A176653 Triangle, read by rows, defined by T(n, m) = f(n-m)*f(n) - f(n-0)*f(0) + 1, where f(n) is 1 if n = 0 and Prime(n) otherwise.

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 4, 3, 4, 1, 1, 4, 5, 5, 4, 1, 1, 10, 9, 13, 9, 10, 1, 1, 10, 17, 19, 19, 17, 10, 1, 1, 16, 21, 37, 31, 37, 21, 16, 1, 1, 16, 29, 43, 55, 55, 43, 29, 16, 1, 1, 18, 29, 57, 63, 93, 63, 57, 29, 18, 1

Offset: 0

Author

Roger L. Bagula, Apr 22 2010

Keywords

• nonn
• tabl

Comments

Row sums are: {1, 2, 4, 6, 13, 20, 53, 94, 181, 288, 429, ...}.

Examples

			Triangle begins as:
  1;
  1,  1;
  1,  2,  1;
  1,  2,  2,  1;
  1,  4,  3,  4,  1;
  1,  4,  5,  5,  4,  1;
  1, 10,  9, 13,  9, 10,  1;
  1, 10, 17, 19, 19, 17, 10,  1;
  1, 16, 21, 37, 31, 37, 21, 16,  1;
  1, 16, 29, 43, 55, 55, 43, 29, 16,  1;
  1, 18, 29, 57, 63, 93, 63, 57, 29, 18, 1;
		

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened

Crossrefs

Cf. A146985

Programs

Magma

f:= func< n | n eq 0 select 1 else NthPrime(n) >;
[[f(n-k)*f(k) - f(n) + 1: k in [0..n]]: n in [0..12]]; // G. C. Greubel, May 07 2019

Mathematica

f[n_]:= If[n==0,1, Prime[n]]; T[n_, m_] = f[n-m]*f[m] - f[n]*f[0] + 1; Table[T[n, m], {n,0,12}, {m,0,n}]//Flatten (* modified by G. C. Greubel, May 07 2019 *)

PARI

{f(n) = if(n==0, 1, prime(n))};
{T(n,k) = f(n-k)*f(k) - f(k) + 1};
for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, May 07 2019

Sage

def f(n):
    if (n==0): return 1
    else: return nth_prime(n)
def T(n, k): return f(n-k)*f(k) - f(n) +1
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, May 07 2019

Formula

T(n, m) = f(n-m)*f(n) - f(n-0)*f(0) + 1, where f(n) is 1 if n = 0 and Prime(n) otherwise.

Extensions

Edited by G. C. Greubel, May 07 2019