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A141067 List of different primes in Pascal-like triangles with index of asymmetry y = 2 and index of obliquity z = 0 or z = 1.

%I A141067 #65 Jan 28 2024 14:32:03
%S A141067 2,19,41,83,3719,5431,1873,3989,8641,18517,38303,79153,136963,2264749,
%T A141067 394969,1748039,6633577,14820521,18051277,3807953189,126558214721,
%U A141067 2710968363511,803233671719,1723473449197,1725438080929,7942459030543,145539180603829,77442861984547
%N A141067 List of different primes in Pascal-like triangles with index of asymmetry y = 2 and index of obliquity z = 0 or z = 1.
%C A141067 For the Pascal-like triangle G(n, k) with index of asymmetry y = 2 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, and G(n+4, k) = G(n+1, k-1) + G(n+1, k) + G(n+2, k) + G(n+3, k) for n >= 0 and k = 1..(n+1). (This is array A140997.)
%C A141067 For the Pascal-like triangle G(n, k) with index of asymmetry y=1 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, and G(n+4, k) = G(n+1, k-2) + G(n+1, k-3) + G(n+2, k-2) + G(n+3, k-1) for n >= 0 and k = 3..(n+3). (This is array A140994.)
%C A141067 From _Petros Hadjicostas_, Jun 12 2019: (Start)
%C A141067 The two triangular arrays A140997 and A140994, which are described above, are mirror images of each other.
%C A141067 To make the current sequence uniquely defined, we follow the suggestion of _R. J. Mathar_ for sequence A141064. For each row of array A140997, 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 A140994 instead.
%C A141067 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 (and it does not matter whether we use e = 0 or e = 1 in the general formulas in the attached photograph).
%C A141067 (End)
%H A141067 Petros Hadjicostas, <a href="/A141067/b141067.txt">Table of n, a(n) for n = 1..64</a>
%H A141067 Juri-Stepan Gerasimov, <a href="/A140998/a140998.jpg">Stepan's triangles and Pascal's triangle are connected by the recurrence relation ...</a>
%e A141067 Pascal-like triangle with y = 2 and z = 0 (i.e., A140997) begins as follows:
%e A141067   1, so no primes.
%e A141067   1 1, so no primes.
%e A141067   1 2 1, then a(1) = 2.
%e A141067   1 4 2 1, so no new primes.
%e A141067   1 8 4 2 1, so no new primes.
%e A141067   1 15 9 4 2 1, so no new primes.
%e A141067   1 28 19 9 4 2 1, so a(2) = 19.
%e A141067   1 52 40 19 9 4 2 1, so no new primes.
%e A141067   1 96 83 41 19 9 4 2 1, so a(3) = 41 and a(4) = 83.
%e A141067   1 177 170 88 41 19 9 4 2 1, so no new primes.
%e A141067   1 326 345 188 88 41 19 9 4 2 1, so no new primes.
%e A141067   1 600 694 400 189 88 41 19 9 4 2 1, so no new primes.
%e A141067   ... [edited by _Petros Hadjicostas_, Jun 12 2019]
%e A141067   Terms a(5) = 3719 and a(6) = 5431 appear in row k = 14, while terms a(7) = 1873 and a(8) = 3989 appear in row k = 15.
%p A141067 # This is a modification of _R. J. Mathar_'s program for A141031 (for the case y = 4 and z = 0).
%p A141067 # Construction of array A140997 (y = 2 and z = 0):
%p A141067 A140997 := 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; else procname(n - 1, k) + procname(n - 2, k) + procname(n - 3, k) + procname(n - 3, k - 1); end if; end proc;
%p A141067 # Construction of the current sequence:
%p A141067 A141067 := 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 := A140997(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 A141067 # Generation of the current sequence:
%p A141067 A141067(50);
%p A141067 # If one wishes to get the primes sorted, then he or she should replace RETURN(a) in the above Maple code with RETURN(sort(a)). In such a case, however, the sequence is not uniquely defined because it depends on the maximum n. - _Petros Hadjicostas_, Jun 15 2019
%Y A141067 Cf. A007318, A140993, A140994, A140995, A140996, A140997, A140998, A141020, A141021, A141031, A141066.
%K A141067 nonn
%O A141067 1,1
%A A141067 _Juri-Stepan Gerasimov_, Jul 14 2008
%E A141067 Partially edited by _N. J. A. Sloane_, Jul 18 2008
%E A141067 Comments and Example edited by _Petros Hadjicostas_, Jun 12 2019
%E A141067 More terms from _Petros Hadjicostas_, Jun 12 2019