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A141068 List of different primes in Pascal-like triangles with index of asymmetry y = 3 and index of obliquity z = 0 or z = 1.

%I A141068 #48 Jan 28 2024 14:31:50
%S A141068 2,17,31,149,11587,49429,15701951,21304973,3846277,251375273,
%T A141068 5449276159,296410704409,750391353973,205109154121,875366796349,
%U A141068 72210869205443,139884035510017,79014319582741129,94461530406533783,2562508045902551
%N A141068 List of different primes in Pascal-like triangles with index of asymmetry y = 3 and index of obliquity z = 0 or z = 1.
%C A141068 For the Pascal-like triangle with index of asymmetry y = 3 and index of obliqueness z = 0, which is read by rows, we have G(n, 0) = G(n+1, n+1) = 1, G(n+2, n+1) = 2, G(n+3, n+1) = 4, G(n+4, n+1) = 8, and G(n+5, k) = G(n+1, k-1) + G(n+1, k) + G(n+2, k) + G(n+3, k) + G(n+4, k) for k = 1..(n+1). (This is array A140996.)
%C A141068 For the Pascal-like triangle with index of asymmetry y = 3 and index of obliqueness z = 1, which is read by rows, we have G(n, n) = G(n+1, 0) = 1, G(n+2, 1) = 2, G(n+3, 2) = 4, G(n+4, 3) = 8, and G(n+5, k) = G(n+1, k-3) + G(n+1, k-4) + G(n+2, k-3) + G(n+3, k-2) + G(n+4, k-1) for k = 4..(n+4). (This is array A140995.)
%C A141068 From _Petros Hadjicostas_, Jun 13 2019: (Start)
%C A141068 The two triangular arrays A140995 and A140996, which are described above, are mirror images of each other.
%C A141068 To make the current sequence uniquely defined, we follow the suggestion of R. J. Mathar for sequence A141064. For each row of array A140996, the primes not appearing in earlier rows are collected, sorted, and added to the sequence. We get exactly the same sequence by working with array A140995 instead.
%C A141068 Finally, we mention that in the attached picture about the connection between Stepan's triangles and the Pascal triangle, the letter s is used to describe the index of asymmetry and the letter e is used to describe the index of obliqueness (instead of the letters y and z, respectively). The Pascal triangle A007318 has index of asymmetry s = y = 0.
%C A141068 (End)
%H A141068 Juri-Stepan Gerasimov, <a href="/A140998/a140998.jpg">Pascal-like triangles and Pascal's triangle are connected by the recurrence relation ...</a>
%e A141068 Pascal-like triangle with y = 3 and z = 0 (i.e., A140996) begins as follows:
%e A141068   1, so no primes.
%e A141068   1 1, so no primes
%e A141068   1 2 1, so a(1) = 2.
%e A141068   1 4 2 1, so no new primes.
%e A141068   1 8 4 2 1, so no new primes.
%e A141068   1 16 8 4 2 1, so new primes.
%e A141068   1 31 17 8 4 2 1, so a(2) = 17 and a(3) = 31.
%e A141068   1 60 35 17 8 4 2 1, so no new primes.
%e A141068   1 116 72 35 17 8 4 2 1, so no new primes.
%e A141068   1 224 148 72 35 17 8 4 2 1, so new primes.
%e A141068   1 432 303 149 72 35 17 8 4 2 1, so a(4) = 149.
%e A141068   ...
%p A141068 # This is a modification of _R. J. Mathar_'s program for A141031 (for the case y = 4 and z = 0).
%p A141068 # Definition of sequence A140996 (y = 3 and z = 0):
%p A141068 A140996 := proc(n, k) option remember; if k < 0 or n < k then 0; elif k = 0 or k = n then 1; elif k = n - 1 then 2; elif k = n - 2 then 4; elif k = n - 3 then 8; else procname(n - 1, k) + procname(n - 2, k) + procname(n - 3, k) + procname(n - 4, k) + procname(n - 4, k - 1); end if; end proc;
%p A141068 # Definition of the current sequence:
%p A141068 A141068 := proc(nmax) local a, b, n, k, new; a := []; for n from 0 to nmax do b := []; for k from 0 to n do new := A140996(n, k); if not (new = 1 or not isprime(new) or new in a or new in b) then b := [op(b), new]; end if; end do; a := [op(a), op(sort(b))]; end do; RETURN(a); end proc;
%p A141068 # Generation of the current sequence:
%p A141068 A141068(80);
%p A141068 # If one wishes to get the primes sorted (as R. J. Mathar does in A141031), then replace RETURN(a) in the code above with RETURN(sort(a)). In such a case, however, the output sequence is not uniquely defined because it depends on the maximum n. - _Petros Hadjicostas_, Jun 15 2019
%Y A141068 Cf. A007318, A140993, A140994, A140995, A140996, A140997, A140998, A141020, A141021, A141031, A141064, A141065, A141066, A141067, A141069, A141070, A141072, A141073.
%K A141068 nonn
%O A141068 1,1
%A A141068 _Juri-Stepan Gerasimov_, Jul 16 2008
%E A141068 Partially edited by _N. J. A. Sloane_, Jul 18 2008
%E A141068 More terms from _Petros Hadjicostas_, Jun 13 2019