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

0,3

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

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.

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)

Example

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;
  ...

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 *)

Crossrefs

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

Sequence in context: A269845 A124258 A001638 * A209565 A122185 A350000

Adjacent sequences: A133823 A133824 A133825 * A133827 A133828 A133829

Keyword

easy,nonn,tabf

Author

Peter Bala, Sep 25 2007

Status

approved