cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A141064 List of different primes in Pascal-like triangles with index of asymmetry y = 1 and index of obliquity z = 0 or z = 1.

Original entry on oeis.org

2, 5, 7, 11, 23, 29, 89, 137, 311, 367, 1021, 3217, 5441, 2377, 12619, 65761, 5741, 144593, 13859, 78511, 1462397, 33461, 469957, 2552939, 11096497, 5930669, 6343133, 26512597, 470831, 127626137, 372222703, 15955507, 538270693, 531077333, 11401285549, 38613943, 15433507333, 92554537183, 113828092793
Offset: 1

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Author

Juri-Stepan Gerasimov, Jul 14 2008

Keywords

Comments

For the Pascal-like triangle G(n, k) with index of asymmetry y = 1 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, k) = G(n+1, k-1) + G(n+1, k) + G(n+2, k) for n >= 0 and k = 1..(n+1).
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, k) = G(n+1, k-1) + G(n+1, k-2) + G(n+2, k-1) for n >= 0 and k = 2..(n+2).
In each row of A140998, the primes not appearing in earlier rows are collected, sorted, and added to the sequence. [R. J. Mathar, Apr 28 2010]
From Petros Hadjicostas, Jun 10 2019: (Start)
For the triangle with index of asymmetry y = 1 and index of obliqueness z = 0, read by rows, we have G(n, k) = A140998(n, k) for 0 <= k <= n.
For the triangle with index of asymmetry y = 1 and index of obliqueness z = 1, read by rows, we have G(n, k) = A140993(n+1, k+1) for 0 <= k <= n.
Thus, except for the (unfortunate) shifting of the indices by 1, triangular arrays A140998 and A140993 are mirror images of each other.
Hence, instead of working with A140998, we may work with A140993: in each row of A140993, the primes not appearing in earlier rows may be collected, sorted, and added to the sequence (paraphrasing R. J. Mathar above!).
(End)

Examples

			Pascal-like triangle with y = 1 and z = 0 (i.e, A140998) begins as follows:
  1, so no prime.
  1 1, so no primes.
  1 2 1, so a(1) = 2.
  1 4 2 1, so no new primes.
  1 7 5 2 1, so a(2) = 5 and a(3) = 7.
  1 12 11 5 2 1, so a(4) = 11.
  1 20 23 12 5 2 1, so a(5) = 23.
  1 33 46 28 12 5 2 1, so no new primes.
  1 54 89 63 29 12 5 2 1, so a(6) = 29 and a(7) = 89.
  1 88 168 137 69 29 12 5 2 1, so a(8) = 137.
  1 143 311 289 161 70 29 12 5 2 1, so a(9) = 311.
  1 232 567 594 367 168 70 29 12 5 2 1, so a(10) = 367.
...
[edited by _Petros Hadjicostas_, Jun 11 2019]

Links

Crossrefs

Cf. A140993, A140994, A140995, A140996, A140997, A140998, A141021, A141022, A141031, A141066, A141067.

Programs

  • Maple
    # This is a modification R. J. Mathar's program from A141031 (for the case y = 4 and z = 0).
# Construct array A140998 (y = 1 and z = 0):
A140998 := 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; else procname(n - 1, k) + procname(n - 2, k) + procname(n - 2, k - 1); end if; end proc;
# Construct the current sequence:
A141064 := 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 := A140998(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;
# Generate terms of the current sequence:
A141064(38);
# If one wants to get the primes sorted, then replace RETURN(a) in the Maple 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

Extensions

Partially edited by N. J. A. Sloane, Jul 18 2008
More terms from R. J. Mathar, Apr 28 2010
More terms from Petros Hadjicostas, Jun 11 2019

A308808 Limiting row sequence of Pascal-like triangle A141021 (with index of asymmetry s = 4).

Original entry on oeis.org

1, 2, 4, 8, 16, 33, 67, 136, 276, 561, 1140, 2316, 4705, 9559, 19421, 39457, 80163, 162864, 330885, 672247, 1365779, 2774802, 5637462, 11453422, 23269491, 47275758, 96048397, 195137952, 396454511, 805461867, 1636426882, 3324667561, 6754603284, 13723075972, 27880662448, 56644103708
Offset: 0

Views

Author

Petros Hadjicostas, Jun 25 2019

Keywords

Comments

In the attached photograph, we see that the index of asymmetry is denoted by s and the index of obliqueness by e. The general recurrence is 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) for n >= 0 with k = 1..(n+1) when e = 0 and k = (s+1)..(n+s+1) when e = 1. The initial conditions are G(n+x+1, n-e*n+e*x-e+1) = 2^x for x=0..s and n >= 0. There is one more initial condition, namely, G(n, e*n) = 1 for n >= 0.
For s = 0, we get Pascal's triangle A007318. For s = 1, we get A140998 (e = 0) and A140993 (e = 1). For s = 2, we get A140997 (e = 0) and A140994 (e = 1). For s = 3, we get A140996 (e = 0) and A140995 (e = 1). For s = 4, we have arrays A141020 (with e = 0) and A141021 (with e = 1). In some of these arrays, the indices n and k are shifted.
For the triangular array G(n, k) = A141021(n, k), we have G(n+6, k) = G(n+1, k-4) + G(n+1, k-5) + G(n+2, k-4) + G(n+3, k-3) + G(n+4, k-2) + G(n+5, k-1) for n >= 0 and k = 5..(n+5) with G(n+x+1, x) = 2^x for x = 0..4 and n >= 0.
With G(n, k) = A141021(n, k), the current sequence (a(k): k >= 0) is defined by a(k) = lim_{n -> infinity} G(n, k) for k >= 0. Then a(k) = a(k-5) + 2*a(k-4) + a(k-3) + a(k-2) + a(k-1) for k >= 5 with a(x) = 2^x for x = 0..4.

Links

Crossrefs

Cf. A007318, A140993, A140994, A140995, A140996, A140997, A140998, A141015, A141018, A141020, A141021, A141031, A141065, A141066, A141067.

Formula

a(k) = a(k-5) + 2*a(k-4) + a(k-3) + a(k-2) + a(k-1) for k >= 5 with a(k) = 2^k for k = 0..4.
G.f.: -(x + 1)*(x^2 + 1)/(x^5 + 2*x^4 + x^3 + x^2 + x - 1).

A309462 Limiting row sequence for Pascal-like triangle A140995 (Stepan's triangle with index of asymmetry s = 3).

Original entry on oeis.org

1, 2, 4, 8, 17, 35, 72, 149, 308, 636, 1314, 2715, 5609, 11588, 23941, 49462, 102188, 211120, 436173, 901131, 1861732, 3846329, 7946496, 16417420, 33918306, 70075047, 144774689, 299103768, 617946857, 1276675050, 2637604132, 5449276664, 11258177753, 23259337731
Offset: 0

Views

Author

Petros Hadjicostas, Aug 03 2019

Keywords

Comments

In the attached photograph, we see that the index of asymmetry is denoted by s and the index of obliqueness by e. The general recurrence is 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) for n >= 0 with k = 1..(n+1) when e = 0 and k = (s+1)..(n+s+1) when e = 1. The initial conditions are G(n+x+1, n-e*n+e*x-e+1) = 2^x for x=0..s and n >= 0. There is one more initial condition, namely, G(n, e*n) = 1 for n >= 0.
For s = 0, we get Pascal's triangle A007318. For s = 1, we get A140998 (e = 0) and A140993 (e = 1). For s = 2, we get A140997 (e = 0) and A140994 (e = 1). For s = 3, we get A140996 (e = 0) and A140995 (e = 1). For s = 4, we have arrays A141020 (with e = 0) and A141021 (with e = 1). In some of these arrays, the indices n and k are shifted.
For the triangular array G(n, k) = A140995(n, k), 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).
With G(n, k) = A140995(n, k), the current sequence (a(k): k >= 0) is defined by a(k) = lim_{n -> infinity} G(n, k) for k >= 0. Then a(k) = a(k-4) + 2*a(k-3) + a(k-2) + a(k-1) for k >= 4 with a(x) = 2^x for x = 0..3.

Links

Crossrefs

Cf. A007318, A140993, A140994, A140995, A140996, A140997, A140998, A141020, A141021, A141031, A141065, A141066, A141067, A141068, A141069, A141070, A141072, A141073, A308808.

Formula

a(0) = 1, a(1) = 2, a(2) = 4, a(3) = 8, and a(k) = a(k-1) + a(k-2) + 2*a(k-3) + a(k-4) for k >= 4.
G.f.: (x^2 + x + 1)/(-x^4 - 2*x^3 - x^2 - x + 1).
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