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A133823
Triangle whose rows are sequences of increasing and decreasing cubes:1; 1,8,1; 1,8,27,8,1; ... .
4
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, 64, 125, 216, 125, 64, 27, 8, 1, 1, 8, 27, 64, 125, 216, 343, 216, 125, 64, 27, 8, 1, 1, 8, 27, 64, 125, 216, 343, 512, 343, 216, 125, 64, 27, 8, 1, 1, 8, 27, 64, 125, 216, 343, 512, 729
OFFSET
0,3
COMMENTS
Reading the triangle by rows produces the sequence 1,1,8,1,1,8,27,8,1,..., analogous to A004737.
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
LINKS
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
FORMULA
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) + ... .
From Boris Putievskiy, Jan 13 2013: (Start)
a(n) = (A004737(n))^3.
a(n) = (floor(sqrt(n-1)) - |n- floor(sqrt(n-1))^2- floor(sqrt(n-1))-1| +1)^3. (End)
EXAMPLE
Triangle starts
1;
1, 8, 1;
1, 8, 27, 8, 1;
1, 8, 27, 64, 27, 8, 1;
From Boris Putievskiy, Jan 13 2013: (Start)
The start of the sequence as table:
1...1...1...1...1...1...
1...8...8...8...8...8...
1...8..27..27..27..27...
1...8..27..64..64..64...
1...8..27..64.125.125...
1...8..27..64.125.216...
. . .
The start of the sequence as triangle array read by rows:
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;
1,8,27,64,125,216,125,64,27,8,1;
. . .
Row number k contains 2*k-1 numbers 1,8,...,(k-1)^3,k^3,(k-1)^3,...,8,1. (End)
MATHEMATICA
Table[Join[Range[n]^3, Range[n-1, 1, -1]^3], {n, 10}]//Flatten (* Harvey P. Dale, May 29 2019 *)
CROSSREFS
Sequence in context: A197110 A109571 A366494 * A168643 A173742 A146881
KEYWORD
easy,nonn,tabf
AUTHOR
Peter Bala, Sep 25 2007
STATUS
approved