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
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
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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
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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
Comments