A157128 -id:A157128 - cp's OEIS frontend

A157127 A transform of the Catalan numbers with a simple Hankel transform.

1, 0, -1, -1, 1, 2, 1, -3, -5, -1, 11, 15, -5, -44, -41, 51, 169, 85, -303, -611, -15, 1513, 2021, -1271, -6827, -5673, 10361, 28301, 10141, -61102, -107015, 20109, 310387, 356515, -354349, -1421913, -936805, 2567875, 5930631, 972819, -14696329

Offset: 0

Paul Barry, Feb 23 2009

Hankel transform is A157128. Partial sums of A157125.

G.f.: (sqrt(1+x^2+4x^3)-sqrt(1+x^2))/(2*sqrt(1+x^2));

a(n)=sum{k=0..n, A157125(k)};

A308293 Lexicographically earliest sequence of positive terms such that a(1) = 1, a(2) = 2, and for any n > 0, (abs(a(n+2)-a(n)), abs(a(n+2)-a(n+1))) is unique.

Original entry on oeis.org

1, 2, 1, 1, 1, 2, 3, 1, 1, 3, 1, 4, 1, 1, 4, 5, 1, 1, 5, 1, 2, 4, 6, 1, 1, 6, 1, 4, 6, 2, 1, 6, 2, 7, 1, 1, 7, 1, 8, 1, 1, 8, 3, 1, 7, 8, 1, 2, 5, 8, 1, 9, 1, 1, 9, 3, 1, 8, 9, 1, 3, 6, 10, 1, 1, 10, 1, 5, 8, 2, 10, 1, 8, 10, 1, 11, 1, 1, 11, 4, 1, 9, 10, 1

Offset: 1

Rémy Sigrist, May 19 2019

nonn

This sequence shows chaotic behavior (see scatterplot in Links section).
This behavior is determined by the choice of the two leading terms.
The variant, say b, with b(1) = b(2) = 1, corresponds to the natural numbers interspersed with pairs of ones: 1,1,1, 2,1,1, 3,1,1, etc. (b(n) = abs(A157128(n))).

			The first terms, alongside (abs(a(n+2)-a(n)), abs(a(n+2)-a(n+1))), are:
  n   a(n)  (abs(a(n+2)-a(n)),abs(a(n+2)-a(n+1)))
  --  ----  -------------------------------------
   1     1  (0,1)
   2     2  (1,0)
   3     1  (0,0)
   4     1  (1,1)
   5     1  (2,1)
   6     2  (1,2)
   7     3  (2,0)
   8     1  (2,2)
   9     1  (0,2)
  10     3  (1,3)
  11     1  (0,3)
  12     4  (3,0)
  13     1  (3,3)
  14     1  (4,1)
  15     4  (3,4)

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Colored scatterplot of (abs(a(n+2)-a(n)), abs(a(n+2)-a(n+1))) for n = 1..32702782 (where the hue is function of n)
Rémy Sigrist, C program for A308293

See A080427 for a simpler variant.

Cf. A157128.

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cp's OEIS Frontend

A157127 A transform of the Catalan numbers with a simple Hankel transform.

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A308293 Lexicographically earliest sequence of positive terms such that a(1) = 1, a(2) = 2, and for any n > 0, (abs(a(n+2)-a(n)), abs(a(n+2)-a(n+1))) is unique.

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