A177894 -id:A177894 - cp's OEIS frontend

A299792 Numbers k such that A177894(k) = 0.

0, 11, 22, 33, 44, 55, 66, 77, 88, 99, 111, 222, 333, 444, 555, 666, 777, 888, 999, 1001, 1010, 1012, 1023, 1034, 1045, 1056, 1067, 1078, 1089, 1100, 1111, 1122, 1133, 1144, 1155, 1166, 1177, 1188, 1199, 1210, 1212, 1221, 1232, 1243, 1254, 1265, 1276, 1287, 1298, 1313

Offset: 1

Jianing Song, Jan 21 2019

A one-digit number a is in this sequence if and only if a = 0.
A two-digit number ab is in this sequence if and only if a = b.
A three-digit number abc is in this sequence if and only if a = b = c.
A four-digit number abcd is in this sequence if and only if a + c = b + d or (a = c and b = d)
A239019 is trivially a subsequence (because the corresponding circular matrices each contains at least two identical rows or columns). {a(n)} \ A239019 is given as A317291.

			1452 is a term because the value of the following determinant is 0:
  | 1 4 5 2 |
  | 4 5 2 1 |
  | 5 2 1 4 |
  | 2 1 4 5 |

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Circulant Matrix
Wikipedia, Circulant matrix

Cf. A177894, A239019, A317291.

A299792Q[k_] := k == 0 || Det[NestList[RotateLeft, IntegerDigits[k], IntegerLength[k]-1]] == 0; Select[Range[0, 2000], A299792Q] (* Paolo Xausa, Mar 11 2024 *)

for(n=0, 1500, if(!A177894(n), print1(n, ", "))) \\ See A177894 for its program

A317291 Numbers k not in A239019 such that A177894(k) = 0.

Original entry on oeis.org

0, 1001, 1012, 1023, 1034, 1045, 1056, 1067, 1078, 1089, 1100, 1122, 1133, 1144, 1155, 1166, 1177, 1188, 1199, 1210, 1221, 1232, 1243, 1254, 1265, 1276, 1287, 1298, 1320, 1331, 1342, 1353, 1364, 1375, 1386, 1397, 1430, 1441, 1452, 1463, 1474, 1485, 1496

Offset: 1

Jianing Song, Jan 21 2019

Of course A177894(A239019(n)) = 0 because the corresponding circular matrices each contains at least two identical rows or columns. This sequence gives the other numbers.

A four-digit number abcd is in this sequence if and only if a + c = b + d and (a != c or b != d).

			2178 is a term because the value of the following determinant is 0, although the determinant itself contains no identical rows or columns:
| 2 1 7 8 |
| 1 7 8 2 |
| 7 8 2 1 |
| 8 2 1 7 |

Eric Weisstein's World of Mathematics, Circulant Matrix.
Wikipedia, Circulant matrix.

Cf. A177894, A239019, A299792.

for(n=0, 1500, if(!n ||(n&!A177894(n)&&!A239019(n)),print1(n, ", "))) \\ See A177894 and A239019 for their programs

A306593 Least number k such that the determinant of the circulant matrix formed by its decimal digits is equal to n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 334, 65, 42, 76, 455, 41, 40, 98, 123, 667, 64, 52, 778, 788, 51, 50, 899, 63, 86, 7787, 2025885, 8788, 62, 74, 46996, 61, 60, 66898, 67997, 85, 73, 78998, 88899, 88999, 335, 72, 4579975, 878888, 71, 70, 10243, 5354, 355, 989999, 114

Offset: 0

Paolo P. Lava, Feb 27 2019

Here only the least numbers are listed: e.g., a(75) = 1031, even if 10002110 also produces 75.

The sequence is infinite because any number of the form (91*10^n - 10) / 90 for n > 0 (A267623 or A283508) has the determinant of the circulant matrix equal to n but, in general, it is not the least possible term. - Giovanni Resta, Mar 06 2019

			                        | 3 3 4 |
a(10) = 334 because det | 4 3 3 | = 10
                        | 3 4 3 |
.
and 334 is the least number to have this property.
.
                          | 4 6 9 9 6 |
                          | 6 4 6 9 9 |
a(34) = 46996 because det | 9 6 4 6 9 | = 34
                          | 9 9 6 4 6 |
                          | 6 9 9 6 4 |
.
and 46996 is the least number to have this property.

Paolo P. Lava, Table of n, a(n) for n = 0..100

Cf. A177894, A219324, A267623, A283508, A306662, A323485, A323486.

with(linalg): P:=proc(q) local a,b,c,d,j,k,i,n,t;
print(0); for i from 1 to q do for n from 1 to q do
d:=ilog10(n)+1; a:=convert(n, base, 10); c:=[];
for k from 1 to nops(a) do c:=[op(c), a[-k]]; od; t:=[op([]), c];
for k from 2 to d do b:=[op([]), c[nops(c)]];
for j from 1 to nops(c)-1 do
b:=[op(b), c[j]]; od;  c:=b; t:=[op(t), c]; od;
if i=det(t) then print(n); break; fi; od; od; end: P(10^7);

md(n) = my(d = if (n, digits(n), [0])); matdet(matrix(#d, #d, i, j, d[1+lift(Mod(j-i, #d))]));
a(n) = my(k=0); while(md(k) != n, k++); k; \\ Michel Marcus, Mar 20 2019

A177894(a(n)) = n when a(n) >= 0. - Rémy Sigrist, Feb 27 2019

A257587 If n = abcd... in decimal, a(n) = a^2 - b^2 + c^2 - d^2 + ...

Original entry on oeis.org

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 1, 0, -3, -8, -15, -24, -35, -48, -63, -80, 4, 3, 0, -5, -12, -21, -32, -45, -60, -77, 9, 8, 5, 0, -7, -16, -27, -40, -55, -72, 16, 15, 12, 7, 0, -9, -20, -33, -48, -65, 25, 24, 21, 16, 9, 0, -11, -24, -39, -56, 36, 35, 32

Offset: 0

N. J. A. Sloane, May 10 2015

Michel Marcus, Table of n, a(n) for n = 0..10000

First 100 terms coincide with those of A177894, but then they diverge.

Cf. A257588, A257796, A352535 (indices of zeros).

A257587[n_] := Total[-(-1)^Range[Max[IntegerLength[n], 1]]*IntegerDigits[n]^2];
Array[A257587, 100, 0] (* Paolo Xausa, Mar 11 2024 *)

a(n) = my(d=digits(n)); sum(k=1, #d, (-1)^(k+1)*d[k]^2); \\ Michel Marcus, Jul 12 2022

def a(n): return sum(int(d)**2*(-1)**i for i, d in enumerate(str(n)))
print([a(n) for n in range(63)]) # Michael S. Branicky, Jul 11 2022

a(A352535(n)) = 0. - Bernard Schott, Jul 12 2022

A306595 Determinant of the circulant matrix whose first column corresponds to the binary digits of n.

Original entry on oeis.org

0, 1, 1, 0, 1, 2, 2, 0, 1, 0, 0, 3, 0, -3, 3, 0, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 0, 1, 0, 4, 0, 0, -9, 9, 0, 4, 9, 0, 8, 9, 0, 8, 5, 0, 0, 9, 0, -9, -8, 0, -5, 0, 0, 8, 5, 0, -5, 5, 0, 1, 2, 2, 3, 2, 24, 24, 4, 2, 3, 3, 32, 3, 4, 32, 5, 2, 24, 3

Offset: 0

Rémy Sigrist, Feb 27 2019

This sequence is the binary variant of A177894.
From Robert Israel, Mar 05 2019: (Start)
a(n) is divisible by A000120(n).
If A070939(n) is even then n is divisible by A000120(n)*A065359(n). (End)

			For n = 13:
- the binary representation of 13 is "1101",
- the corresponding circulant matrix is:
    [1 1 0 1]
    [1 1 1 0]
    [0 1 1 1]
    [1 0 1 1]
- its determinant is -3,
- hence a(13) = -3.

Robert Israel, Table of n, a(n) for n = 0..10000
Wikipedia, Circulant matrix
Index entries for sequences related to binary expansion of n

Cf. A000120, A065359, A070939, A121016, A177894, A219325, A306714.

a:= n-> `if`(n=1, 1, (l-> LinearAlgebra[Determinant](Matrix(nops(l),
       shape=Circulant[l[-i]$i=1..nops(l)])))(convert(n, base, 2))):
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 05 2019

a(n) = my (d=if (n, binary(n), [0])); my (m=matrix(#d, #d, i,j, d[1+(i-j)%#d])); return (matdet(m))

a(A121016(n)) = 0 for any n > 0.
a(2^k) = 1 for any k >= 0.
a(A219325(n)) = A219325(n) for any n > 0.

A307651 a(n) is the determinant of the Vandermonde matrix of the digits of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, -7, -6, -5, -4, -3, -2, -1, 0

Offset: 0

Rémy Sigrist, Apr 20 2019

			             | 2^0 2^1 2^2 |
a(234) = det | 3^0 3^1 3^2 | = 2.
             | 4^0 4^1 4^2 |

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Wikipedia, Vandermonde matrix

See A307710 for the factorial base variant.

Cf. A007376, A010784, A177894, A307586, A307587.

a(n) = my (d=digits(n)); matdet(matrix(#d,#d,r,c,d[r]^(c-1)))

a(n) != 0 iff n belongs to A010784.

a(n) = 0 for any n > 9876543210.

A306598 Determinant of the circulant matrix whose first column corresponds to the divisors of n.

Original entry on oeis.org

1, -3, -8, 49, -24, -960, -48, -3375, 676, -8640, -120, -2247392, -168, -34560, -46080, 923521, -288, -28789488, -360, -54867456, -184320, -216000, -528, -89384770560, 15376, -423360, -512000, -438939648, -840, -558786571200, -960, -992436543, -1152000

Offset: 1

Rémy Sigrist, Feb 27 2019

From Robert Israel, Mar 06 2019: (Start)
a(n) is divisible by A000203(n).
If n is not a square, a(n) is divisible by A000203(n)*A071324(n).
(End)

			For n = 12:
- the divisors of 12 are: 1, 2, 3, 4, 6, 12,
- the corresponding circulant matrix is:
    [ 1 12  6  4  3  2]
    [ 2  1 12  6  4  3]
    [ 3  2  1 12  6  4]
    [ 4  3  2  1 12  6]
    [ 6  4  3  2  1 12]
    [12  6  4  3  2  1]
- its determinant is -2247392,
- hence, a(12) = -2247392.

Robert Israel, Table of n, a(n) for n = 1..10000
Wikipedia, Circulant matrix

Cf. A027750, A086459, A177894.

f:= proc(n) local F,d; uses numtheory, LinearAlgebra;
  F:= sort(convert(divisors(n),list));
  d:= nops(F);
  Determinant(Matrix(d,d,shape=Circulant[F]))
end proc:
map(f, [$1..100]); # Robert Israel, Mar 06 2019

a[n_] := Module[{dd = Divisors[n], m, r}, m = Length[dd]; r = E^(2 Pi I/m); Product[Sum[dd[[j+1]] r^(j k), {j, 0, m-1}], {k, 0, m-1}] // FullSimplify];
Array[a, 100] (* Jean-François Alcover, Oct 17 2020 *)

a(n) = my (d=divisors(n)); my (m=matrix(#d, #d, i,j, d[1+(i-j)%#d])); return (matdet(m))

Apparently, a(n) > 0 iff n is a square.
a(p) = p^2 - 1 for any prime number p.
a(p^2) = p^6 - 2*p^3 + 1 for any prime number p.
a(2^k) = A086459(k+1) for any k >= 0.
If p < q are primes, a(p*q) = -(p^4-1)*(q^2-1)^2. - Robert Israel, Mar 06 2019

A348428 Positive integers m that are equal to the determinant of the left circulant matrix formed by the decimal digits of m.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1547, 26027, 26933, 45018, 69781, 80487, 154791, 23203827, 257059332, 278945612, 456790123, 469135802, 493827160, 494376160, 506172839, 530864197, 543209876, 897163795, 1662971175, 2293668391, 3880266075, 6473710191

Offset: 1

Chai Wah Wu, Oct 18 2021

A left circulant matrix is also called a anti-circulant or (-1)-circulant matrix.
Subsequence of A219327.
Fixed points of A177894. - John Keith, Oct 24 2021

			           ⎡1  5  4  7⎤
1547 = det ⎢5  4  7  1⎥
           ⎢4  7  1  5⎥
           ⎣7  1  5  4⎦.

Cf. A219324, A219326, A219327.

Cf. A177894.

Select[Range[10^6], Equal[Det[NestList[RotateLeft, #2, #3 - 1]], #1] & @@ {#1, #2, Length[#2]} & @@ {#, IntegerDigits[#]} &] (* Michael De Vlieger, Oct 18 2021 *)

isok(m) = {my(d=digits(m), x); matdet(matrix(#d, #d, i, j, if (i==1, d[j], x = lift(Mod(j+i-1, #d)); if (!x, x += #d); d[x]))) == m;} \\ Michel Marcus, Oct 19 2021

from sympy import Matrix
A348428_list = []
for n in range(1,10**6):
    s = [int(d) for d in str(n)]
    m = len(s)
    if n == Matrix(m, m, lambda i, j: s[(i+j) % m]).det():
        A348428_list.append(n)

cp's OEIS Frontend

A299792 Numbers k such that A177894(k) = 0.

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A317291 Numbers k not in A239019 such that A177894(k) = 0.

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A306593 Least number k such that the determinant of the circulant matrix formed by its decimal digits is equal to n.

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A257587 If n = abcd... in decimal, a(n) = a^2 - b^2 + c^2 - d^2 + ...

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A306595 Determinant of the circulant matrix whose first column corresponds to the binary digits of n.

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A307651 a(n) is the determinant of the Vandermonde matrix of the digits of n.

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A306598 Determinant of the circulant matrix whose first column corresponds to the divisors of n.

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