A133820 -id:A133820 - cp's OEIS frontend

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

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

Peter Bala, Sep 25 2007

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

			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)

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], 2012.

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

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

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)

A133821 Triangle whose rows are sequences of increasing fourth powers: 1; 1,16; 1,16,81; ... .

Original entry on oeis.org

1, 1, 16, 1, 16, 81, 1, 16, 81, 256, 1, 16, 81, 256, 625, 1, 16, 81, 256, 625, 1296, 1, 16, 81, 256, 625, 1296, 2401, 1, 16, 81, 256, 625, 1296, 2401, 4096, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 10000

Offset: 1

Peter Bala, Sep 25 2007

Reading the triangle by rows produces the sequence 1,1,16,1,16,81,1,16,81,256,..., analogous to A002260.

			Triangle starts
1;
1, 16;
1, 16; 81;
1, 16, 81, 256;
1, 16, 81, 256, 625;

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened

Cf. A000538 (row sums), A002260, A133819, A133820, A133824.

a133821 n k = a133821_tabl !! (n-1) !! (k-1)
a133821_row n = a133821_tabl !! (n-1)
a133821_tabl = map (`take` (tail a000583_list)) [1..]
-- Reinhard Zumkeller, Nov 11 2012

Module[{nn=10,fp},fp=Range[(nn(nn+1))/2]^4;Table[TakeList[fp,{n}],{n,nn}]]//Flatten (* Requires Mathematica version 11 or later *) (* Harvey P. Dale, Mar 29 2020 *)

O.g.f.: (1+11qx+11q^2x^2+q^3x^3)/((1-x)(1-qx)^5) = 1 + x(1 + 16q) + x^2(1 + 16q + 81q^2) + ... . Cf. 4th row of A008292.

Offset changed by Reinhard Zumkeller, Nov 11 2012

cp's OEIS Frontend

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

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