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%I #75 Jan 28 2024 14:21:28
%S 4,8,16,35,60,72,116,148,224,303,432,308,618,833,636,1257,1606,1313,
%T 2550,3096,1314,2709,5160,5968,2715,5584,10418,11504,5609,11499,20991,
%U 22175,23655,42215,42744,11588,23934,48607,82392,84752,23941,99763,158816,169880
%N List of different composites in Pascal-like triangles with index of asymmetry y = 3 and index of obliqueness z = 0 or z = 1.
%C 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 n >= 0 and k = 1..(n+1). (This is array A140996.)
%C 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 n >= 0 and k = 4..(n+4). (This is array A140995.)
%C From _Petros Hadjicostas_, Jun 13 2019: (Start)
%C The arrays A140995 and A140996, which are described above, are mirror images of one another.
%C 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 composites 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 Finally, we explain the meaning of the double recurrence in the attached photograph. It concerns the connection between Stepan's triangles and Pascal's triangles. The creator of the stone slab uses the notation G_n^k to denote the double array G(n, k), where 0 <= k <= n.
%C On the stone slab, the letter s is used to denote the "index of asymmetry" (denoted by y here) and the letter e is used to denote the 0-1 "index of obliqueness" (denoted by z here). Thus, as described above, there are two kinds of Stepan-Pascal triangles depending on whether e is equal to 0 or 1. (The case s = 0 corresponds to Pascal's triangle A007318.)
%C If e = 0, the value of k goes from 1 to n + 1, whereas if e = 1 the value of k goes from s + 1 to n + s + 1.
%C The "index of asymmetry" s = y can take any (fixed) integer value from 0 to infinity. The fixed value of s = y determines the number of initial conditions: G(n + x + 1, n - e*n + e*x - e + 1) = 2^x for x = 0, 1, ..., s. In addition, there is one more initial condition: G(n, e*n) = 1.
%C The "index of asymmetry" s = y also determines the order of the recurrence (which is probably s + 2 = y + 2): G(n + s + 2, k) = G(n + 1, k - e*s + e - 1) + Sum_{1 <= m <= s + 1} G(n + m, k - e*s + m*e - 2*e).
%C (End)
%H Petros Hadjicostas, <a href="/A141069/b141069.txt">Table of n, a(n) for n = 1..110</a>
%H Juri-Stepan Gerasimov, <a href="/A140998/a140998.jpg">Stepan's triangles and Pascal's triangle are connected by the recurrence relation ...</a>
%e Pascal-like triangle with y = 3 and z = 0 (i.e., A140996) begins as follows:
%e 1, so no composites.
%e 1 1, so no composites.
%e 1 2 1, so no composites.
%e 1 4 2 1, so a(1) = 4.
%e 1 8 4 2 1, so a(2) = 8.
%e 1 16 8 4 2 1, so a(3) = 16.
%e 1 31 17 8 4 2 1, so no new composites.
%e 1 60 35 17 8 4 2 1, so a(4) = 35 and a(5) = 60.
%e 1 116 72 35 17 8 4 2 1, so a(6) = 72 and a(7) = 116.
%e 1 224 148 72 35 17 8 4 2 1, so a(8) = 148 and a(9) = 224.
%e 1 432 303 149 72 35 17 8 4 2 1, so a(10) = 303 and a(11) = 432.
%e ... [edited by _Petros Hadjicostas_, Jun 13 2019]
%p # This is a modification of _R. J. Mathar_'s program from sequence A141031 (for the case y = 4 and z = 0).
%p # Definition of sequence A140996 (y = 3 and z = 0):
%p 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 # Definition of current sequence:
%p A141069 := 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 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 # Generation of current sequence until row n = 30:
%p A141069(30);
%p # If one wishes the composites to be sorted, then replace RETURN(a) with RETURN(sort(a)) in the above Maple code. In such a case, however, the output may not necessarily be uniquely defined (because it changes with the value of n). - _Petros Hadjicostas_, Jun 15 2019
%Y Cf. A007318, A140993, A140994, A140995, A140996, A140997, A140998, A141020, A141021, A141031, A141064, A141065, A141066, A141067, A141069, A141070, A141072, A141073.
%K nonn
%O 1,1
%A _Juri-Stepan Gerasimov_, Jul 16 2008
%E Partially edited by _N. J. A. Sloane_, Jul 18 2008
%E More terms from _Petros Hadjicostas_, Jun 13 2019
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