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A309229 Square array read by upwards antidiagonals: T(n,k) = Sum_{i=1..n} A191898(i,k). 6
1, 2, 1, 3, 0, 1, 4, 1, 2, 1, 5, 0, 0, 0, 1, 6, 1, 1, 1, 2, 1, 7, 0, 2, 0, 3, 0, 1, 8, 1, 0, 1, 4, -2, 2, 1, 9, 0, 1, 0, 0, -3, 3, 0, 1, 10, 1, 2, 1, 1, -2, 4, 1, 2, 1, 11, 0, 0, 0, 2, 0, 5, 0, 0, 0, 1, 12, 1, 1, 1, 3, 1, 6, 1, 1, 1, 2, 1, 13, 0, 2, 0, 4, 0, 0, 0, 2, 0, 3, 0, 1, 14, 1, 0, 1, 0, -2, 1, 1, 0, -4, 4, -2, 2, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

log(A003418(n)) = Sum_{k>=1} (T(n, k)/k - 1/k).

Partial sums of the symmetric matrix A191898. - Mats Granvik, Apr 12 2020

1 + Sum_{k=1..2*n} sign((sign(n+Sum_{j=2..k}-|T(n,j)|)+1)) appears to be asymptotic to sqrt(8*n). - Mats Granvik, Jun 08 2020

LINKS

Table of n, a(n) for n=1..105.

Mats Granvik, Mathematica MatrixPlot of 1000 times 1000 size matrix

Mats Granvik, Mathematica program for the recurrence

Mats Granvik, Mathematica program to compute the sequence with the conjectured asymptotic sqrt(8*n)

Mathematics Stack Exchange, Do these series converge to the von Mangoldt function?

FORMULA

Recurrence:

T(n, 1) = [n >= 1]*n;

T(1, k) = 1;

T(n, k) = [n > k]*T(n - k, k) + [n <= k](Sum_{i=0..n-1} T(n - 1, k - i) - Sum_{i=1..n-1} T(n, k - i)). - Mats Granvik, Jun 19 2020

T(n,k) = Sum_{i=1..n} A191898(i,k).

EXAMPLE

   1, 1, 1, 1, 1,  1, 1, 1, 1,  1,  1,  1,  1,  1, ...

   2, 0, 2, 0, 2,  0, 2, 0, 2,  0,  2,  0,  2,  0, ...

   3, 1, 0, 1, 3, -2, 3, 1, 0,  1,  3, -2,  3,  1, ...

   4, 0, 1, 0, 4, -3, 4, 0, 1,  0,  4, -3,  4,  0, ...

   5, 1, 2, 1, 0, -2, 5, 1, 2, -4,  5, -2,  5,  1, ...

   6, 0, 0, 0, 1,  0, 6, 0, 0, -5,  6,  0,  6,  0, ...

   7, 1, 1, 1, 2,  1, 0, 1, 1, -4,  7,  1,  7, -6, ...

   8, 0, 2, 0, 3,  0, 1, 0, 2, -5,  8,  0,  8, -7, ...

   9, 1, 0, 1, 4, -2, 2, 1, 0, -4,  9, -2,  9, -6, ...

  10, 0, 1, 0, 0, -3, 3, 0, 1,  0, 10, -3, 10, -7, ...

  11, 1, 2, 1, 1, -2, 4, 1, 2,  1,  0, -2, 11, -6, ...

  12, 0, 0, 0, 2,  0, 5, 0, 0,  0,  1,  0, 12, -7, ...

  13, 1, 1, 1, 3,  1, 6, 1, 1,  1,  2,  1,  0, -6, ...

  14, 0, 2, 0, 4,  0, 0, 0, 2,  0,  3,  0,  1,  0, ...

  ...

MATHEMATICA

f[n_] := DivisorSum[n, MoebiusMu[#] # &]; nn = 14; A = Accumulate[Table[Table[f[GCD[n, k]], {k, 1, nn}], {n, 1, nn}]]; Flatten[Table[Table[A[[n - k + 1, k]], {k, 1, n}], {n, 1, nn}]] (* Mats Granvik, Jun 09 2020 *)

CROSSREFS

Cf. A003418, A191898.

Sequence in context: A127094 A221642 A158906 * A143239 A158951 A126988

Adjacent sequences:  A309226 A309227 A309228 * A309230 A309231 A309232

KEYWORD

tabl,sign

AUTHOR

Mats Granvik, Aug 10 2019

STATUS

approved

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Last modified January 26 02:37 EST 2021. Contains 340429 sequences. (Running on oeis4.)