A081131
a(n) = n^(n-2) * binomial(n,2).
Original entry on oeis.org
0, 0, 1, 9, 96, 1250, 19440, 352947, 7340032, 172186884, 4500000000, 129687123005, 4086546038784, 139788510734886, 5159146026151936, 204350482177734375, 8646911284551352320, 389289535005334947848, 18580248257778920521728, 937146152681201173795569
Offset: 0
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[n lt 2 select 0 else n^(n-2)*Binomial(n,2): n in [0..20]]; // G. C. Greubel, May 18 2021
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Join[{0},Table[n^(n-2) Binomial[n, 2], {n, 1, 20}]] (* Vladimir Joseph Stephan Orlovsky, Apr 19 2011 *)
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[0 if (n<2) else n^(n-2)*binomial(n,2) for n in (0..20)] # G. C. Greubel, May 18 2021
A219324
Positive integers n that are equal to the determinant of the circulant matrix formed by the decimal digits of n.
Original entry on oeis.org
1, 2, 3, 4, 5, 6, 7, 8, 9, 247, 370, 378, 407, 481, 518, 592, 629, 1360, 3075, 26027, 26933, 45018, 69781, 80487, 154791, 1920261, 2137616, 2716713, 3100883, 3480140, 3934896, 4179451, 4830936, 5218958, 11955168, 80651025, 95738203, 257059332, 278945612, 456790123, 469135802, 493827160, 494376160
Offset: 1
| 2 4 7 |
247 = det | 7 2 4 |
| 4 7 2 |
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f[n_] := Det[ NestList[ RotateRight@# &, IntegerDigits@ n, Floor[ Log10[n] + 1] - 1]]; k = 1; lst = {}; While[k < 1120000000, a = f@ k; If[a == k, AppendTo[lst, k]]; k++]; lst (* Robert G. Wilson v, Nov 20 2012 *)
Select[Range[53*10^5],Det[Table[RotateRight[IntegerDigits[#],d],{d,0,IntegerLength[ #]-1}]]==#&] (* The program generates the first 34 terms of the sequence. To generate more, increase the Range constant, but the program will take a long time to run. *) (* Harvey P. Dale, Jul 05 2021 *)
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{ isA219324(n) = local(d,m,r); d=eval(Vec(Str(n))); m=#d; r=Mod(x,polcyclo(m)); prod(j=1,m,sum(i=1,m,d[i]*r^((i-1)*j)))==n }
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from sympy import Matrix
A219324_list = []
for n in range(1,10**4):
s = [int(d) for d in str(n)]
m = len(s)
if n == Matrix(m, m, lambda i, j: s[(i-j) % m]).det():
A219324_list.append(n) # Chai Wah Wu, Oct 18 2021
A303261
Numbers having n digits in base n+1, and equal to the determinant of a circulant matrix based on these digits.
Original entry on oeis.org
1, 28, 35, 1936, 2761, 3421, 3732, 4043, 4354, 281048, 289820, 333293, 420239, 428752, 430686, 437554, 500380, 500888, 736600, 941578, 984377, 1027176, 1069975, 1112774, 1155573, 1662216, 1776201, 2087008, 5331625, 6825024, 7014400
Offset: 1
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for(n=1, 10, for(k=(n+1)^(n-1), (n+1)^n-1, d=Vec(digits(k, n+1)); abs(matdet(matrix(n, n, i, j, d[(j-i)%n+1])))==k&&print1(k", ")))
A303262
Table where row n lists numbers N equal to the determinant of an n X n circulant having as a row the base n+1 digits of N.
Original entry on oeis.org
1, 1, 1, 8, 9, 28, 35, 1, 65, 80, 91, 1, 44, 99, 550, 854, 1936, 2761, 3421, 3732, 4043, 4354, 1, 63, 65, 2527, 3311, 3969, 4095, 13545, 13889, 1, 128, 129, 145, 6066, 16384, 16385, 16512, 16513, 16641, 18560, 18577, 18669, 18705, 90738, 103759, 103965, 109220, 120142, 121920
Offset: 1
The table starts
(n=1) 1,
(n=2) 1,
(n=3) 1, 8, 9, 28, 35,
(n=4) 1, 65, 80, 91,
(n=5) 1, 44, 99, 550, 854, 1936, 2761, 3421, 3732, 4043, 4354,
(n=6) 1, 63, 65, 2527, 3311, 3969, 4095, 13545, 13889,
(n=7) 1, 128, 129, 145, 6066, 16384, 16385, 16512, 16513, 16641, 18560, 18577, 18669, 18705, 90738, 103759, 103965, ...
For example, T(3,1) = 1 because the determinant of the circulant starting with [0, 0, 1] is 1. For the same reason each row starts with 1.
T(3,2) = 8 = 020[4] (digits in base 4) = det(circulant([0, 2, 0])).
T(3,5) = 35 = 203[4] = det(circulant([2, 0, 3])).
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for(n=1,7,for(k=1,(n+1)^n-1,d=Vec(digits(k,n+1),-n);abs(matdet(matrix(n,n,i,j,d[(j-i)%n+1])))==k&&print1(k",")))
Showing 1-4 of 4 results.
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