A193678 -id:A193678 - cp's OEIS frontend

A342545 a(n)^2 is the least square that has exactly n 0's in base n.

2, 24, 16, 280, 216, 3430, 4096, 19683, 100000, 4348377, 2985984, 154457888, 105413504, 4442343750, 4294967296, 313909084845, 198359290368, 8712567840033, 10240000000000, 500396429346030, 584318301411328, 38112390316557080, 36520347436056576, 298023223876953125

Offset: 2

Hugo Pfoertner, Apr 07 2021

			   n    a(n)         a(n)^2   in base n
   2       2              4   100
   3      24            576   210100
   4      16            256   10000
   5     280          78400   10002100
   6     216          46656   1000000
   7    3430       11764900   202000000
   8    4096       16777216   100000000
   9   19683      387420489   1000000000
  10  100000    10000000000   10000000000
  11 4348377 18908382534129   6030000000000
  12 2985984  8916100448256   1000000000000

Chai Wah Wu, Table of n, a(n) for n = 2..702

Cf. A000290, A062971, A193678, A342260, A342546.

for(b=2,12,for(k=1,oo,my(s=k^2,v=digits(s,b));if(sum(k=1,#v,v[k]==0)==b,print1(k,", ");break)))

from numba import njit
@njit # works with 64 bits through a(14)
def digits0(n, b):
  count0 = 0
  while n >= b:
    n, r = divmod(n, b)
    count0 += (r==0)
  return count0 + (n==0)
from sympy import integer_nthroot
def a(n):
  an = integer_nthroot(n**n, 2)[0]
  while digits0(an*an, n) != n: an += 1
  return an
print([a(n) for n in range(2, 13)]) # Michael S. Branicky, Apr 07 2021

from itertools import product
from functools import reduce
from sympy.utilities.iterables import multiset_permutations
from sympy import integer_nthroot
def A342545(n):
    for a in range(1,n):
        p, q = integer_nthroot(a*n**n,2)
        if q: return p
    l = 1
    while True:
        cmax = n**(l+n+1)
        for a in range(1,n):
            c = cmax
            for b in product(range(1,n),repeat=l):
                for d in multiset_permutations((0,)*n+b):
                    p, q = integer_nthroot(reduce(lambda c, y: c*n+y, [a]+d),2)
                    if q: c = min(c,p)
            if c < cmax:
                return c
        l += 1 # Chai Wah Wu, Apr 07 2021

a(2*n) = A062971(n) = 2*A193678(n).

More terms from Chai Wah Wu, Apr 07 2021

A384075 a(n) = neg(M(n)), where M(n) is the n X n circulant matrix with (row 1) = (1,3,5,7, ..., 2n - 1), and neg(M(n)) is the negative part of the determinant of M(n); see A380661.

Original entry on oeis.org

0, -9, -45, -4716, -200200, -20916552, -2462535768, -406262340288, -84096850828032, -21708790967664000, -6808563893605222144, -2552145158372103507456, -1126589571631974396251136, -578462264691449080954733568, -341831891354409385226121600000

Offset: 1

Clark Kimberling, May 22 2025

sign

			The rows of M(4) are (1,3,5,7), (7,1,3,5), (5,7,1,3), (3,5,7,1); determinant(M(4)) = -4716; permanent(M(4)) = 2668, so neg(M(4)) = (-2048 - 7384)/2 = -4716 and pos(M(4)) = (-2048 + 7384)/2 = 2668.

Cf. A193678 (determinant), A384074 (permanent), A380661, A384076, A384077, A384078.

z = 19;
v[n_] := Table[2 k + 1, {k, 0, n - 1}];
u[n_] := Table[RotateRight[#, k - 1], {k, 1, Length[#]}] &[v[n]];
p = Table[Simplify[Permanent[u[n]]], {n, 1, z}]   (* A384074  *)
d = Table[Simplify[Det[u[n]]], {n, 1, z}]  (* A193678, with alternating signs *)
neg = (d - p)/2   (* A384075 *)
pos = (d + p)/2   (* A384076 *)

a(n) = (1/2)*((-1)^n*A193678(n) - A384074(n)).

A193681 Discriminant of minimal polynomial of 2*cos(Pi/n) (see A187360).

Original entry on oeis.org

1, 1, 1, 8, 5, 12, 49, 2048, 81, 2000, 14641, 2304, 371293, 1075648, 1125, 2147483648, 410338673, 1259712, 16983563041, 1024000000, 453789, 2414538435584, 41426511213649, 1358954496, 762939453125, 7340688973975552, 31381059609, 4739148267126784, 10260628712958602189, 324000000

Offset: 1

Wolfdieter Lang, Sep 13 2011

For the discriminant of a polynomial in terms of the square of a determinant of a Vandermonde matrix build from the zeros of the polynomial see, e.g., A127670.
  The zeros of the polynomials C(n,x) with coefficient triangle A187360 are given there in a comment.
The discriminant of the monic C(n,x) polynomial can also be computed from its zeros x_i and the derivative of C, via (-1)^binomial(delta(n),2)*product(C'(n,x)|_{x=x_i},i=1..delta(n)), with the degree delta(n)=A055034(n). For a reference see, e.g., Rivlin, p. 218, quoted in A127670.

Robert Israel, Table of n, a(n) for n = 1..500
Ed Pegg Jr, Table illustrating A193681 (Each box gives n, degree (A055034), and determinant (this sequence).)

Cf. A055034, A127670, A187360, A193678.

g:= proc(n) local P,z,j;
   P:= factor(evala(Norm(z-convert(2*cos(Pi/n),RootOf))));
   if type(P,`^`) then P:= op(1,P) fi;
   discrim(P,z)
end proc:
map(g, [$1..100]); # Robert Israel, Aug 04 2015

Table[NumberFieldDiscriminant[Cos[Pi/m]], {m, 1, z}]  (* Clark Kimberling, Aug 03 2015 *)

a(n) = discriminant(C(n,x)), n>=1, with C(n,x):=sum(A187360(n,m)*x^m,m=0..A055034(n)), the minimal polynomial of 2*cos(Pi/n).

A317403 a(n)=(-1)^((n-2)(n-1)/2)2^(n-1)*n^(n-3).

Original entry on oeis.org

1, 1, -4, -32, 400, 6912, -153664, -4194304, 136048896, 5120000000, -219503494144, -10567230160896, 564668382613504, 33174037869887488, -2125764000000000000, -147573952589676412928, 11034809241396899282944, 884295678882933431599104, -75613185918270483380568064

Offset: 1

Rigoberto Florez, Aug 26 2018

sign

Discriminant of Fibonacci polynomials.

Fibonacci polynomials are defined as F(0)=0, F(1)=1 and F(n)=x*F(n-1)+F(n-2) for n>1. Coefficients are given in triangle A168561 with offset 1.

Rigoberto Flórez, Robinson Higuita, and Alexander Ramírez, The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials, arXiv:1808.01264 [math.NT], 2018.
Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Star of David and other patterns in the Hosoya-like polynomials triangles, 2018.
R. Flórez, R. Higuita and A. Mukherjees, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.
R. Flórez, N. McAnally, and A. Mukherjees, Identities for the generalized Fibonacci polynomial, Integers, 18B (2018), Paper No. A2.
Eric Weisstein's World of Mathematics, Discriminant
Eric Weisstein's World of Mathematics, Fibonacci Polynomial

Cf. A006645, A001629, A001871, A006645, A007701, A045618, A045925, A093967, A168561, A193678, A317404, A317405, A317408, A317451, A318184, A318197.

Essentially the same as A127670.

[(-1)^((n-2)*(n-1) div 2)*2^(n-1)*n^(n-3): n in [1..20]]; // Vincenzo Librandi, Aug 27 2018

Array[(-1)^((#-2)*(#-1)/2)*2^(#-1)*#^(#-3)&,20]

concat([1], [poldisc(p) | p<-Vec(x/(1-x^2-y*x) - x + O(x^20))]) \\ Andrew Howroyd, Aug 26 2018

A317450 a(n)=(-1)^((n-2)(n-1)/2)2^((n-1)^2)*n^(n-3).

Original entry on oeis.org

1, 1, -16, -2048, 1638400, 7247757312, -164995463643136, -18446744073709551616, 9803356117276277820358656, 24178516392292583494123520000000, -271732164163901599116133024293512544256, -13717048991958695477963985711266803110069141504, 3074347100178259797134292590832254504315406543889629184

Offset: 1

Rigoberto Florez, Aug 26 2018

sign

Discriminant of Pell polynomials.

Pell polynomials are defined as P(0)=0, P(1)=1 and P(n)=2xP(n-1)+P(n-2) for n>1.

Rigoberto Flórez, Robinson Higuita, and Alexander Ramírez, The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials, arXiv:1808.01264 [math.NT], 2018.
Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Star of David and other patterns in the Hosoya-like polynomials triangles, 2018.
R. Flórez, N. McAnally, and A. Mukherjees, Identities for the generalized Fibonacci polynomial, Integers, 18B (2018), Paper No. A2.
R. Flórez, R. Higuita and A. Mukherjees, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.
Eric Weisstein's World of Mathematics, Discriminant
Eric Weisstein's World of Mathematics, Pell Polynomial

Cf. A006645, A001629, A001871, A006645, A007701, A045618, A045925, A093967, A193678, A317404, A317405, A317408, A317451, A318184, A318197.

Essentially the same as A086804.

Array[(-1)^((#-2)*(#-1)/2)* 2^((#-1)^2)*#^(#-3)&,15]

cp's OEIS Frontend

A342545 a(n)^2 is the least square that has exactly n 0's in base n.

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A384075 a(n) = neg(M(n)), where M(n) is the n X n circulant matrix with (row 1) = (1,3,5,7, ..., 2n - 1), and neg(M(n)) is the negative part of the determinant of M(n); see A380661.

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A193681 Discriminant of minimal polynomial of 2*cos(Pi/n) (see A187360).

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A317403 a(n)=(-1)^((n-2)*(n-1)/2)*2^(n-1)*n^(n-3).

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A317450 a(n)=(-1)^((n-2)*(n-1)/2)*2^((n-1)^2)*n^(n-3).

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Mathematica

A317403 a(n)=(-1)^((n-2)(n-1)/2)2^(n-1)*n^(n-3).

A317450 a(n)=(-1)^((n-2)(n-1)/2)2^((n-1)^2)*n^(n-3).