A191898 Symmetric square array read by antidiagonals: T(n,1)=1, T(1,k)=1, T(n,k) = -Sum_{i=1..k-1} T(n-i,k) for n >= k, -Sum_{i=1..n-1} T(k-i,n) for n < k.
1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, -1, -2, -1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 1, -1, 1, -1, 1, 1, 1, -2, 1, 1, -2, 1, 1, 1, -1, 1, -1, -4, -1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 1, -1, 1, -1, -6, -1, 1, -1, 1, -1, 1
Offset: 1
Examples
Array starts: n\k | 1 2 3 4 5 6 7 8 9 10 ----+----------------------------------------------------- 1 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... 2 | 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, ... 3 | 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, ... 4 | 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, ... 5 | 1, 1, 1, 1, -4, 1, 1, 1, 1, -4, ... 6 | 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, ... 7 | 1, 1, 1, 1, 1, 1, -6, 1, 1, 1, ... 8 | 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, ... 9 | 1, 1, -2, 1, 1, -2, 1, 1, -2, 1, ... 10 | 1, -1, 1, -1, -4, -1, 1, -1, 1, 4, ...
Links
- Antti Karttunen, Table of n, a(n) for n = 1..22155; the first 210 antidiagonals of the array
- Mats Granvik, Is this similarity to the Fourier transform of the von Mangoldt function real?
- Mats Granvik, Is this Riemann zeta function product equal to the Fourier transform of the von Mangoldt function?
- Mats Granvik, Primes approximated by eigenvalues?
- Mats Granvik, Are the primes found as a subset in this sequence a(n)?
- Mats Granvik, Will every eigenvalue in this type of matrix eventually be a common eigenvalue to infinitely many subsequent larger matrices of the same form?
- Mats Granvik, How write Dirichlet character sums for the terms of the von Mangoldt function?
- Mats Granvik, Do these series converge to the von Mangoldt function?
- Mats Granvik, Is this sum equal to the Möbius function?
- Mats Granvik, Can the Riemann hypothesis be relaxed to say that this matrix A consists of square roots?
- Mats Granvik, Elementary proof of the prime number theorem?
- Mats Granvik, Is this Dirichlet series generating function of the von Mangoldt function matrix correct?
- Mats Granvik, Question about ratios of polynomials evaluated at x=1
Crossrefs
Programs
-
Mathematica
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}]]; (* Michael Somos, Jul 18 2011 *) (* Conjectured expression for the matrix as Dirichlet characters *) s = RandomInteger[{1, 3}]; c = RandomInteger[{1, 3}]; nn = 12; b = Table[Exp[MangoldtLambda[Divisors[n]]]^-MoebiusMu[Divisors[n]], {n, 1, nn^Max[s, c]}]; j = 1; MatrixForm[Table[Table[Product[(b[[n^s]][[m]]*DirichletCharacter[b[[n^s]][[m]], j, k^c] - (b[[n^s]][[m]] - 1)), {m, 1, Length[Divisors[n]]}], {n, 1, nn}], {k, 1, nn}]] (* Mats Granvik, Nov 23 2013 and Aug 09 2016 *) -
PARI
{T(n, k) = if( n<1 || k<1, 0, n==1 || k==1, 1, k>n, T(k, n), k -
Python
from sympy.core.cache import cacheit @cacheit def T(n, k): return 0 if n<1 or k<1 else 1 if n==1 or k==1 else T(k, n) if k>n else T(k, (n - 1)%k + 1) if n>k else -sum([T(n, i) for i in range(1, n)]) for n in range(1, 21): print([T(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, Oct 23 2017
Formula
T(n,1)=1, T(1,k)=1, n>=k: -Sum_{i=1..k-1} T(n-i,k), n
T(n, n) = A023900(n). - Michael Somos, Jul 18 2011
T(n, k) = A023900(gcd(n,k)). - Mats Granvik, Jun 18 2012
Dirichlet generating function for sequence in the n-th row: zeta(s)*Sum_{ d divides n } mu(d)/d^(s-1). - Mats Granvik, Jun 18 2012 & Jun 19 2016
From Mats Granvik, Jun 19 2016: (Start)
Dirichlet generating function for the whole matrix: Sum_{k>=1} (Sum_{n>=1} T(n,k)/(n^c*k^s)) = Sum_{n>=1} (zeta(s)*Sum_{ d divides n } mu(d)/d^(s-1))/n^c = zeta(s)*zeta(c)/zeta( c + s - 1 ).
T(n,k) = A127093(n,k)^(1/2-i*a(k))*transpose(A008683(k)*(A127093(n,k)^(1/2+i*a(n)))) where a(x) is some real number. An example would be T(n,k) = A127093(n,k)^(zetazero(k))*transpose(A008683(k)*(A127093(n,k)^(zetazero(-k)))) but this is of course not special for only the zeta zeros.
Recurrence for a subset of A191898 that is a cross-directional variant of the recurrence in A051731: T(1,1)=1, T(1,2..k)=0, T(2..n,1)=0, n >= k: -Sum_{i=1..k-1} T(n-i,k) - T(n-i,k-1), n < k: -Sum_{i=1..n-1} T(k-i,n) - T(k-i,n-1). Notice that the identity matrix in linear algebra satisfies a similar recurrence:
T(1,1)=1, T(1,2..k)=0, T(2..n,1)=0, n >= k: -Sum_{i=1..n-1} T(n-i,k) - T(n-i,k-1), n < k: -Sum_{i=1..k-1} T(k-i,n) - T(k-i,n-1).
(End)
Dirichlet generating function for absolute values: Sum_{k>=1} (Sum_{n>=1} abs(T(n,k))/(n^c*k^s)) = zeta(s)*zeta(c)*zeta(s + c - 1)/zeta(2*(s + c - 1))*Product_{k>=1} (1 - 2/(prime(k) + prime(k)^(s + c))). After Vaclav Kotesovec in A173557. - Mats Granvik, Apr 25 2021
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
Comments
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.
Examples
1.2807764064044151374553524639935192562...
Programs
-
Mathematica
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 *)
-
PARI
(sqrt(17)+1)/4 \\ Charles R Greathouse IV, May 12 2016
A260238 Denominators of the characteristic polynomials of the von Mangoldt function matrix.
1, 1, 1, 2, 1, 1, 6, 6, 1, 1, 2, 4, 12, 1, 1, 2, 20, 3, 60, 1, 1, 2, 60, 360, 360, 60, 1, 1, 2, 420, 2520, 1260, 504, 420, 1, 1, 1, 4, 120, 5040, 2520, 5040, 840, 1, 1, 1, 1, 3, 360, 15120, 7560, 15120, 2520, 1
Offset: 1
Comments
The von Mangoldt function matrix is the symmetric Greatest Common Divisor (GCD) type 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 second diagonal appears to be A003418.
Examples
{
{0},
{1, 1},
{1, 2, 1},
{1, 6, 6, 1},
{1, 2, 4, 12, 1},
{1, 2, 20, 3, 60, 1},
{1, 2, 60, 360, 360, 60, 1},
{1, 2, 420, 2520, 1260, 504, 420, 1},
{1, 1, 4, 120, 5040, 2520, 5040, 840, 1},
{1, 1, 1, 3, 360, 15120, 7560, 15120, 2520, 1}
}
Programs
-
Mathematica
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[Denominator[b]]
Comments