

A141068


List of different primes in Pascallike triangles with index of asymmetry y = 3 and index of obliquity z = 0 or z = 1.


9



2, 17, 31, 149, 11587, 49429, 15701951, 21304973, 3846277, 251375273, 5449276159, 296410704409, 750391353973, 205109154121, 875366796349, 72210869205443, 139884035510017, 79014319582741129, 94461530406533783, 2562508045902551
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OFFSET

1,1


COMMENTS

For the Pascallike 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, k1) + 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.)
For the Pascallike 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, k3) + G(n+1, k4) + G(n+2, k3) + G(n+3, k2) + G(n+4, k1) for k = 4..(n+4). (This is array A140995.)
From Petros Hadjicostas, Jun 13 2019: (Start)
The two triangular arrays A140995 and A140996, which are described above, are mirror images of each other.
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.
Finally, we mention that in the attached picture about the connection between Stepan's triangles with the Pascal triangle, the letter s is used to describe the index of asymmetry and the letter e is used to describe he index of obliqueness
(instead of the letters y and z, respectively). The Pascal triangle A007318 has index of asymmetry s = y = 0.
(End)


LINKS

Table of n, a(n) for n=1..20.
JuriStepan Gerasimov, Pascallike triangles and Pascal's triangle are connected by the recurrence relation ...


EXAMPLE

Pascallike triangle with y = 3 and z = 0 (i.e., A140996) begins as follows:
1, so no primes.
1 1, so no primes
1 2 1, so a(1) = 2.
1 4 2 1, so no new primes.
1 8 4 2 1, so no new primes.
1 16 8 4 2 1, so new primes.
1 31 17 8 4 2 1, so a(2) = 17 and a(3) = 31.
1 60 35 17 8 4 2 1, so no new primes.
1 116 72 35 17 8 4 2 1, so no new primes.
1 224 148 72 35 17 8 4 2 1, so new primes.
1 432 303 149 72 35 17 8 4 2 1, so a(4) = 149.
...


MAPLE

# This is a modification of R. J. Mathar's program for A141031 (for the case y = 4 and z = 0).
# Definition of sequence A140996 (y = 3 and z = 0):
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;
# Definition of the current sequence:
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;
# Generation of the current sequence:
A141068(80);
# 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


CROSSREFS

Cf. A007318, A140993, A140994, A140995, A140996, A140997, A140998, A141020, A141021, A141031, A141064, A141065, A141066, A141067, A141069, A141070, A141072, A141073.
Sequence in context: A197186 A063118 A267540 * A162622 A078164 A060387
Adjacent sequences: A141065 A141066 A141067 * A141069 A141070 A141071


KEYWORD

nonn


AUTHOR

JuriStepan Gerasimov, Jul 16 2008


EXTENSIONS

Partially edited by N. J. A. Sloane, Jul 18 2008
More terms by Petros Hadjicostas, Jun 13 2019


STATUS

approved



