0), A057427 (k=1).
Cf. A180870.

Programs

  • PARI
    A(n, k) = 2^k*sum(j=1, n, (cos((2*j-1)*Pi/(2*n+1)))^k, x=0)

Formula

A(n + k, 2*k - 1) = A(k, 2*k-1) = 4^(k-1).
Let P_n(x) be the polynomial: Sum_{k=0..n} x^k*A180870(n, k). Let R_n(x) be the polynomial Product_{k=0..n} x-Roots(P_n, k)^m. A(n, k) = abs([x^1] R_n(x))/2^(m*(n-1)), for n > 0. - Thomas Scheuerle, Oct 07 2024

A376484 Array read by ascending antidiagonals: A(n,k)=4^k*Sum_{j=1..n} sin(2*j*Pi/(2*n+1))^(2*k).

Original entry on oeis.org

0, 1, 0, 2, 3, 0, 3, 5, 9, 0, 4, 7, 15, 27, 0, 5, 9, 21, 50, 81, 0, 6, 11, 27, 70, 175, 243, 0, 7, 13, 33, 90, 245, 625, 729, 0, 8, 15, 39, 110, 315, 882, 2250, 2187, 0, 9, 17, 45, 130, 385, 1134, 3234, 8125, 6561, 0, 10, 19, 51, 150, 455, 1386, 4158, 12005, 29375, 19683, 0
Offset: 0

Views

Author

Cheng-Jun Li, Sep 24 2024

Keywords

Comments

It is only a conjecture that A(n,k) is always an integer.
It appears that A(n,k) is divisible by 2*n+1 when n, k are positive integers.

Examples

			For n = 0 to 10 and k = 0 to 10, A(n, k) shows as below :
  0  0  0   0   0    0    0     0      0      0       0
  1  3  9  27  81  243  729  2187   6561  19683   59049
  2  5 15  50 175  625 2250  8125  29375 106250  384375
  3  7 21  70 245  882 3234 12005  44933 169099  638666
  4  9 27  90 315 1134 4158 15444  57915 218781  831222
  5 11 33 110 385 1386 5082 18876  70785 267410 1016158
  6 13 39 130 455 1638 6006 22308  83655 316030 1200914
  7 15 45 150 525 1890 6930 25740  96525 364650 1385670
  8 17 51 170 595 2142 7854 29172 109395 413270 1570426
  9 19 57 190 665 2394 8778 32604 122265 461890 1755182
 10 21 63 210 735 2646 9702 36036 135135 510510 1939938

Crossrefs

Conjectures: This array is related to existing sequences as follows: (Start)
Rows: A000004, A000244, A020876, A322459 (with alternate signs).
Columns: A001477, A005408, A016945.
Main Diagonals: A033876.
A(0,k) = A000004, A(1,k) = A000244, A(2,k) = A020876, A(3,k) = (-1)^k * A322459 (First 4 rows of the array).
A(n,0) = A001477(n > 0), A(n,1) = A005408(n > 0), A(n,2) = A016945(n > 0) (First 3 columns of the array).
A(n,n) = A033876(n > 0) (Main diagonal from top left corner). (End)

Programs

  • PARI
    A(n,k) = 4^k*sum(j=1,n,(sin(2*j*Pi/(2*n+1)))^(2*k))

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.