A308782 Index of first occurrence of n appearing twice in succession in van Eck's sequence (A181391), or 0 if it never occurs.

2, 0, 8, 50, 132, 414, 171, 659, 96, 361, 12311, 7224, 5896, 2954, 5804, 72387, 12756, 1292, 4332, 3715, 2704, 1887, 5780, 9837, 11721, 1094, 70067, 32610, 57658, 26146, 167389, 94957, 36588, 19663, 35588, 9627, 108296, 51653, 38147, 54788, 81871, 15502

Offset: 0

Author

Deron Stewart, Jun 24 2019

Keywords

• nonn

Comments

With V=A181391, a(n) is the smallest number m such that V(m) = V(m-1) = n.

Since van Eck's sequence is generated by considering the gap between identical terms reappearing, it is of interest to consider terms of value n which repeat with a gap of length n.

When this happens the term is repeated in succession.

Some observations that follow from the definition of V:

V(a(n)-1-n) = n. The value n has to appear exactly n terms apart in V to make the following term equal to n, e.g., for n=3: "..., 3, 8, 0, 3, 3, ...".

V(a(n)+1) = 1. Since the term n appeared twice in a row, the following term of V must be 1.

V(a(n)-2) = V(a(n)-n-2) = V(a(n)-2*n-2). The number preceding the repeated terms appears three times with gaps of n.

V(a(n)+2) = the number of terms since the previous repeated value of some number (though it may not be the first time it is repeated). So V(a(n)-V(a(n)+2)) = V(a(n)-V(a(n)+2)-1).

Examples

			With V=A181391 and n=8:
V(95) = V(96) = 8. Therefore, a(8) = 96.
---
V(a(n)-1-n) = n:
a(8) - 1 - 8 = 87.
V(87) = 8.
---
V(a(n)+1) = 1:
a(8) + 1 = 97.
V(97) = 1.
---
V(a(n)-2) = V(a(n)-n-2) = V(a(n)-2*n-2):
a(8) - 2 = 94.
a(8) - 8 - 2 = 86.
a(8) - 2*8 - 2 = 78.
V(94) = V(86) = V(78) = 3.
---
V(a(8)+2) = 46. a(8) - 46 = 50. The previous repeated terms in V are V(50) = V(49) = 5.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..999
• Rémy Sigrist, C++ program for A308782

Crossrefs

Cf. A181391 (van Eck's sequence).