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 A133825 Triangle whose rows are sequences of increasing and decreasing triangular numbers: 1; 1,3,1; 1,3,6,3,1; ... . 3
 1, 1, 3, 1, 1, 3, 6, 3, 1, 1, 3, 6, 10, 6, 3, 1, 1, 3, 6, 10, 15, 10, 6, 3, 1, 1, 3, 6, 10, 15, 21, 15, 10, 6, 3, 1, 1, 3, 6, 10, 15, 21, 28, 21, 15, 10, 6, 3, 1, 1, 3, 6, 10, 15, 21, 28, 36, 28, 21, 15, 10, 6, 3, 1, 1, 3, 6, 10, 15, 21, 28, 36, 45, 36, 28, 21, 15, 10, 6, 3, 1, 1, 3, 6, 10 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Reading the triangle by rows produces the sequence 1,1,3,1,1,3,6,3,1,..., analogous to A004737. T(n,k) =  min(n*(n+1)/2,k*(k+1)/2), n, k >0.  The order of the list T(n,k) is by sides of squares from T(1,n) to T(n,n), then from T(n,n) to T(n,1). - Boris Putievskiy, Jan 13 2013 LINKS Harvey P. Dale, Table of n, a(n) for n = 0..1000 Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO] FORMULA O.g.f.: (1+qx)/((1-x)(1-qx)^2(1-q^2x)) = 1 + x(1 + 3q + q^2) + x^2(1 + 3q + 6q^2 + 3q^3 + q^4) + ... . From Boris Putievskiy, Jan 13 2013: (Start) a(n) = A004737(n)*(A004737(n)+1)/2. a(n) = z*(z+1)/2, where z = floor(sqrt(n-1)) - |n- floor(sqrt(n-1))^2- floor(sqrt(n-1))-1| +1. (End) EXAMPLE Triangle starts 1; 1, 3, 1; 1, 3, 6, 3, 1; 1, 3, 6, 10, 6, 3, 1; 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... . . . The start of the sequence as triangle array read by rows: 1, 1, 3, 1, 1, 3, 6, 3, 1, 1, 3, 6, 10, 6, 3, 1, 1, 3, 6, 10, 15, 10, 6, 3, 1, 1, 3, 6, 10, 15, 21, 15, 10, 6, 3, 1, 1, 3, 6, 10, 15, 21, 28, 21, 15, 10, 6, 3, 1, . . . Row number k contains 2*k-1 numbers 1,3,...,k*(k-1)/2,k*(k+1)/2,k*(k-1)/2,...,3,1. (End) MATHEMATICA Module[{nn=10, ac}, ac=Accumulate[Range[nn]]; Table[Join[Take[ ac, n], Reverse[ Take[ac, n-1]]], {n, nn}]]//Flatten (* Harvey P. Dale, Apr 18 2019 *) CROSSREFS Cf. A000330 (row sums), A004737, A124258, A133826, A106255. Sequence in context: A245541 A209563 A308624 * A156710 A114588 A253223 Adjacent sequences:  A133822 A133823 A133824 * A133826 A133827 A133828 KEYWORD easy,nonn,tabf AUTHOR Peter Bala, Sep 25 2007 STATUS approved

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Last modified July 23 11:44 EDT 2019. Contains 325254 sequences. (Running on oeis4.)