A124171 -id:A124171 - cp's OEIS frontend

A124148 Fibonacci triangle read by rows; the triangles below read by rows. Analog of A124171.

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 3, 5, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 3, 5, 1, 1, 2, 3, 5, 8, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 3, 5, 1, 1, 2, 3, 5, 8, 1, 1, 2, 3, 5, 8, 13, 1, 1

Offset: 1

Jonathan Vos Post, Dec 13 2006

The function is slow-growing at first. The smallest n such that a(n) > n occurs when a(816) = 987. But eventually, the superpolynomial Fibonacci dominates the merely cubic tetrahedral numbers and the mean value of a(n)/n exceeds any fixed bound. There is a slower-starting such analog that starts with F(0) = 0 and F(1) = 1, the triangles beginning: 0 0 0, 1 0 0, 1 0, 1, 1 0 0, 1 0, 1, 1 0, 1, 1, 2 0 0, 1 0, 1, 1 0, 1, 1, 2 0, 1, 1, 2, 3; reading by rows gives offset 0,36 and many zeros.

			1
1
1, 1
1
1, 1
1, 1, 2
1
1, 1
1, 1, 2
1, 1, 2, 3
1
1, 1
1, 1, 2
1, 1, 2, 3
1, 1, 2, 3, 5

Cf. A000045, A000292, A124171.

Flatten[((Fibonacci@ Range@ # &) /@ Range@# &) /@ Range[10]] (* Giovanni Resta, Jun 16 2016 *)

a(n) = F(A124171(n)) = A000045(A124171(n)).
For k>0, max(row(T(k))) = F(k) where T(n) = A000217(k), F(k) = A000045(k).
Records for a(n) after a(1) = 1 are given by a(A000292(n)) = C(n+2,3) = n(n+1)(n+2)/6 = F(n+1) = A000045(n+1).

Data corrected by Giovanni Resta, Jun 16 2016

A333516 Irregular triangle read by rows in which row n lists the first A000217(n) terms of A002260, n >= 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 2, 3, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6

Offset: 1

Andrew Slattery, Mar 25 2020

a(n) equals the difference between n and the largest number less than n that can be expressed as the sum of the i-th triangular number and the j-th tetrahedral number for integers i < j.

			Triangle begins:
  1;
  1, 1, 2;
  1, 1, 2, 1, 2, 3;
  1, 1, 2, 1, 2, 3, 1, 2, 3, 4;
  1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5;
  1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6;
  1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7;
  ...

Row sums give A000292.
Right border gives A000027.
Cf. A000217, A002260, A124171.

T:= n-> seq([$1..i][], i=1..n):
seq(T(n), n=1..7);  # Alois P. Heinz, Apr 10 2020

from math import comb, isqrt
from sympy import integer_nthroot
def A333516(n): return (r:=n-1-comb((m:=integer_nthroot(6*n,3)[0])+(n>comb(m+2,3))+1,3))-comb((k:=isqrt(m:=r+1<<1))+(m>k*(k+1)),2)+1 # Chai Wah Wu, Nov 10 2024

a(n) = A002260(A124171(n)).

A115215 Sequence obtained by reading the triangles shown below by diagonals.

Original entry on oeis.org

1, 1, 3, 2, 1, 3, 6, 2, 5, 4, 1, 3, 6, 10, 2, 5, 9, 4, 8, 7, 1, 3, 6, 10, 15, 2, 5, 9, 14, 4, 8, 13, 7, 12, 11, 1, 3, 6, 10, 15, 21, 2, 5, 9, 14, 20, 4, 5, 8, 13, 19, 7, 12, 18, 11, 17, 16, 1, 3, 6, 10, 15, 21, 28, 2, 5, 9, 14, 20, 27, 4, 8, 13, 19, 26, 7, 12, 18, 25, 11, 17, 24, 16, 23, 22

Offset: 1

Colm Mulcahy, Dec 05 2006

nonn

			1
1
2 3
1
2 3
4 5 6
1
2 3
4 5 6
7 8 9 10
1
2 3
4 5 6
7 8 9 10
11 12 13 14 15

See A124171 for another version.

More terms from Jonathan R. Love (japanada11(AT)yahoo.ca), Mar 07 2007

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A124148 Fibonacci triangle read by rows; the triangles below read by rows. Analog of A124171.

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A333516 Irregular triangle read by rows in which row n lists the first A000217(n) terms of A002260, n >= 1.

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A115215 Sequence obtained by reading the triangles shown below by diagonals.

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