A127648 Triangle read by rows: row n consists of n zeros followed by n+1.

1, 0, 2, 0, 0, 3, 0, 0, 0, 4, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15

Offset: 0

Author

Gary W. Adamson, Jan 22 2007

Keywords

• nonn
• easy
• tabl

Comments

Alternatively, a(n) = k if n+1 is the k-th triangular number and 0 otherwise.

Triangle T(n,k), 0<=k<=n, read by rows, given by (0,0,0,0,0,0,0,0,0,0,...) DELTA (2,-1/2,1/2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 27 2011

Examples

			First few rows of the triangle:
  1;
  0, 2;
  0, 0, 3;
  0, 0, 0, 4;
  0, 0, 0, 0, 5;
  0, 0, 0, 0, 0, 6;
  0, 0, 0, 0, 0, 0, 7;
  ...
		

Links

• Antti Karttunen, Rows n = 0..360 of the triangle, flattened (Rows n = 0..100 from G. C. Greubel)

Crossrefs

Cf. A003506, A007318, A084938, A010054, A103406, A142150.

Programs

Haskell

a127648 n k = a127648_tabl !! n !! k
a127648_row n = a127648_tabl !! n
a127648_tabl = map reverse $ iterate (\(x:xs) -> x + 1 : 0 : xs) [1]
a127648_list = concat a127648_tabl
-- Reinhard Zumkeller, Jul 13 2013

Magma

[k eq n select n+1 else 0: k in [0..n], n in [0..20]]; // G. C. Greubel, Mar 12 2024

Maple

A127648 := proc(n)
    for i from 0 do
        if A000217(i) = n+1 then
            return i ;
        elif A000217(i) >n then
            return 0 ;
        end if;
    end do;
end proc: # R. J. Mathar, Apr 23 2013

Mathematica

Flatten[Table[{n,Table[0,{n}]},{n,15}]] (* Harvey P. Dale, Jul 27 2011 *)

PARI

A127648(n) = if(ispolygonal(1+n,3), (sqrtint(1+((1+n)*8))-1)/2, 0); \\ Antti Karttunen, Jan 19 2025

Python

for i in range(1,15):
    print(i, end=", ")
    for j in range(i):
        print("0", end=", ") # Mohammad Saleh Dinparvar, May 11 2020

Python

from math import isqrt
from sympy.ntheory.primetest import is_square
def A127648(n): return (m:=isqrt(k:=n<<1))+(k>m*(m+1)) if is_square((n<<3)+1) else 0 # Chai Wah Wu, Jun 09 2025

SageMath

def A127648(n): return (sqrt(9+8*n)-1)//2 if ((sqrt(9+8*n)-3)/2).is_integer() else 0
[A127648(n) for n in range(153)] # G. C. Greubel, Mar 12 2024

Formula

Infinite lower triangular matrix with (1, 2, 3, ...) in the main diagonal and the rest zeros.

This sequence * A007318 (Pascal's Triangle) = A003506.

A007318 * this sequence = A103406.

G.f.: 1/(x*y-1)^2. - R. J. Mathar, Aug 11 2015

a(n) = (1/2) (round(sqrt(4 + 2 n)) - round(sqrt(2 + 2 n))) (-1 + round(sqrt(2 + 2 n)) + round(sqrt(4 + 2 n))). - Brian Tenneson, Jan 27 2017

From G. C. Greubel, Mar 13 2024: (Start)

T(n, n) = n+1.

Sum_{k=0..n} T(n, k) = n+1.

Sum_{k=0..n} (-1)^k*T(n, k) = (-1)^n*(n+1).

Sum_{k=0..floor(n/2)} T(n-k, k) = A142150(n+2).

Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (-1)^floor(n/2)*A142150(n+2). (End)