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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; ... .

%I A133825 #27 Nov 29 2023 06:58:30
%S A133825 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,
%T A133825 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,
%U A133825 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
%N A133825 Triangle whose rows are sequences of increasing and decreasing triangular numbers: 1; 1,3,1; 1,3,6,3,1; ... .
%C A133825 Reading the triangle by rows produces the sequence 1,1,3,1,1,3,6,3,1,..., analogous to A004737.
%C A133825 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
%H A133825 Harvey P. Dale, <a href="/A133825/b133825.txt">Table of n, a(n) for n = 0..1000</a>
%H A133825 Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations [of] Integer Sequences And Pairing Functions</a> arXiv:1212.2732 [math.CO], 2012.
%F A133825 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) + ... .
%F A133825 From _Boris Putievskiy_, Jan 13 2013: (Start)
%F A133825 a(n) = A004737(n)*(A004737(n)+1)/2.
%F A133825 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)
%e A133825 Triangle starts
%e A133825   1;
%e A133825   1, 3, 1;
%e A133825   1, 3, 6, 3, 1;
%e A133825   1, 3, 6, 10, 6, 3, 1;
%e A133825 From _Boris Putievskiy_, Jan 13 2013: (Start)
%e A133825 The start of the sequence as table:
%e A133825   1...1...1...1...1...1...
%e A133825   1...3...3...3...3...3...
%e A133825   1...3...6...6...6...6...
%e A133825   1...3...6..10..10..10...
%e A133825   1...3...6..10..15..15...
%e A133825   1...3...6..10..15..21...
%e A133825   1...3...6..10..15..21...
%e A133825   . . .
%e A133825 The start of the sequence as triangle array read by rows:
%e A133825   1,
%e A133825   1, 3, 1,
%e A133825   1, 3, 6, 3, 1,
%e A133825   1, 3, 6, 10, 6, 3, 1,
%e A133825   1, 3, 6, 10, 15, 10, 6, 3, 1,
%e A133825   1, 3, 6, 10, 15, 21, 15, 10, 6, 3, 1,
%e A133825   1, 3, 6, 10, 15, 21, 28, 21, 15, 10, 6, 3, 1,
%e A133825   . . .
%e A133825 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)
%t A133825 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 *)
%Y A133825 Cf. A000330 (row sums), A004737, A124258, A133826, A106255.
%K A133825 easy,nonn,tabf
%O A133825 0,3
%A A133825 _Peter Bala_, Sep 25 2007