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 A106255 Triangle composed of triangular numbers, row sums = A006918. 5
 1, 1, 1, 1, 3, 1, 1, 3, 3, 1, 1, 3, 6, 3, 1, 1, 3, 6, 6, 3, 1, 1, 3, 6, 10, 6, 3, 1, 1, 3, 6, 10, 10, 6, 3, 1, 1, 3, 6, 10, 15, 10, 6, 3, 1, 1, 3, 6, 10, 15, 15, 10, 6, 3, 1, 1, 3, 6, 10, 15, 21, 15, 10, 6, 3, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS Row sums = A006918, (deleting the zero): 1, 2, 5, 8, 14, 20, 30 ,... Row sums are: {1, 2, 5, 8, 14, 20, 30, 40, 55, 70, 91, ...}. T(n,k) = min(n*(n+1)/2,k*(k+1)/2), read by antidiagonals. - Boris Putievskiy, Jan 13 2013 LINKS Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012. FORMULA Perform the operation Q * R; Q = infinite lower triangular matrix with 1, 2, 3...in each column (offset, fill in spaces with zeros). Q = upper right triangular matrix of the form:   1 1 1 1 ...   0 1 1 1 ...   0 0 1 1 ...   0 0 0 1 ... Q * R generates an array:   1 1 1  1 ...   1 3 3  3 ...   1 3 6  6 ...   1 3 6 10 ..   ... ... from which we take antidiagonals forming the rows of triangle A106255. From Boris Putievskiy, Jan 13 2013: (Start) a(n) = min(A002260(n)*(A002260(n)+1)/2,A004737(n)*(A004737(n)+1)/2). a(n) = min(i*(i+1)/2,j*(j+1)/2), 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 From Boris Putievskiy, Jan 13 2013: (Start) The start of the sequence as table: 1...1...1...1...1...1... 1...3...3...3...3...3... 1...3...6...6...6...6... 1...3...6..10..10..10... 1...3...6..10..15..15... 1...3...6..10..15..21... 1...3...6..10..15..21... . . . (End) Triangle rows or columns can be generated by following the triangle format: 1; 1, 1; 1, 3, 1; 1, 3, 3, 1; 1, 3, 6, 3, 1; 1, 3, 6, 6, 3, 1; ... {1}, {1, 1}, {1, 3, 1}, {1, 3, 3, 1}, {1, 3, 6, 3, 1}, {1, 3, 6, 6, 3, 1}, {1, 3, 6, 10, 6, 3, 1}, {1, 3, 6, 10, 10, 6, 3, 1}, {1, 3, 6, 10, 15, 10, 6, 3, 1}, {1, 3, 6, 10, 15, 15, 10, 6, 3, 1}, {1, 3, 6, 10, 15, 21, 15, 10, 6, 3, 1} (End) MATHEMATICA p[x_, n_] = Sum[x^i*If[i ==Floor[n/2] && Mod[n, 2] == 0, 0, If[i <= Floor[n/2], 2*(i + 1), -(2*((n + 1) - i))]], {i, 0, n}]/(2*(1 - x)); Table[CoefficientList[FullSimplify[p[x, n]], x], {n, 1, 11}]; Flatten[%] CROSSREFS Cf. A006918, A002260, A004736. Sequence in context: A046540 A123191 A157454 * A143086 A327481 A152714 Adjacent sequences:  A106252 A106253 A106254 * A106256 A106257 A106258 KEYWORD nonn,tabl AUTHOR Gary W. Adamson, Apr 28 2005 EXTENSIONS Additional comments from Roger L. Bagula and Gary W. Adamson, Apr 02 2009 STATUS approved

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Last modified April 21 11:19 EDT 2021. Contains 343150 sequences. (Running on oeis4.)