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 A077049 Left summatory matrix, T, by antidiagonals. 15
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 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 1 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) LINKS Michael De Vlieger, Table of n, a(n) for n = 1..11325 (Rows 1 <= n <= 150). C. Kimberling, Matrix Transformations of Integer Sequences, J. Integer Seqs., Vol. 6, 2003. FORMULA T(n, k)=1 if k|n, else 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) EXAMPLE 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 =   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) 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 *) PROG (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 xrange(1, 11): print [T(n - k + 1, k) for k in xrange(1, n + 1)] # Indranil Ghosh, Jul 22 2017 CROSSREFS Cf. A051731, A077050, A077051, A077052, A000005 (row sums). Cf. A032741, A002033. [Gary W. Adamson, Apr 28 2010] Sequence in context: A157926 A263243 A131377 * A124895 A089885 A190233 Adjacent sequences:  A077046 A077047 A077048 * A077050 A077051 A077052 KEYWORD nonn,tabl AUTHOR Clark Kimberling, Oct 22 2002 STATUS approved

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Last modified May 19 04:06 EDT 2019. Contains 323377 sequences. (Running on oeis4.)