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A079505
The last number for which a determinant of base-n numbers is nonzero.
1
14, 25, 61, 121, 211, 337, 505, 721, 991, 1321, 1717, 2185, 2731, 3361, 4081, 4897, 5815, 6841, 7981, 9241, 10627, 12145, 13801, 15601, 17551, 19657, 21925, 24361, 26971, 29761, 32737, 35905, 39271, 42841, 46621, 50617, 54835, 59281, 63961, 68881
OFFSET
2,1
COMMENTS
Suppose the number k written in base b requires N digits. We build A_k, a square N X N matrix with the digits of k, k-1,...,k-N+1 in base b. The number Det[A_k] is 0 for k greater than b^3-b+1 (except if b=2).
|Det[A_k]| is at most (b-1)^2. The last nonzero value is 1-b, which occurs for k = b^3-b+1 (cf. A061600) except for b=2, though I did not prove it.
FORMULA
a(n) = n^3 - n + 1 (except for n=2, a(2)=14).
From Chai Wah Wu, Nov 30 2018: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 6.
G.f.: x^2*(7*x^4 - 29*x^3 + 45*x^2 - 31*x + 14)/(x - 1)^4. (End)
EXAMPLE
a(3)=25 because the determinant sequence in base 3 is 1, 2, 2, -1, -1, 4, -2, -2, -2, 2, 0, 1, -1, 0, 1, -1, 0, -4, 4, 0, 2, -2, 0, 2, -2, 0, 0, 0, 0, 0, 0, 0, .... and Det[A_k]=0 for k > 25.
MAPLE
seq(coeff(series(x^2*(7*x^4-29*x^3+45*x^2-31*x+14)/(x-1)^4, x, n+1), x, n), n = 2 .. 35); # Muniru A Asiru, Nov 30 2018
MATHEMATICA
Table[ls = {}; Do[nt = Length[IntegerDigits[k, b]]; Ak = Table[IntegerDigits[k - i, b, nt], {i, 0, nt - 1}]; AppendTo[ls, Det[Ak]], {k, 1, b^4}]; Position[ls, _?(#!=0&)][[ -1, 1]], {b, 2, 10}]
CoefficientList[Series[(7x^4 -29x^3 +45x^2 -31x +14)/(x-1)^4, {x, 0, 50}], x] (* Vincenzo Librandi, Dec 01 2018 *)
LinearRecurrence[{4, -6, 4, -1}, {14, 25, 61, 121, 211}, 40] (* Harvey P. Dale, Feb 12 2023 *)
PROG
(Magma) [14] cat [n^3-n+1: n in [3..50]]; // Vincenzo Librandi, Dec 01 2018
(PARI) vector(50, n, n++; if(n==2, 14, n^3-n+1)) \\ G. C. Greubel, Jan 18 2019
(Sage) [14] + [n^3-n+1 for n in (3..50)] # G. C. Greubel, Jan 18 2019
(GAP) Concatenation([14], List([3..50], n -> n^3-n+1)); # G. C. Greubel, Jan 18 2019
CROSSREFS
Cf. A061600.
Sequence in context: A164403 A164398 A174519 * A039604 A199405 A256573
KEYWORD
base,nonn,easy
AUTHOR
Carlos Alves, Jan 21 2003
EXTENSIONS
Edited by T. D. Noe, Jun 24 2009
STATUS
approved