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A077049
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Left summatory matrix, T, by antidiagonals upwards.
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15
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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
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OFFSET
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1,1
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COMMENTS
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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
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)
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LINKS
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FORMULA
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T(n,k)=1 if k|n, otherwise T(n,k)=0, k >= 1, n >= 1.
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)
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EXAMPLE
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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
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)
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MAPLE
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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 ;
end do:
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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