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%I #25 May 16 2019 14:57:31
%S 0,1,2,3,4,5,6,7,8,9,302,65,42,76,10320,41,40,98,522,413,64,52,354,
%T 645,51,50,142,63,86,1534,13112,1387,62,74,514,61,60,635,978,85,73,
%U 1431,502,2677,152,72,746,625,71,70,378,2415,254,475,366,83,95,263,33442
%N Least number k such that the determinant of the symmetric Hankel matrix formed by its decimal digits is equal to n.
%C The first nontrivial values for which a(n) = n are at n = 288 and n = 26825.
%C When a(n) < n: 168, 182, 232, 234, 252, 272, 280, 300, 304, 320, 324, 325, etc. - _Robert G. Wilson v_, May 14 2019
%C 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
%H Robert G. Wilson v, <a href="/A308110/b308110.txt">Table of n, a(n) for n = 0..10000</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Hankel_matrix">Hankel matrix</a>
%e | 3 0 2 |
%e a(10) = 302 because det | 0 2 0 | = 10.
%e | 2 0 3 |
%e .
%e | 1 0 3 2 0 |
%e | 0 3 2 0 2 |
%e a(14)= 10320 because det | 3 2 0 2 3 | = 14.
%e | 2 0 2 3 0 |
%e | 0 2 3 0 1 |
%p with(numtheory): with(linalg): P:=proc(q) local c,d,i,k,n,t: print(0);
%p for i from 1 to q do for n from 1 to q do c:=convert(n, base, 10): t:=[]:
%p for k from 1 to nops(c) do t:=[op(t),0]: od: d:=t: t:=[]:
%p for k from 1 to nops(c) do t:=[op(t),d]: t[k,-k]:=1: od:
%p if det(evalm(toeplitz(c) &* t))=i then print(n); break: fi:
%p od: od: end: P(10^8);
%t 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 *)
%Y Cf. A306593, A307887, A308126.
%K nonn,base
%O 0,3
%A _Paolo P. Lava_, May 13 2019
%E Offset corrected by _Robert G. Wilson v_, May 14 2019