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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 (list; graph; refs; listen; history; text; internal format)
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

Harvey P. Dale, Table of n, a(n) for n = 0..9999

Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO]

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

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

Sequence in context: A178163 A197110 A109571 * A168643 A173742 A146881

Adjacent sequences:  A133820 A133821 A133822 * A133824 A133825 A133826

KEYWORD

easy,nonn,tabf

AUTHOR

Peter Bala, Sep 25 2007

STATUS

approved

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Last modified June 24 02:56 EDT 2021. Contains 345415 sequences. (Running on oeis4.)