cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A307887 Least number k such that the determinant of the symmetric Toeplitz 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, 433, 65, 42, 76, 455, 41, 40, 98, 457, 766, 64, 52, 243, 788, 51, 50, 413, 63, 86, 142, 21024, 3055, 62, 74, 645, 61, 60, 524, 25624, 85, 73, 756, 20031, 514, 412, 72, 23688, 152, 71, 70, 641, 364, 355, 2542, 245, 83, 95, 798, 625
Offset: 0

Views

Author

Paolo P. Lava, May 03 2019

Keywords

Examples

			                        | 4 3 3 |
a(10) = 433 because det | 3 4 3 | = 10.
                        | 3 3 4 |
.
                         | 2 5 6 2 4 |
                         | 5 2 5 6 2 |
a(38)= 25624 because det | 6 5 2 5 6 | = 38.
                         | 2 6 5 2 5 |
                         | 4 2 6 5 2 |

Links

Crossrefs

Cf. A306593.

Programs

  • Maple
    with(numtheory): with(linalg): P:=proc(q) local a,c,i,k,n; print(0);
for i from 1 to q do for n from 1 to q do a:=convert(n,base,10):
c:=[]: for k from 1 to nops(a) do c:=[op(c), a[-k]]: od:
if det(toeplitz(c))=i then print(n); break: fi: od: od: end: P(10^6);

A306662 Least number k such that the determinant of the circulant matrix of its representation in base 2 is equal to n.

Original entry on oeis.org

0, 1, 5, 11, 23, 47, 95, 191, 43, 38, 1535, 3071, 571, 12287, 24575, 137, 269, 196607, 393215, 786431, 295, 687, 6291455, 12582911, 69, 155, 100663295, 134, 293, 805306367, 1610612735, 3221225471, 75, 518, 25769803775, 301, 8874
Offset: 0

Views

Author

Paolo P. Lava, Mar 04 2019

Keywords

Comments

Here only the least numbers are listed: e.g., a(10) = 1531, even if 1791, 1919, 1983, 2015, 2031, 2039, 2043, etc. also produce 10.
The sequence is infinite because any number of the form 3*2^(n-1) - 1 (A083329) has the determinant of the circulant matrix of its representation in base 2 equal to n but, in general, it is not the least possible term.
It would be nice to characterize the values of n where k < A083329(n).

Examples

			                                      | 1 0 1 1 |
a(3) = 11 because 11 = 1011_2 and det | 1 1 0 1 | = 3
                                      | 1 1 1 0 |
                                      | 0 1 1 1 |
.
and 11 is the least number to have this property.
.
                                       | 1 0 1 1 1 |
                                       | 1 1 0 1 1 |
a(4) = 23 because 23 = 10111_2 and det | 1 1 1 0 1 | = 4
                                       | 1 1 1 1 0 |
                                       | 0 1 1 1 1 |
.
and 23 is the least number to have this property.

Crossrefs

Cf. A083329, A219324, A306593, A323485, A323486.

Programs

  • Maple
    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
a:=convert(n, base, 2); d:=nops(a); 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);

Extensions

a(31)-a(36) from Giovanni Resta, Mar 05 2019

A306853 Positive integers equal to the permanent of the circulant matrix formed by their decimal digits.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 261, 370, 407, 52036, 724212, 223123410
Offset: 1

Views

Author

Paolo P. Lava, Mar 13 2019

Keywords

Comments

1, 2, 3, 4, 5, 6, 7, 8, 9, 370 and 407 are also equal to the determinant of the circulant matrix formed by their decimal digits.

Examples

			     | 2 6 1 |
perm | 1 2 6 | = 2*2*2 + 6*6*6 + 1*1*1 + 1*2*6 + 6*1*2 + 2*6*1 = 261.
     | 6 1 2 |
.
     | 2 2 3 1 2 3 4 1 0 |
     | 0 2 2 3 1 2 3 4 1 |
     | 1 0 2 2 3 1 2 3 4 |
     | 4 1 0 2 2 3 1 2 3 |
perm | 3 4 1 0 2 2 3 1 2 | = 223123410
     | 2 3 4 1 0 2 2 3 1 |
     | 1 2 3 4 1 0 2 2 3 |
     | 3 1 2 3 4 1 0 2 2 |
     | 2 3 1 2 3 4 1 0 2 |

Links

Crossrefs

Cf. A219324, A219327, A306662, A306593, A306714.
Up to n=110 the permanent of the circulant matrix of the digits of n is equal to A101337 but from n=111 on it can differ.

Programs

  • Maple
    with(linalg): P:=proc(q) local a, b, c, d, i, j, k, n, t;
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 n=permanent(t)
then print(n); fi; od; end: P(10^7);
  • PARI
    mpd(n) = {my(d = digits(n)); matpermanent(matrix(#d, #d, i, j, d[1+lift(Mod(j-i, #d))]));}
isok(n) = mpd(n) == n; \\ Michel Marcus, Mar 14 2019
  • Python
    from sympy import Matrix
A306853_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]).per():
        A306853_list.append(n) # Chai Wah Wu, Oct 18 2021

Extensions

a(15) from Vaclav Kotesovec, Aug 19 2021

A308110 Least number k such that the determinant of the symmetric Hankel 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, 302, 65, 42, 76, 10320, 41, 40, 98, 522, 413, 64, 52, 354, 645, 51, 50, 142, 63, 86, 1534, 13112, 1387, 62, 74, 514, 61, 60, 635, 978, 85, 73, 1431, 502, 2677, 152, 72, 746, 625, 71, 70, 378, 2415, 254, 475, 366, 83, 95, 263, 33442
Offset: 0

Views

Author

Paolo P. Lava, May 13 2019

Keywords

Comments

The first nontrivial values for which a(n) = n are at n = 288 and n = 26825.
When a(n) < n: 168, 182, 232, 234, 252, 272, 280, 300, 304, 320, 324, 325, etc. - Robert G. Wilson v, May 14 2019
Records: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 302, 10320, 13112, 33442, 53242, 55262, 58493, 74959, 1021310, 1124232, 1272626, 1400230, 2034050, 2514162, 3043724, 4986388, 5604351, 106071534, 108162262, 117200232, 128580276, 134314966, 163332550, 165244716, 166811088, 225231732, 229330425, etc. - Robert G. Wilson v, May 15 2019

Examples

			                        | 3 0 2 |
a(10) = 302 because det | 0 2 0 | = 10.
                        | 2 0 3 |
.
                         | 1 0 3 2 0 |
                         | 0 3 2 0 2 |
a(14)= 10320 because det | 3 2 0 2 3 | = 14.
                         | 2 0 2 3 0 |
                         | 0 2 3 0 1 |

Links

Crossrefs

Cf. A306593, A307887, A308126.

Programs

  • Maple
    with(numtheory): with(linalg): P:=proc(q) local c,d,i,k,n,t: print(0);
for i from 1 to q do for n from 1 to q do c:=convert(n, base, 10): t:=[]:
for k from 1 to nops(c) do t:=[op(t),0]: od: d:=t: t:=[]:
for k from 1 to nops(c) do t:=[op(t),d]: t[k,-k]:=1: od:
if det(evalm(toeplitz(c) &* t))=i then print(n); break: fi:
od: od: end: P(10^8);
  • Mathematica
    f[n_] := Block[{k = 0}, While[id = IntegerDigits@ k; Det[HankelMatrix[id, Reverse@ id]] != n, k++]; k]; Array[f, 60, 0] (* Robert G. Wilson v, May 14 2019 *)

Extensions

Offset corrected by Robert G. Wilson v, May 14 2019
Showing 1-4 of 4 results.

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