A172225 -id:A172225 - cp's OEIS frontend

A201510 Number of ways to place 9 nonattacking wazirs on an n X n board.

0, 0, 0, 0, 2578, 1247116, 97284860, 2817340064, 44218721793, 457851259868, 3506596268191, 21355746900992, 108582220087480, 477032549147428, 1857084405493128, 6529640029479296, 21044674478336823, 62903854631232636, 176034055470126073, 464793685059669728

Offset: 1

Vaclav Kotesovec, Dec 02 2011

nonn

Wazir is a leaper [0,1].

V. Kotesovec, Non-attacking chess pieces

Cf. A172225, A172226, A172227, A172228, A178409, A201507, A201508.

Explicit formula: n^18/362880 - n^16/2016 + n^15/2520 + 349*n^14/8640 - 23*n^13/360 - 277*n^12/144 + 163*n^11/36 + 199529*n^10/3456 - 4381*n^9/24 - 313811*n^8/288 + 1622087*n^7/360 + 1073654363*n^6/90720 - 12207881*n^5/180 - 24979477*n^4/504 + 72278641*n^3/126 - 11491519*n^2/45 - 6271604*n/3 + 2530368, n>=8.
G.f.: x^5*(14*x^21 - 226*x^20 + 2514*x^19 - 15414*x^18 + 54363*x^17 - 241813*x^16 + 1440666*x^15 - 4412622*x^14 - 2699713*x^13 + 64333547*x^12 - 202456488*x^11 + 209746960*x^10 + 407620979*x^9 - 1743413585*x^8 + 2469587594*x^7 - 1465834094*x^6 - 9995512037*x^5 - 6126508561*x^4 - 1179686478*x^3 - 74030494*x^2 - 1198134*x - 2578)/(x-1)^19.
a(n) = A232833(n,9). - R. J. Mathar, Apr 11 2024

A239352 van Heijst's upper bound on the number of squares inscribed by a real algebraic curve in R^2 of degree n, if the number is finite.

Original entry on oeis.org

0, 0, 1, 12, 48, 130, 285, 546, 952, 1548, 2385, 3520, 5016, 6942, 9373, 12390, 16080, 20536, 25857, 32148, 39520, 48090, 57981, 69322, 82248, 96900, 113425, 131976, 152712, 175798, 201405, 229710, 260896, 295152, 332673, 373660, 418320, 466866, 519517

Offset: 0

Jonathan Sondow, Mar 21 2014

In 1911 Toeplitz conjectured the Square Peg (or Inscribed Square) Problem: Every continuous simple closed curve in the plane contains 4 points that are the vertices of a square. The conjecture is still open. Many special cases have been proved; see Matschke's beautiful 2014 survey.

Recently van Heijst proved that any real algebraic curve in R^2 of degree d inscribes either at most (d^4 - 5d^2 + 4d)/4 or infinitely many squares. He conjectured that a generic complex algebraic plane curve inscribes exactly (d^4 - 5d^2 + 4d)/4 squares.

			A point or a line has no inscribed squares, so a(0) = a(1) = 0.
A circle has infinitely many inscribed squares, and an ellipse that is not a circle has exactly one, agreeing with a(2) = 1.
G.f. = x^2 + 12*x^3 + 48*x^4 + 130*x^5 + 285*x^6 + 546*x^7 + 952*x^8 + ...

Otto Toeplitz, Über einige Aufgaben der Analysis situs, Verhandlungen der Schweizerischen Naturforschenden Gesellschaft in Solothurn, 4 (1911), 197.

G. C. Greubel, Table of n, a(n) for n = 0..2500
Wouter van Heijst, The algebraic square peg problem, arXiv:1403.5979 [math.AG], 2014.
Wouter van Heijst, The algebraic square peg problem, Master’s thesis, Aalto University, 2014.
Benjamin Matschke, A Survey on the Square Peg Problem, AMS Notices, 61 (2014), 346-352.
Benjamin Matschke, Extended Survey on the Square Peg Problem, Max Planck Institute for Mathematics, 2014.
Sequences related to inscribed squares
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A088544, A089058, A123673, A123697, A209432, A231739.

[(n^4 - 5*n^2 + 4*n)/4: n in [0..50]]; // G. C. Greubel, Aug 07 2018

Table[(n^4 - 5 n^2 + 4 n)/4, {n, 0, 38}]

for(n=0,50, print1((n^4 - 5*n^2 + 4*n)/4, ", ")) \\ G. C. Greubel, Aug 07 2018

a(n) = (n^4 - 5*n^2 + 4*n)/4 = n*(n - 1)*(n^2 + n - 4)/4 = A000217(n-1)*A034856(n-1), which shows the formula is an integer.
G.f.: x^2 * (1 + 7*x - 2*x^2) / (1 - x)^5. - Michael Somos, Mar 21 2014
a(n) = A172225(n)/2. - R. J. Mathar, Jan 09 2018

A381854 Triangle read by rows: T(n, k) is the number of invertible n X n matrices over GF(2) that can be optimally row-reduced in k steps, n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 1, 6, 24, 51, 60, 24, 2, 1, 12, 96, 542, 2058, 5316, 7530, 4058, 541, 6, 1, 20, 260, 2570, 19680, 117860, 540470, 1769710, 3571175, 3225310, 736540, 15740, 24, 1, 30, 570, 8415, 101610, 1026852, 8747890, 61978340, 355193925, 1561232840, 4753747050, 8111988473, 4866461728, 437272014, 949902, 120

Offset: 0

Søren Fuglede Jørgensen, Mar 08 2025

Using transvections as the generating set of the matrix group, this is the number of inequivalent minimal words in k generators; the number of elements at distance k from the identity in the corresponding Cayley graph.

Also the number of different elements that can be represented by minimal quantum circuits of k CNOT gates on n qubits.

			Triangle begins:
   n\k  0    1    2    3    4    5    6    7    8    9
   0:   1
   1:   1
   2:   1    2    2    1
   3:   1    6   24   51   60   24    2
   4:   1   12   96  542 2058 5316 7530 4058  541    6
   ...
For n = 2, k = 1, the two matrices are [[1, 1], [0, 1]] and [[1, 0], [1, 1]].
For n = 2, k = 2, the two matrices are [[1, 1], [1, 0]] and [[0, 1], [1, 1]].
For n = 2, k = 3, the only matrix is [[0, 1], [1, 0]].

Søren Fuglede Jørgensen, Table of n, a(n) for n = 0..83
Marc Bataille, Quantum Circuits of CNOT gates: Optimization and Entanglement, Quantum Information Processing, 21(7):269 (2022).
Jens Emil Christensen, Søren Fuglede Jørgensen, Andreas Pavlogiannis, and Jaco van de Pol, On Exact Sizes of Minimal CNOT Circuits, RC 2025, LNCS, vol 15716, pp. 71-88; arXiv:2503.01467 [quant-ph] (2025).
Ketan N. Patel, Igor L. Markov, and John P. Hayes, Optimal synthesis of linear reversible circuits, Quantum Info. Comput., 8(3) (2008), pp. 282-294.

Cf. A002378 (column 1), A172225 (column 2), A002884 (row sums).

T(n, 0) = 1.
T(n, 1) = n^2 - n.
T(n, 2) = (1/2)*(n^4 - 5*n^2 + 4*n).
T(n, 3) = (1/6)*(n^6 + 3*n^5 - 9*n^4 - 63*n^3 + 179*n^2 - 111*n).
Sum_{k>=0} T(n,k) = A002884(n).

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A201510 Number of ways to place 9 nonattacking wazirs on an n X n board.

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A239352 van Heijst's upper bound on the number of squares inscribed by a real algebraic curve in R^2 of degree n, if the number is finite.

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A381854 Triangle read by rows: T(n, k) is the number of invertible n X n matrices over GF(2) that can be optimally row-reduced in k steps, n >= 0, k >= 0.

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