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 A106255 Triangle composed of triangular numbers, row sums = A006918. 4
 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 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 this triangle. LINKS Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012. FORMULA From Boris Putievskiy, Jan 13 2013: (Start) T(n,k) = min(n*(n+1)/2,k*(k+1)/2), read by antidiagonals. 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, 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;   ... 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 (row sums, without the zero), 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 June 30 16:05 EDT 2022. Contains 354943 sequences. (Running on oeis4.)