A133826 Triangle whose rows are sequences of increasing and decreasing tetrahedral numbers: 1; 1,4,1; 1,4,10,4,1; ... .

1, 1, 4, 1, 1, 4, 10, 4, 1, 1, 4, 10, 20, 10, 4, 1, 1, 4, 10, 20, 35, 20, 10, 4, 1, 1, 4, 10, 20, 35, 56, 35, 20, 10, 4, 1, 1, 4, 10, 20, 35, 56, 84, 56, 35, 20, 10, 4, 1, 1, 4, 10, 20, 35, 56, 84, 120, 84, 56, 35, 20, 10, 4, 1, 1, 4, 10, 20, 35, 56, 84, 120, 165, 120, 84, 56, 35, 20, 10, 4, 1

Offset: 0

Author

Peter Bala, Sep 25 2007

Keywords

• easy
• nonn
• tabf

Comments

Reading the triangle by rows produces the sequence 1,1,4,1,1,4,10,4,1,..., analogous to A004737.

T(n,k) = min(n*(n+1)*(n+2)/6, k*(k+1)*(k+2)/6) n, k > 0. 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

Examples

			Triangle T(n,k) starts:
  1;
  1,  4,  1;
  1,  4, 10,  4,  1;
  1,  4, 10, 20, 10,  4,  1;
  1,  4, 10, 20, 35, 20, 10,  4,  1;
  1,  4, 10, 20, 35, 56, 35, 20, 10,  4,  1;
  1,  4, 10, 20, 35, 56, 84, 56, 35, 20, 10,  4,  1;
  ...
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Crossrefs

Cf. A000292, A002415 (row sums), A004737, A124258, A133825.

Programs

Maple

T:= n-> (f-> (f(i)$i=1..n, f(n-i)$i=1..n-1))(t-> t*(t+1)*(t+2)/6):
seq(T(n), n=1..10);  # Alois P. Heinz, Feb 17 2022

Mathematica

Module[{nn=10,tet},tet=Table[(n(n+1)(n+2))/6,{n,nn}];Table[Join[Take[ tet,k], Reverse[ Take[tet,k-1]]],{k,nn}]]//Flatten (* Harvey P. Dale, Oct 22 2017 *)
Table[Series[(1-h^(2*N+4))^2/(1-h^2)^4-((2+N)^2 *h^(2N+2))/(1-h^2)^2, {h, 0, 4*N}], {N,0,5}] // Normal (* Sergii Voloshyn, Sep 09 2022 *)

Formula

O.g.f.: (1+q*x)/((1-x)*(1-q*x)^3*(1-q^2x)) = 1 + x*(1 + 4*q + q^2) + x^2*(1 + 4*q + 10*q^2 + 4*q^3 + q^4) + ... .

From Boris Putievskiy, Jan 13 2013: (Start)

a(n) = A004737(n)*(A004737(n)+1)*(A004737(n)+2)/2.

a(n) = z*(z+1)*(z+2)/6, where z = floor(sqrt(n-1)) - |n - floor(sqrt(n-1))^2 - floor(sqrt(n-1)) - 1| + 1. (End)