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%I #72 Mar 26 2022 09:53:18
%S 1,3,1,6,2,1,10,4,2,1,15,6,3,2,1,21,9,5,3,2,1,28,12,7,4,3,2,1,36,16,9,
%T 6,4,3,2,1,45,20,12,8,5,4,3,2,1,55,25,15,10,7,5,4,3,2,1,66,30,18,12,9,
%U 6,5,4,3,2,1,78,36,22,15,11,8,6,5,4,3,2,1,91,42,26,18,13,10,7,6,5,4,3,2,1
%N Lower triangular matrix multiplication: A004736 * A051731.
%C Row sums = A078567: (1, 4, 9, 17, 27, 41, 57, ...).
%C T(n ,1) = A000217(n), n >= 1.
%C From _Bob Selcoe_, Aug 08 2016: (Start)
%C Columns are partial sums of k-repeating increasing positive integers:
%C Column 1 is {1+2+3+4+5+...} = A000217 (triangular numbers);
%C Column 2 is {1+1+2+2+3+3+4+4+...} = A002620 (quarter-squares);
%C Column 3 is {1+1+1+2+2+2+3+3+3+...} = A130518.
%C Columns k = 4..7 are A130519, A130520, A174709 and A174738, respectively.
%C Row sums are A078567(n+1).
%C T(n,k) is the number of positive multiples of k which can be expressed as i-j, {i=1..n; j=0..n-1}. So for example, T(5,2)=6 because there are 6 ways to express i-j {i<=5} as a multiple of 2: {5-3, 4-2, 3-1, 2-0, 5-1 and 4-0}.
%C (End)
%C Conjecture: For T(n,k) n >= k^(3/2), there is at least one prime in the interval [T(n-1,k+1), T(n,k)]. - _Bob Selcoe_, Aug 21 2016
%C Theorem: For n >= 3k, T(n,k) is composite. - _Daniel Hoying_, Jul 08 2020
%H Michael De Vlieger, <a href="/A134546/b134546.txt">Table of n, a(n) for n = 1..11325</a> (1 <= n <= 150)
%F A004736 * A051731 as infinite lower triangular matrices: T(n, k) = Sum_{m=k..n} A004736(n, m)*A051731(m, k), T(n, k) = 0 if n < k.
%F T(n, k) = 0 if n < k, T(1, 1) = 1, and T(n, k) = T(n-1, k) + floor(n/k), for n >= 2. - _Richard R. Forberg_, Jan 17 2015
%F T(n,k) = k*floor(n/k)*floor((n+k)/k)/2 - floor(n/k)*(k-1-(n mod k)). - _Bob Selcoe_, Aug 21 2016
%F T(n,k) = k*A000217(b) + (b+1)*[(n +1)-(b + 1)*k] for 1 <= k <= floor[(n + 1) / 2] where b = floor[(n - k + 1) / k], T(n,k) = n-k+1 for floor[(n + 1) / 2] < k <= n and T(n,k) = 0 for k > n. - _Henri Gonin_, May 12 2020
%F T(n,k) = (-k/2)*floor(n/k)^2+(n-k/2+1)*floor(n/k). - _Daniel Hoying_, May 25 2020
%F From _Daniel Hoying_, Jul 06 2020: (Start)
%F T(m + 2*n - 1, m + n) = n for n > 0, m >= 0.
%F T(3*m + 3*ceiling((n-3)/6) + (n+1)/2, 2*m + 2*ceiling((n-3)/6) + 1) = n for n > 0, n odd, 0 <= m <= floor(n/3).
%F T(3*m + 3*ceiling(n/6) + n/2 - 1, 2*m + 2*ceiling(n/6)) = n for n > 0, n even, 0 <= m <= floor(n/3). (End)
%e The triangle T(n, k) begins:
%e n\k 1 2 3 4 5 6 7 8 9 10 ...
%e 1: 1
%e 2: 3 1
%e 3: 6 2 1
%e 4: 10 4 2 1
%e 5: 15 6 3 2 1
%e 6: 21 9 5 3 2 1
%e 7: 28 12 7 4 3 2 1
%e 8: 36 16 9 6 4 3 2 1
%e 9: 45 20 12 8 5 4 3 2 1
%e 10: 55 25 15 10 7 5 4 3 2 1
%e ... Reformatted. - _Wolfdieter Lang_, Feb 04 2015
%e T(10,3) = 15: 3*floor(10/3)*floor(13/3)/2 - floor(10/3)*(3-1 - 13 mod 3) = 3*3*4/2 - 3*(3-1-1) = 18-3 = 15. - _Bob Selcoe_, Aug 21 2016
%p T := proc(n, k) option remember: `if`(n = k, 1, T(n-1, k) + iquo(n,k)) end:
%p seq(seq(T(n,k), k=1..n),n=1..16); # _Peter Luschny_, May 26 2020
%t nn = 12; f[w_] := Map[PadRight[#, nn] &, w]; MapIndexed[Take[#1, First@ #2] &, f@ Table[Reverse@ Range@ n, {n, nn}].f@ Table[Boole@ Divisible[n, #] & /@ Range@ n, {n, nn}]] // Flatten (* _Michael De Vlieger_, Aug 10 2016 *)
%o (PARI) t(n, k) = if (k>n, 0, if (n==1, 1, t(n-1, k) + n\k));
%o tabl(nn) = {m = matrix(nn, nn, n , k, t(n,k)); for (n=1, nn, for (k=1, n, print1(m[n, k], ", ");); print(););} \\ _Michel Marcus_, Jan 18 2015
%o (PARI) trg(nn) = {ma = matrix(nn, nn, n, k, if (k<=n, n-k+1, 0)); mb = matrix(nn, nn, n, k, if (k<=n, !(n%k), 0)); ma*mb;} \\ _Michel Marcus_, Jan 20 2015
%Y Cf. A004736, A051731, A078567.
%Y Cf. A000217, A002620, A078567, A130518, A130519, A130520, A174709, A174738.
%K nonn,easy,tabl
%O 1,2
%A _Gary W. Adamson_, Oct 31 2007
%E Edited. Name clarified. Formulas rewritten. - _Wolfdieter Lang_, Feb 04 2015
%E Corrected and extended by _Michael De Vlieger_, Aug 10 2016
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