A260238 -id:A260238 - cp's OEIS frontend

A188934 Decimal expansion of (1+sqrt(17))/4.

1, 2, 8, 0, 7, 7, 6, 4, 0, 6, 4, 0, 4, 4, 1, 5, 1, 3, 7, 4, 5, 5, 3, 5, 2, 4, 6, 3, 9, 9, 3, 5, 1, 9, 2, 5, 6, 2, 8, 6, 7, 9, 9, 8, 0, 6, 3, 4, 3, 4, 0, 5, 1, 0, 8, 5, 9, 9, 6, 5, 8, 3, 9, 3, 2, 7, 3, 7, 3, 8, 5, 8, 6, 5, 8, 4, 4, 0, 5, 3, 9, 8, 3, 9, 6, 9, 6, 5, 9, 1, 2, 7, 0, 2, 6, 7, 1, 0, 7, 4, 1, 7, 1, 1, 3, 6, 0, 1, 0, 2, 3, 4, 8, 0, 3, 5, 3, 5, 4, 0, 3, 8, 2, 5, 3, 5, 5, 2, 1, 0

Offset: 1

Clark Kimberling, Apr 13 2011

Decimal expansion of the length/width ratio of a (1/2)-extension rectangle.  See A188640 for definitions of shape and r-extension rectangle.
A (1/2)-extension rectangle matches the continued fraction [1,3,1,1,3,1,1,3,1,1,3,...] for the shape L/W=(1+sqrt(17))/4.  This is analogous to the matching of a golden rectangle to the continued fraction [1,1,1,1,1,1,1,1,...].  Specifically, for the (1/2)-extension rectangle, 1 square is removed first, then 3 squares, then 1 square, then 1 square,..., so that the original rectangle of shape (1+sqrt(17))/4 is partitioned into an infinite collection of squares.
Conjecture: This number is an eigenvalue to infinitely many n*n submatrices of A191898, starting in the upper left corner, divided by the row index. For the first few characteristic polynomials see A260237 and A260238. - Mats Granvik, May 12 2016.

			1.2807764064044151374553524639935192562...

Cf. A188640, A188935.

Essentially the same as A188485.

r = 1/2; t = (r + (4 + r^2)^(1/2))/2; RealDigits[ N[ FullSimplify@ t, 111]][[1]]
(* for the continued fraction *) ContinuedFraction[t, 120]
RealDigits[(1 + Sqrt@ 17)/4, 10, 111][[1]] (* Or *)
RealDigits[Exp@ ArcSinh[1/4], 10, 111][[1]] (* Robert G. Wilson v, Aug 17 2011 *)

(sqrt(17)+1)/4 \\ Charles R Greathouse IV, May 12 2016

A260237 Numerators of the characteristic polynomials of the von Mangoldt function matrix.

Original entry on oeis.org

0, 1, -1, -1, -1, 1, 1, 11, -1, -1, 0, -3, -9, 5, 1, 0, 3, 81, 7, -73, -1, 0, 3, 73, -1261, -1183, 53, 1, 0, -3, -1231, 5251, 8989, 1451, -731, -1, 0, 0, 7, 397, -12491, -19877, -15047, 1567, 1, 0, 0, 0, -7, -1483, 50111, 69761, 45959, -5261, -1

Offset: 1

Mats Granvik, Jul 20 2015

The von Mangoldt function matrix is the symmetric matrix A191898 divided by either the row index or the column index.

Every eigenvalue of a smaller von Mangoldt function matrix appears to be common to infinitely many larger von Mangoldt matrices. The eigenvalues of smaller von Mangoldt function matrices also repeat within larger von Mangoldt function matrices.

			The first term set to zero is not part of the characteristic polynomials. It is only there for the formatting of the table.
{
{0},
{1, -1},
{-1, -1, 1},
{1, 11, -1, -1},
{0, -3, -9, 5, 1},
{0, 3, 81, 7, -73, -1},
{0, 3, 73, -1261, -1183, 53, 1},
{0, -3, -1231, 5251, 8989, 1451, -731, -1},
{0, 0, 7, 397, -12491, -19877, -15047, 1567, 1},
{0, 0, 0, -7, -1483, 50111, 69761, 45959, -5261,-1}
}

Denominators in A260238.

Clear[nnn, nn, T, n, k, x]; nnn = 9; T[n_, k_] := T[n, k] = Which[n < 1 || k < 1, 0, n == 1 || k == 1, 1, k > n, T[k, n], n > k, T[k, Mod[n, k, 1]], True, -Sum[T[n, i], {i, n - 1}]]; b = Table[CoefficientList[CharacteristicPolynomial[Table[Table[T[n, k]/n, {k, 1, nn}], {n, 1, nn}], x], x], {nn, 1, nnn}]; Flatten[{0,Numerator[b]}]

cp's OEIS Frontend

A188934 Decimal expansion of (1+sqrt(17))/4.

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