A077049 Left summatory matrix, T, by antidiagonals upwards.

1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Clark Kimberling, Oct 22 2002

Keywords

• nonn
• tabl

Comments

If S = (s(1), s(2), ...) is a sequence written as a column vector, then T*S is the summatory sequence of S; i.e., its n-th term is Sum_{k|n} s(k). T is the inverse of the left Moebius transformation matrix, A077050. Except for the first term in some cases, column 1 of T^(-2) is A007427, column 1 of T^(-1) is A008683, Column c of T^2 is A000005, column 1 of T^3 is A007425.

This is essentially the same as A051731, which includes only the triangle. Note that the standard in the OEIS is left to right antidiagonals, which would make this the right summatory matrix, and A077051 the left one. - Franklin T. Adams-Watters, Apr 08 2009

From Gary W. Adamson, Apr 28 2010: (Start)

As defined with antidiagonals of the array = the triangle shown in the example section. Row sums of this triangle = A032741 (with a different offset): 1, 1, 2, 1, 3, 1, 3, ...

Let the triangle = M. Then lim_{n->inf} M^n = A002033, the left-shifted vector considered as a sequence: (1, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 8, ...). (End)

Examples

			T(4,2) = 1 since 2 divides 4. Northwest corner:
  1 0 0 0 0 0
  1 1 0 0 0 0
  1 0 1 0 0 0
  1 1 0 1 0 0
  1 0 0 0 1 0
  1 1 1 0 0 1
From _Gary W. Adamson_, Apr 28 2010: (Start)
First few rows of the triangle (when T is read by antidiagonals upwards):
  1;
  1, 0;
  1, 1, 0;
  1, 0, 0, 0;
  1, 1, 1, 0, 0;
  1, 0, 0, 0, 0, 0;
  1, 1, 0, 1, 0, 0, 0;
  1, 0, 1, 0, 0, 0, 0, 0;
  1, 1, 0, 0, 1, 0, 0, 0, 0;
  ... (End)
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..11325 (Rows 1 <= n <= 150).
• Clark Kimberling, Matrix Transformations of Integer Sequences, J. Integer Seqs., Vol. 6, 2003.

Crossrefs

Cf. A051731, A077050, A077051, A077052, A000005 (row sums).

Cf. A032741, A002033. - Gary W. Adamson, Apr 28 2010

Programs

Maple

A077049 := proc(n,k)
    if modp(n,k) = 0 then
        1;
    else
        0 ;
    end if;
end proc:
for d from 2 to 10 do
    for k from 1 to d-1 do
        n := d-k ;
        printf("%d,",A077049(n,k)) ;
    end do:
end do: # R. J. Mathar, Jul 22 2017

Mathematica

With[{nn = 14}, DeleteCases[#, -1] & /@ Transpose@ Table[Take[#, nn] &@ Flatten@ Join[ConstantArray[-1, k - 1], ConstantArray[Reverse@ IntegerDigits[2^(k - 1), 2], Ceiling[(nn - k + 1)/k]]], {k, nn}]] // Flatten (* Michael De Vlieger, Jul 22 2017 *)

PARI

nn=10; matrix(nn, nn, n, k, if (n % k, 0, 1)) \\ Michel Marcus, May 21 2015

Python

def T(n, k):
    return 1 if n%k==0 else 0
for n in range(1, 11): print([T(n - k + 1, k) for k in range(1, n + 1)]) # Indranil Ghosh, Jul 22 2017

Formula

T(n,k)=1 if k|n, otherwise T(n,k)=0, k >= 1, n >= 1.

From Boris Putievskiy, May 08 2013: (Start)

As table T(n,k) = floor(k/n) - floor((k-1)/n).

As linear sequence a(n) = floor(A004736(n)/A002260(n)) - floor((A004736(n)-1)/A002260(n)); a(n) = floor(j/i)-floor((j-1)/i), where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). (End)

Extensions

Name edited by Petros Hadjicostas, Jul 27 2019