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A141066 List of different composites in Pascal-like triangles with index of asymmetry y = 2 and index of obliquity z = 0 or z = 1.

%I A141066 #71 Jan 28 2024 14:31:35
%S A141066 4,8,9,15,28,40,52,96,88,170,177,188,326,345,189,400,600,694,406,846,
%T A141066 1104,1386,871,1779,2031,2751,872,1866,3736,6872,7730,10672,4022,8505,
%U A141066 12640,15979,20885,4023,8633,18079,23249,32859,40724,42762,67240,18559,39677,78652,80866,153402
%N A141066 List of different composites in Pascal-like triangles with index of asymmetry y = 2 and index of obliquity z = 0 or z = 1.
%C A141066 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, G(n+4, k) = G(n+1, k-1) + G(n+1, k) + G(n+2, k) + G(n+3, k) for k = 1..(n+1). (This is array A140997.)
%C A141066 For the Pascal-like triangle 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, 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 k = 3..(n+3). (This is array A140994.)
%C A141066 From _Petros Hadjicostas_, Jun 12 2019: (Start)
%C A141066 The two triangular arrays A140997 and A140994, which are described above, are mirror images of each other.
%C A141066 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 composites 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 A141066 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 A141066 (End)
%H A141066 Petros Hadjicostas, <a href="/A141066/b141066.txt">Table of n, a(n) for n = 1..191</a> [b-file format amended by _Georg Fischer_, Jun 22 2019]
%H A141066 Juri-Stepan Gerasimov, <a href="/A140998/a140998.jpg">Stepan's triangles and Pascal's triangle are connected by the recurrence relation ...</a>
%e A141066 Pascal-like triangle with y = 2 and z = 0 (i.e., A140997) begins as follows:
%e A141066   1, so no composite.
%e A141066   1 1, so no composite.
%e A141066   1 2 1, so no composite.
%e A141066   1 4 2 1, so a(1) = 4.
%e A141066   1 8 4 2 1, so a(2) = 8.
%e A141066   1 15 9 4 2 1, so a(3) = 9 and a(4) = 15.
%e A141066   1 28 19 9 4 2 1, so a(5) = 28.
%e A141066   1 52 40 19 9 4 2 1, so a(6) = 40 and a(7) = 52.
%e A141066   1 96 83 41 19 9 4 2 1, so a(8) = 96.
%e A141066   1 177 170 88 41 19 9 4 2 1, so a(9) = 88, a(10) = 170, and a(11) = 177.
%e A141066   1 326 345 188 88 41 19 9 4 2 1, so a(12) = 188, a(13) = 326, and a(14) = 345.
%e A141066   1 600 694 400 189 88 41 19 9 4 2 1, so a(15) = 189, a(16) = 400, a(17) = 600, and a(18) = 694.
%e A141066 ... [example edited by _Petros Hadjicostas_, Jun 11 2019]
%p A141066 # This is a modification of _R. J. Mathar_'s program for A141031 (for the case y = 4 and z = 0).
%p A141066 # Construction of array A140997 (y = 2 and z = 0):
%p A141066 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 A141066 # Construction of the current sequence:
%p A141066 A141066 := 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 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 A141066 # Generation of numbers in the current sequence:
%p A141066 A141066(19);
%p A141066 # If one wishes to sort the numbers, then replace RETURN(a) with RETURN(sort(a)) in the above Maple code. In this case, however, the sequence is not uniquely defined because it depends on the maximum n. - _Petros Hadjicostas_, Jun 15 2019
%Y A141066 Cf. A007318 (y = 0), A140993 (y = 1 and z = 1), A140994 (y = 2 and z = 1), A140995 (y = 3 and z = 1), A140996 (y = 3 and z = 0), A140997 (y = 2 and z = 0), A140998 (y = 1 and z = 0), A141020 (y = 4 and z = 0), A141021 (y = 4 and z = 1), A141064 (has primes when y = 1), A141065 (has composites when y = 1), A141067 (has primes when y = 2), A141068 (has primes when y = 3), A141069 (has composites when y = 3).
%K A141066 nonn
%O A141066 1,1
%A A141066 _Juri-Stepan Gerasimov_, Jul 14 2008
%E A141066 Partially edited by _N. J. A. Sloane_, Jul 18 2008
%E A141066 Comments and Example edited by _Petros Hadjicostas_, Jun 12 2019
%E A141066 More terms from _Petros Hadjicostas_, Jun 12 2019