A133823 - OEIS

%I #27 Nov 29 2023 06:58:22

%S 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 64,125,216,125,64,27,8,1,1,8,27,64,125,216,343,216,125,64,27,8,1,1,8,

%U 27,64,125,216,343,512,343,216,125,64,27,8,1,1,8,27,64,125,216,343,512,729

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

%C Reading the triangle by rows produces the sequence 1,1,8,1,1,8,27,8,1,..., analogous to A004737.

%C 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 Harvey P. Dale, <a href="/A133823/b133823.txt">Table of n, a(n) for n = 0..9999</a>

%H 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 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 From _Boris Putievskiy_, Jan 13 2013: (Start)

%F a(n) = (A004737(n))^3.

%F a(n) = (floor(sqrt(n-1)) - |n- floor(sqrt(n-1))^2- floor(sqrt(n-1))-1| +1)^3. (End)

%e Triangle starts

%e   1;

%e   1, 8, 1;

%e   1, 8, 27, 8, 1;

%e   1, 8, 27, 64, 27, 8, 1;

%e From _Boris Putievskiy_, Jan 13 2013: (Start)

%e The start of the sequence as table:

%e   1...1...1...1...1...1...

%e   1...8...8...8...8...8...

%e   1...8..27..27..27..27...

%e   1...8..27..64..64..64...

%e   1...8..27..64.125.125...

%e   1...8..27..64.125.216...

%e   . . .

%e The start of the sequence as triangle array read by rows:

%e   1;

%e   1,8,1;

%e   1,8,27,8,1;

%e   1,8,27,64,27,8,1;

%e   1,8,27,64,125,64,27,8,1;

%e   1,8,27,64,125,216,125,64,27,8,1;

%e   . . .

%e Row number k contains 2*k-1 numbers 1,8,...,(k-1)^3,k^3,(k-1)^3,...,8,1. (End)

%t Table[Join[Range[n]^3,Range[n-1,1,-1]^3],{n,10}]//Flatten (* _Harvey P. Dale_, May 29 2019 *)

%Y Cf. A004737, A037270 (row sums), A133820, A124258, A133824, A003983.

%K easy,nonn,tabf

%O 0,3

%A _Peter Bala_, Sep 25 2007