A133824 Triangle whose rows are sequences of increasing and decreasing fourth powers: 1; 1,16,1; 1,16,81,16,1; ... .

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

Offset: 0

Author

Peter Bala, Sep 25 2007

Keywords

• easy
• nonn
• tabf

Comments

Reading the triangle by rows produces the sequence 1,1,16,1,1,16,81,16,1,..., analogous to A004737.

From - Boris Putievskiy, Jan 13 2013: (Start)

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).

Row number k contains 2*k-1 numbers 1,16,...,(k-1)^4,k^4,(k-1)^4,...,16,1. (End)

Examples

			Triangle starts:
  1;
  1, 16, 1;
  1, 16, 81, 16, 1;
  1, 16, 81, 256, 81, 16, 1;
  ...
From _Boris Putievskiy_, Jan 13 2013: (Start)
The start of the sequence as table:
  1...1...1...1...1.. .1...
  1..16..16..16..16...16...
  1..16..81..81..81...81...
  1..16..81.256.256..256...
  1..16..81.256.625..625...
  1..16..81.256.625.1296...
  ...
(End)
		

Links

• Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Crossrefs

Cf. A004737, A061803 (row sums), A133821, A124258, A133823, A003983.

Programs

Mathematica

p4[n_]:=Module[{c=Range[n]^4},Join[c,Rest[Reverse[c]]]]; Flatten[p4/@ Range[10]] (* Harvey P. Dale, Dec 08 2014 *)

Formula

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

From Boris Putievskiy, Jan 13 2013: (Start)

T(n,k) = min(n,k)^4.

a(n) = (A004737(n))^4.

a(n) = (A124258(n))^2.

a(n) = (floor(sqrt(n-1)) - |n- floor(sqrt(n-1))^2- floor(sqrt(n-1))-1| +1)^4. (End)