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Triangle read by rows: T(n,0) = T(n,n) = 1, 0 < k < n: T(n,k) = smallest prime not less than T(n-1,k) + T(n-1,k-1).
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%I #21 Oct 24 2024 03:11:57

%S 1,1,1,1,2,1,1,3,3,1,1,5,7,5,1,1,7,13,13,7,1,1,11,23,29,23,11,1,1,13,

%T 37,53,53,37,13,1,1,17,53,97,107,97,53,17,1,1,19,71,151,211,211,151,

%U 71,19,1,1,23,97,223,367,431,367,223,97,23,1,1,29,127

%N Triangle read by rows: T(n,0) = T(n,n) = 1, 0 < k < n: T(n,k) = smallest prime not less than T(n-1,k) + T(n-1,k-1).

%C T(n,k) = T(n,n-k);

%C T(n,0) = 1, cf. A000012;

%C T(n,1) = A008578(n), n > 0;

%C A199424(n) = first row in triangle A199302 containing n-th prime;

%C A199425(n) = number of distinct primes in rows 0 through n;

%C large terms in the b-file are probable primes only, row number > 50.

%H Reinhard Zumkeller, <a href="/A199333/b199333.txt">Rows n = 0..150 of triangle, flattened</a>

%H <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>

%F T(n,k) = A007918(T(n-1,k) + T(n-1,k-1)), 0 < k < n, T(n,0) = T(n,n) = 1.

%e 0: 1

%e 1: 1 1

%e 2: 1 2 1

%e 3: 1 3 3 1

%e 4: 1 5 7 5 1

%e 5: 1 7 13 13 7 1

%e 6: 1 11 23 29 23 11 1

%e 7: 1 13 37 53 53 37 13 1

%e 8: 1 17 53 97 107 97 53 17 1

%e primes in 8th row:

%e T(7,0) + T(7,1) = 1+13 = 14 --> T(8,1) = T(8,7) = 19;

%e T(7,1) + T(7,2) = 13+37 = 50 --> T(8,2) = T(8,6) = 53, already in row 7;

%e T(7,2) + T(7,3) = 37+53 = 90 --> T(8,3) = T(8,5) = 97;

%e T(7,3) + T(7,4) = 53+53 = 106 --> T(8,4) = 107.

%t T[n_, k_] := T[n, k] = Switch[k, 0|n, 1, _, With[{m = T[n-1, k] + T[n-1, k-1]}, If[PrimeQ[m], m, NextPrime[m]]]];

%t Table[T[n, k], {n, 0, 13}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Sep 19 2021 *)

%o (Haskell)

%o a199333 n k = a199333_tabl !! n !! k

%o a199333_row n = a199333_tabl !! n

%o a199333_list = concat a199333_tabl

%o a199333_tabl = iterate

%o (\row -> map a159477 $ zipWith (+) ([0] ++ row) (row ++ [0])) [1]

%Y Cf. A159477; A199581 & A199582 (central terms), A199694 (row sums), A199695 & A199696 (row products); A007318.

%K nonn,tabl

%O 0,5

%A _Reinhard Zumkeller_, Nov 09 2011