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A133823 Triangle whose rows are sequences of increasing and decreasing cubes:1; 1,8,1; 1,8,27,8,1; ... .

%I A133823 #27 Nov 29 2023 06:58:22
%S A133823 1,1,8,1,1,8,27,8,1,1,8,27,64,27,8,1,1,8,27,64,125,64,27,8,1,1,8,27,
%T A133823 64,125,216,125,64,27,8,1,1,8,27,64,125,216,343,216,125,64,27,8,1,1,8,
%U A133823 27,64,125,216,343,512,343,216,125,64,27,8,1,1,8,27,64,125,216,343,512,729
%N A133823 Triangle whose rows are sequences of increasing and decreasing cubes:1; 1,8,1; 1,8,27,8,1; ... .
%C A133823 Reading the triangle by rows produces the sequence 1,1,8,1,1,8,27,8,1,..., analogous to A004737.
%C A133823 T(n,k) = min(n,k)^3. 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 A133823 Harvey P. Dale, <a href="/A133823/b133823.txt">Table of n, a(n) for n = 0..9999</a>
%H A133823 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 A133823 O.g.f.: (1+qx)(1+4qx+q^2x^2)/((1-x)(1-qx)^3(1-q^2x)) = 1 + x(1 + 8q + q^2) + x^2(1 + 8q + 27q^2 + 8q^3 + q^4) + ... .
%F A133823 From _Boris Putievskiy_, Jan 13 2013: (Start)
%F A133823 a(n) = (A004737(n))^3.
%F A133823 a(n) = (floor(sqrt(n-1)) - |n- floor(sqrt(n-1))^2- floor(sqrt(n-1))-1| +1)^3. (End)
%e A133823 Triangle starts
%e A133823   1;
%e A133823   1, 8, 1;
%e A133823   1, 8, 27, 8, 1;
%e A133823   1, 8, 27, 64, 27, 8, 1;
%e A133823 From _Boris Putievskiy_, Jan 13 2013: (Start)
%e A133823 The start of the sequence as table:
%e A133823   1...1...1...1...1...1...
%e A133823   1...8...8...8...8...8...
%e A133823   1...8..27..27..27..27...
%e A133823   1...8..27..64..64..64...
%e A133823   1...8..27..64.125.125...
%e A133823   1...8..27..64.125.216...
%e A133823   . . .
%e A133823 The start of the sequence as triangle array read by rows:
%e A133823   1;
%e A133823   1,8,1;
%e A133823   1,8,27,8,1;
%e A133823   1,8,27,64,27,8,1;
%e A133823   1,8,27,64,125,64,27,8,1;
%e A133823   1,8,27,64,125,216,125,64,27,8,1;
%e A133823   . . .
%e A133823 Row number k contains 2*k-1 numbers 1,8,...,(k-1)^3,k^3,(k-1)^3,...,8,1. (End)
%t A133823 Table[Join[Range[n]^3,Range[n-1,1,-1]^3],{n,10}]//Flatten (* _Harvey P. Dale_, May 29 2019 *)
%Y A133823 Cf. A004737, A037270 (row sums), A133820, A124258, A133824, A003983.
%K A133823 easy,nonn,tabf
%O A133823 0,3
%A A133823 _Peter Bala_, Sep 25 2007