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A353196 Number of stabilizer states on n qubits.

6, 60, 1080, 36720, 2423520, 315057600, 81284860800, 41780418451200, 42866709330931200, 87876754128408960000, 360118938418219918080000, 2950814581398894008747520000, 48352047730802277227336862720000, 1584496604138390624739828991334400000

Offset: 1

James Rayman, Apr 29 2022

A stabilizer state is a quantum state on n qubits prepared by applying a series of Hadamard, CNOT, and S gates to the all-zero state. There are only a finite number of such states for any n.

			For n = 1, the a(1) = 6 states are |0>, |1>, |+>, |->, |i>, and |-i>.

D. Gross, Hudson's Theorem for finite-dimensional quantum systems, arXiv:quant-ph/0602001, 2006-2007.

Cf. A000079, A003956, A028362.

Table[2^n * QPochhammer[-2, 2, n], {n, 13}] (* Amiram Eldar, Aug 17 2025 *)

def a(n):
    ans = 2 ** n
    for i in range(1, n+1):
        ans *= 2 ** i + 1
    return ans

from math import prod
def A353196(n): return prod((1<Chai Wah Wu, Jun 20 2022

a(n) = 2^n*Product_{i=1..n} (2^i+1).
a(n) = A000079(n)*A028362(n+1).
a(n) ~ c * 2^(n*(n+3)/2) where c = Product_{k>=1} (1 + 1/2^k) = A079555. - Amiram Eldar, Aug 17 2025

A342627 Second differences of the y-coordinates of the vertices of the Babylonian Wurm.

Original entry on oeis.org

0, -1, -1, -1, -1, 0, 0, -1, 0, 1, 1, -1, 2, -1, -2, -1, -1, 0, 0, 1, -1, -1, 0, 1, -1, 0, 1, -2, 0, 0, 2, 3, -2, 1, 4, 1, 1, 3, -2, 1, 3, 1, 2, 0, -1, 1, -2, 2, -1, 2, 0, 0, -3, -1, -1, -2, -4, -1, -4, 3, -5, 4, -3, 2, -5, -1, -2, 1, -2, 0, 1, 2, -1, -1, -1

Offset: 1

James Rayman, Apr 18 2021

Equivalently, the first differences of A342625.

See A256175 for the definition of the Babylonian Wurm.
See A342622 and A342623 for the coordinates of the Wurm.
See A342624 and A342625 for the first differences.
See A342626 for the second differences of the x-coordinates.

for i in range(10000): print(i+1, wurm(i+2)[1] - 2*wurm(i+1)[1] + wurm(i)[1]) # wurm is defined in the program for A256175

a(n) = A342623(n+2) - 2*A342623(n+1) + A342623(n).

A342626 Second differences of the x-coordinates of the vertices of the Babylonian Wurm.

Original entry on oeis.org

1, 1, 0, 0, -2, -1, -1, 2, 1, 2, 1, 0, 1, 0, -1, -1, -3, -1, -1, -2, 1, 3, 1, 3, -2, 1, 2, -5, -1, -1, -4, -2, 1, -1, -1, 0, 0, 1, -1, 0, 2, 1, 5, -1, -3, 2, -4, 3, -2, 6, 1, 1, 6, 1, 1, 1, 1, 0, -1, 1, -2, 2, -1, 1, -3, -1, -3, 2, -5, 1, 3, 3, -1, -1, -2, -5

Offset: 1

James Rayman, Apr 17 2021

Equivalently, the first differences of A342624.

See A256175 for the definition of the Babylonian Wurm.
See A342622 and A342623 for the coordinates of the Wurm.
See A342624 and A342625 for the first differences.
See A342627 for the second differences of the y-coordinates.

for i in range(10000): print(i+1, wurm(i+2)[0] - 2*wurm(i+1)[0] + wurm(i)[0]) # wurm is defined in the program for A256175

a(n) = A342622(n+2) - 2*A342622(n+1) + A342622(n).

A343484 Number of equivalence classes of length n squarefree reduced words over {0, 1, 2, 3} under action of renaming symbols.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 13, 18, 27, 41, 62, 90, 134, 198, 293, 423, 619, 908, 1329, 1938, 2832, 4142, 6061, 8824, 12879, 18794, 27425, 39977, 58333, 85109, 124180, 180994, 263931, 384933, 561402, 818617, 1193841, 1740980, 2538896, 3702022, 5398458, 7872351

Offset: 0

James Rayman, Apr 16 2021

nonn

Any equivalence class of words with length at least 2 has exactly 8 members.

			a(5) = 5 corresponds to the 5 words 01210, 01030, 01230, 01032, and 01232.

Richard A. Dean, A sequence without repeats on x, x^{-1}, y, y^{-1}, Amer. Math. Monthly 72, 1965. pp. 383-385. MR 31 #350.
Tero Harju, Avoiding Square-Free Words on Free Groups, arXiv:2104.06837 [math.CO], 2021.

Cf. A215075, A343421.

a(n) = A343421(n)/8 for n >= 2.

A342625 The y-coordinates of the vectors used to construct the Babylonian Wurm.

Original entry on oeis.org

1, 1, 0, -1, -2, -3, -3, -3, -4, -4, -3, -2, -3, -1, -2, -4, -5, -6, -6, -6, -5, -6, -7, -7, -6, -7, -7, -6, -8, -8, -8, -6, -3, -5, -4, 0, 1, 2, 5, 3, 4, 7, 8, 10, 10, 9, 10, 8, 10, 9, 11, 11, 11, 8, 7, 6, 4, 0, -1, -5, -2, -7, -3, -6, -4, -9

Offset: 1

James Rayman, Apr 09 2021

sign

Equivalently, this sequence gives the differences of the y-coordinates of successive vertices of the Babylonian Wurm.

See A256175 for the definition of the Babylonian Wurm.
See A342624 for the x-coordinates.
Cf. A342622, A342623, A342626, A342627.

for i in range(10000): print(i+1, wurm(i+1)[1] - wurm(i)[1]) # wurm is defined in the program for A256175.

a(n) = A342623(n+1) - A342623(n).

A342624 The x-coordinates of the vectors used to construct the Babylonian Wurm.

Original entry on oeis.org

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

Offset: 1

James Rayman, Apr 09 2021

Equivalently, this sequence gives the differences of the x-coordinates of successive vertices of the Babylonian Wurm.

See A256175 for the definition of the Babylonian Wurm.
See A342625 for the y-coordinates.
Cf. A342622, A342623, A342626, A342627.

for i in range(10000): print(i+1, wurm(i+1)[0] - wurm(i)[0]) # wurm is defined in the program for A256175.

a(n) = A342622(n+1) - A342622(n).

A342622 The x-coordinates of the vertices of the Babylonian Wurm.

Original entry on oeis.org

0, 0, 1, 3, 5, 7, 7, 6, 4, 4, 5, 8, 12, 16, 21, 26, 30, 33, 33, 32, 30, 26, 23, 23, 24, 28, 30, 33, 38, 38, 37, 35, 29, 21, 14, 6, -3, -12, -21, -29, -38, -47, -54, -60, -61, -63, -68, -71, -78, -82, -88, -88, -87, -85, -77, -68, -58, -47, -35

Offset: 1

James Rayman, Mar 16 2021

See A256175 for the definition of the Babylonian Wurm.

See A342623 for the y-coordinates.

for i in range(10000): print(i+1, wurm(i)[0]) # wurm is defined in the program for A256175.

A342623 The y-coordinates of the vertices of the Babylonian Wurm.

Original entry on oeis.org

0, 1, 2, 2, 1, -1, -4, -7, -10, -14, -18, -21, -23, -26, -27, -29, -33, -38, -44, -50, -56, -61, -67, -74, -81, -87, -94, -101, -107, -115, -123, -131, -137, -140, -145, -149, -149, -148, -146, -141, -138, -134, -127, -119, -109, -99, -90, -80

Offset: 1

James Rayman, Mar 17 2021

See A256175 for the definition of the Babylonian Wurm.

See A342622 for the x-coordinates.

for i in range(10000): print(i+1, wurm(i)[1]) # wurm is defined in the program for A256175.

A340727 a(n) is the smallest integer that can be written as a product of n distinct integers > 1 in at least two different ways.

Original entry on oeis.org

12, 48, 240, 1440, 8640, 60480, 604800, 5443200, 59875200, 718502400, 9340531200, 124540416000, 1743565824000, 29640619008000, 502146957312000, 8536498274304000, 162193467211776000, 3406062811447296000, 68121256228945920000, 1498667637036810240000

Offset: 2

James Rayman, Jan 17 2021

nonn

			a(2) = 12 since 12 = 2*6 = 3*4.
a(4) = 240 since 240 = 2*3*4*10 = 2*3*5*8.

James Rayman, Table of n, a(n) for n = 2..400

Cf. A081957.

from heapq import *
import math
def a(n):
    prev, visited, v = 0, set(), list(range(2, n+2))
    pq = [(math.factorial(n+1), v)]
    while True:
        prod, v = heappop(pq)
        if tuple(v) in visited: continue
        visited.add(tuple(v))
        if prev != prod: prev = prod
        else: return prod
        for i in range(n):
            if i == n-1 or v[i] + 1 < v[i+1]:
                u = v[:]
                u[i] += 1
                heappush(pq, (prod // v[i] * u[i], u))

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User: James Rayman

A353196 Number of stabilizer states on n qubits.

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A342627 Second differences of the y-coordinates of the vertices of the Babylonian Wurm.

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A342626 Second differences of the x-coordinates of the vertices of the Babylonian Wurm.

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A343484 Number of equivalence classes of length n squarefree reduced words over {0, 1, 2, 3} under action of renaming symbols.

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A342625 The y-coordinates of the vectors used to construct the Babylonian Wurm.

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A342624 The x-coordinates of the vectors used to construct the Babylonian Wurm.

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A342622 The x-coordinates of the vertices of the Babylonian Wurm.

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A342623 The y-coordinates of the vertices of the Babylonian Wurm.

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