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 A255315 Lower triangular matrix describing the shape of a half hyperbola in the Dirichlet divisor problem. 1
 1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 0, 2, 1, 1, 1, 0, 1, 2, 1, 1, 1, 0, 1, 2, 1, 1, 1, 1, 0, 1, 1, 2, 1, 1, 1, 1, 0, 0, 2, 2, 1, 1, 1, 1, 1, 0, 0, 2, 1, 2, 1, 1, 1, 1, 1, 0, 0, 2, 1, 2, 1, 1, 1, 1, 1, 1, 0, 0, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,8 COMMENTS The sum of terms of row n is n. Length of row n is n. From Mats Granvik, Feb 21 2016: (Start) A006218(n) = (n^2 - ((2*Sum_{kk=1..n} Sum_{k=1..kk} T(n,k)) - n)) + 2*n - round(1 + (1/2)*(-3 + sqrt(n) + sqrt(1 + n))). A006218(n) = -((n^2 - ((2*Sum_{kk=1..n} Sum_{k=1..kk} T(n,n-k+1)) - n)) - 2*n + round(1 + (1/2)*(-3 + sqrt(n) + sqrt(1 + n)))). (End) From Mats Granvik, May 28 2017: (Start) A006218(n) = (n^2 - (2*(Sum_{k=1..n} T(n, k)*(n - k + 1)) - n)) + 2*n - round(1 + (1/2)*(-3 + sqrt(n) + sqrt(1 + n))). A006218(n) = -((n^2 - (2*(Sum_{k=1..n} T(n, n - k + 1)*(n - k + 1)) - n)) - 2*n + round(1 + (1/2)*(-3 + sqrt(n) + sqrt(1 + n)))). (End) From Mats Granvik, Sep 07 2017: (Start) It appears that: The number of 0's in row n is equal to the number of 2's in row n and their number is given by A000196(n) - 1. The number of 1's in column k is given by A152948(k+2). The number of 2's in column k is given by A000096(k-1). The row index of the last nonzero entry in column k is given by A005563(k). (End) From Mats Granvik, Oct 06 2018: (Start) Compare the formula for this table to the formula in A094820. The smallest k such that T(n,k)=2 is given by A079643(n) = floor(n/floor(sqrt(n))). This gives the lower bound: A006218(n) >= A094761(n) + A079643(n)*2*(A000196(n)-1). <=> A006218(n) >= 2*n - (floor(sqrt(n)))^2 + floor(n/floor(sqrt(n)))*2*floor(sqrt(n)-1). The average of k:s such that T(n,k)=2, for n>3 is given by: b(n) = Sum_{k=1..n} (k*floor(abs(T(n, k)-1/2)))/floor(sqrt(n)-1). This gives A006218(n) = 2*n - (floor(sqrt(n)))^2 + b(n)*2*floor(sqrt(n)-1) = 2*n - (floor(sqrt(n)))^2 + (Sum_{k=1..n} (k*floor(abs(T(n, k)-1/2))))*2, for n>3. The largest k such that T(n,k)=2 is given by A004526(n) = floor(n/2). This gives the upper bound: A006218(n) <= A094761(n) + A004526(n)*2*(A000196(n)-1). <=> A006218(n) <= 2*n - (floor(sqrt(n)))^2 + floor(n/2)*2*floor(sqrt(n)-1). The lower bound starts:  1, 3, 5, 8, 10, 14, 16, 20, 21, 23, ... Sequence A006218 starts: 1, 3, 5, 8, 10, 14, 16, 20, 23, 27, ... The upper bound starts:  1, 3, 5, 8, 10, 14, 16, 20, 25, 31, ... (End) LINKS Eric Weisstein's World of Mathematics, Dirichlet Divisor Problem FORMULA T(n,a) = Sum_{b=1..n} [a*b <= n and ((a + 1)*(b + 1) > n or a < b)]. - Mats Granvik, Oct 06 2018 EXAMPLE 1; 1, 1; 1, 1, 1; 0, 2, 1, 1; 0, 2, 1, 1, 1; 0, 1, 2, 1, 1, 1; 0, 1, 2, 1, 1, 1, 1; 0, 1, 1, 2, 1, 1, 1, 1; 0, 0, 2, 2, 1, 1, 1, 1, 1; 0, 0, 2, 1, 2, 1, 1, 1, 1, 1; 0, 0, 2, 1, 2, 1, 1, 1, 1, 1, 1; 0, 0, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1; MATHEMATICA (* From Mats Granvik, Feb 21 2016: (Start) *) nn = 12; T = Table[    Sum[Table[      If[And[If[n*k <= r, If[n >= k, 1, 0], 0] == 1,        If[(n + 1)*(k + 1) <= r, If[n >= k, 1, 0], 0] == 0], 1, 0], {n,        1, r}], {k, 1, r}], {r, 1, nn}]; Flatten[T] A006218a = Table[(n^2 - (2*Sum[Sum[T[[n, k]], {k, 1, kk}], {kk, 1, n}] -         n)) + 2*n - Round[1 + (1/2)*(-3 + Sqrt[n] + Sqrt[1 + n])], {n,      1, nn}]; A006218b = -Table[(n^2 - (2*           Sum[Sum[T[[n, n - k + 1]], {k, 1, kk}], {kk, 1, n}] - n)) -      2*n + Round[1 + (1/2)*(-3 + Sqrt[n] + Sqrt[1 + n])], {n, 1, nn}]; (A006218b - A006218a); (* (End) *) (* From Mats Granvik, May 28 2017: (Start) *) nn = 12; T = Table[    Sum[Table[      If[And[If[n*k <= r, If[n >= k, 1, 0], 0] == 1,        If[(n + 1)*(k + 1) <= r, If[n >= k, 1, 0], 0] == 0], 1, 0], {n,        1, r}], {k, 1, r}], {r, 1, nn}]; Flatten[T] A006218a = Table[(n^2 - (2*Sum[T[[n, k]]*(n - k + 1), {k, 1, n}] - n)) +     2*n - Round[1 + (1/2)*(-3 + Sqrt[n] + Sqrt[1 + n])], {n, 1, nn}]; A006218b = Table[-((n^2 - (2*Sum[T[[n, n - k + 1]]*(n - k + 1), {k, 1, n}] -           n)) - 2*n +       Round[1 + (1/2)*(-3 + Sqrt[n] + Sqrt[1 + n])]), {n, 1, nn}]; (A006218b - A006218a); (* (End) *) CROSSREFS Cf. A006218, A094820. Sequence in context: A282380 A083661 A029369 * A125072 A162642 A139146 Adjacent sequences:  A255312 A255313 A255314 * A255316 A255317 A255318 KEYWORD tabl,nonn AUTHOR Mats Granvik, May 31 2015 STATUS approved

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Last modified September 16 08:36 EDT 2019. Contains 327091 sequences. (Running on oeis4.)