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

Author

Paolo P. Lava, May 13 2019

Keywords

• nonn
• base

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

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

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