A308110 Least number k such that the determinant of the symmetric Hankel matrix formed by its decimal digits is equal to n.

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

0,3

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

Links

Robert G. Wilson v, Table of n, a(n) for n = 0..10000

Wikipedia, Hankel matrix

Example

                        | 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 |

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 *)

Crossrefs

Cf. A306593, A307887, A308126.

Sequence in context: A229761 A004882 A306853 * A306593 A046469 A335151

Adjacent sequences: A308107 A308108 A308109 * A308111 A308112 A308113

Keyword

nonn,base

Author

Paolo P. Lava, May 13 2019

Extensions

Offset corrected by Robert G. Wilson v, May 14 2019

Status

approved