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.

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A101453 Number of inequivalent solutions to toroidal (8n+1)-queen problem under the symmetry operator R45(x,y)=( (x-y)/sqrt(2), (x+y)/sqrt(2) ).

Original entry on oeis.org

1, 0, 4, 0, 0, 192, 1792, 0, 0, 466432, 0, 33658880, 441192448
Offset: 0

Views

Author

Yuh-Pyng Shieh, Yung-Luen Lan, Jieh Hsiang (arping(AT)turing.csie.ntu.edu.tw), Jan 19 2005

Keywords

Comments

The R45 operator is not valid on toroidal N-queen problem if 2 is not a perfect square modulo N. For example, a(3)=0 is because 2 is not a perfect square modulo 25. see A057126. Toroidal N-queen problem has no fixed points under R45 if N is not equal to 8k+1 for some integer k.

Examples

			a(5)=6 because the number of inequivalent solutions to toroidal 41-queen problem under R45 is 192.

References

  • Jieh Hsiang, Yuh-Pyng Shieh and YaoChiang Chen, "The Cyclic Complete Mappings Counting Problems", PaPS: Problems and Problem Sets for ATP Workshop in conjunction with CADE-18 and FLoC 2002, Copenhagen, Denmark, 2002/07/27-08/01.

Links

Crossrefs

Cf. A007705, A057126.

A101454 Number of inequivalent solutions to toroidal (8n+1)-queen problem under the symmetry operator R45(x,y)=( (x-y)/sqrt(2), (x+y)/sqrt(2) ), divided by 2^n.

Original entry on oeis.org

1, 0, 1, 0, 0, 6, 28, 0, 0, 911, 0, 16435, 107713
Offset: 0

Views

Author

Yuh-Pyng Shieh, Yung-Luen Lan, Jieh Hsiang (arping(AT)turing.csie.ntu.edu.tw), Jan 19 2005

Keywords

Comments

The R45 operator is not valid on toroidal N-queen problem if 2 is not a perfect square modulo N. For example, a(3)=0 is because 2 is not a perfect square modulo 25. See A057126. Toroidal N-queen problem has no fixed points under R45 if N is not equal to 8k+1 for some integer k.

Examples

			a(5)=6 because the number of inequivalent solutions to toroidal 41-queen problem under R45 is 192 and 192 / (2^5) = 6.

References

  • Jieh Hsiang, Yuh-Pyng Shieh and YaoChiang Chen, "The Cyclic Complete Mappings Counting Problems", PaPS: Problems and Problem Sets for ATP Workshop in conjunction with CADE-18 and FLoC 2002, Copenhagen, Denmark, 2002/07/27-08/01.

Links

Crossrefs

Cf. A007705, A057126.

A273179 Numbers k for which 2 has exactly four square roots mod k.

Original entry on oeis.org

119, 161, 217, 238, 287, 322, 329, 391, 434, 497, 511, 527, 553, 574, 623, 658, 679, 697, 713, 721, 782, 791, 799, 833, 889, 943, 959, 994, 1022, 1054, 1057, 1081, 1106, 1127, 1169, 1207, 1241, 1246, 1271, 1337, 1343, 1351, 1358, 1393, 1394, 1426, 1442, 1457, 1513
Offset: 1

Views

Author

Dale Taylor, May 17 2016

Keywords

Comments

Sequence is fairly regular; 437 numbers below 1000 have 4 square roots of 2 mod k; 4796 below 10^4; 47276 below 10^5; and 452172 below 10^6.

Examples

			2 has square roots 11, 45, 74, 108 mod 119; e.g., 11^2 == 2 (mod 119).

Crossrefs

Subsequence of A057126 (Numbers n such that 2 is a square mod n).

Programs

  • Mathematica
    Select[Range[3000], Length@PowerModList[2, 1/2, #] == 4 &]

A381884 Triangle read by rows: T(n, k) = 0 if n = 0 or k is not a quadratic residue modulo n, otherwise T(n, k) = k.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, 0, 3, 0, 1, 0, 0, 4, 0, 1, 0, 0, 4, 5, 0, 1, 0, 3, 4, 0, 6, 0, 1, 2, 0, 4, 0, 0, 7, 0, 1, 0, 0, 4, 0, 0, 0, 8, 0, 1, 0, 0, 4, 0, 0, 7, 0, 9, 0, 1, 0, 0, 4, 5, 6, 0, 0, 9, 10, 0, 1, 0, 3, 4, 5, 0, 0, 0, 9, 0, 11, 0, 1, 0, 0, 4, 0, 0, 0, 0, 9, 0, 0, 12
Offset: 0

Views

Author

Peter Luschny, Mar 17 2025

Keywords

Examples

			Triangle T(n, k) starts:
  [0] 0;
  [1] 0, 1;
  [2] 0, 1, 2;
  [3] 0, 1, 0, 3;
  [4] 0, 1, 0, 0, 4;
  [5] 0, 1, 0, 0, 4, 5;
  [6] 0, 1, 0, 3, 4, 0, 6;
  [7] 0, 1, 2, 0, 4, 0, 0, 7;
  [8] 0, 1, 0, 0, 4, 0, 0, 0, 8;
  [9] 0, 1, 0, 0, 4, 0, 0, 7, 0, 9;
.
Array Arow(n) = [T(j, n), j = 0.. ] starts:
  [0] 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
  [1] 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  [2] 0, 2, 2, 0, 0, 0, 0, 2, 0, 0, ...
  [3] 0, 3, 3, 3, 0, 0, 3, 0, 0, 0, ...
  [4] 0, 4, 4, 4, 4, 4, 4, 4, 4, 4, ...
  [5] 0, 5, 5, 0, 5, 5, 0, 0, 0, 0, ...
  [6] 0, 6, 6, 6, 0, 6, 6, 0, 0, 0, ...
  [7] 0, 7, 7, 7, 0, 0, 7, 7, 0, 7, ...
  [8] 0, 8, 8, 0, 8, 0, 0, 8, 8, 0, ...
  [9] 0, 9, 9, 9, 9, 9, 9, 9, 9, 9, ...
.
3 is not a quadratic residue modulo 7, therefore T(7, 3) = 0.
12 is a quadratic residue modulo 23, therefore T(23, 12) = 12.

Crossrefs

Indices of the nonzero terms in row n of the array: A057126 (row 2), A057125 (row 3), A057762 (row 5), A262931 (row 6), A262932 (row 7).

Programs

  • Maple
    QR := (k, n) -> ifelse(n = 0 or NumberTheory:-QuadraticResidue(k, n) < 0, 0, 1):
T := (n, k) -> k*QR(k, n): seq(seq(T(n, k), k = 0..n), n = 0..12);
Arow := (n, len) -> seq(T(j, n), j=0..len): seq(lprint([n], Arow(n, 9), n=0..9);
  • Mathematica
    QR[n_, k_] := Module[{x, y}, If[Reduce[x^2 == n + k*y, {x, y}, Integers] =!= False, 1, -1]];
T[n_, k_] := If[n == 0 || QR[k, n] < 0, 0, k];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten
  • Python
    from sympy.ntheory import is_quad_residue
def QR(n, k): return is_quad_residue(n, k)
def T(n, k): return 0 if n == 0 or not QR(k, n) else k
for n in range(13): print([T(n, k) for k in range(n + 1)])
Previous Showing 21-24 of 24 results.

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