|
%I #11 Sep 08 2022 08:45:53
%S 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,
%T 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,
%U 18,29,57,63,93,63,57,29,18,1
%N 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.
%C Row sums are: {1, 2, 4, 6, 13, 20, 53, 94, 181, 288, 429, ...}.
%H G. C. Greubel, <a href="/A176653/b176653.txt">Rows n = 0..100 of triangle, flattened</a>
%F 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.
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 2, 1;
%e 1, 2, 2, 1;
%e 1, 4, 3, 4, 1;
%e 1, 4, 5, 5, 4, 1;
%e 1, 10, 9, 13, 9, 10, 1;
%e 1, 10, 17, 19, 19, 17, 10, 1;
%e 1, 16, 21, 37, 31, 37, 21, 16, 1;
%e 1, 16, 29, 43, 55, 55, 43, 29, 16, 1;
%e 1, 18, 29, 57, 63, 93, 63, 57, 29, 18, 1;
%t 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 *)
%o (PARI)
%o {f(n) = if(n==0, 1, prime(n))};
%o {T(n,k) = f(n-k)*f(k) - f(k) + 1};
%o for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, May 07 2019
%o (Magma)
%o f:= func< n | n eq 0 select 1 else NthPrime(n) >;
%o [[f(n-k)*f(k) - f(n) + 1: k in [0..n]]: n in [0..12]]; // _G. C. Greubel_, May 07 2019
%o (Sage)
%o def f(n):
%o if (n==0): return 1
%o else: return nth_prime(n)
%o def T(n, k): return f(n-k)*f(k) - f(n) +1
%o [[T(n, k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, May 07 2019
%Y Cf. A146985
%K nonn,tabl
%O 0,5
%A _Roger L. Bagula_, Apr 22 2010
%E Edited by _G. C. Greubel_, May 07 2019
|
