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

0,5

Comments

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

Links

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

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.

Example

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;

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 *)

Prog

(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
(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
(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

Crossrefs

Cf. A146985

Sequence in context: A331447 A352460 A342767 * A174842 A156074 A051287

Adjacent sequences: A176650 A176651 A176652 * A176654 A176655 A176656

Keyword

nonn,tabl

Author

Roger L. Bagula, Apr 22 2010

Extensions

Edited by G. C. Greubel, May 07 2019

Status

approved