A020942 First column of 3rd-order Zeckendorf array A136189.

1, 5, 7, 10, 14, 18, 20, 24, 26, 29, 33, 35, 38, 42, 46, 48, 51, 55, 59, 61, 65, 67, 70, 74, 78, 80, 84, 86, 89, 93, 95, 98, 102, 106, 108, 112, 114, 117, 121, 123, 126, 130, 134, 136, 139, 143, 147, 149, 153, 155, 158, 162, 164, 167, 171, 175, 177, 180, 184

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• easy
• nice

Comments

I would like to get similar sequences where the least term in the representation is 2 [gives 2 8 11 15 21 27 30..., which is now A064105], 3, 4, 6, etc. They are the 2nd, 3rd, etc. columns of the 3rd-order Zeckendorf array. [See cross-references. - N. J. A. Sloane, Apr 29 2024]

These have now been entered in the OEIS as

column 1: A020942.

column 2: A064105.

column 3: A064106.

column 4: A372749.

column 5: A372750.

column 6: A372752.

column 7: A372756.

column 8: A372757.

Examples

			1=1; 5=4+1; 7=6+1; 10=9+1; etc.
		

Links

• A.H.M. Smeets, Table of n, a(n) for n = 1..20000
• Larry Ericksen and Peter G. Anderson, Patterns in differences between rows in k-Zeckendorf arrays, The Fibonacci Quarterly, Vol. 50, February 2012. - _N. J. A. Sloane_, Jun 10 2012
• Clark Kimberling, The Zeckendorf array equals the Wythoff array, Fibonacci Quarterly 33 (1995) 3-8.
• Jeffrey Shallit, The Narayana Morphism and Related Words, arXiv:2503.01026 [math.CO], 2025.

Crossrefs

Cf. A000930, A064105, A064106, A136189, A136496, A202342, A372749, A372750, A372752, A372756, A372757.

Formula

Any number n has unique representation as a sum of terms from {1, 2, 3, 4, 6, 9, 13, 19, ...} (cf. A000930) such that no two terms are adjacent or pen-adjacent; e.g., 7=6+1. Sequence gives all n where that representation involves 1.

Conjecture: a(n) = A202342(n) + n. - Sean A. Irvine, May 05 2019 [proved in corrected form in Shallit (2025); it should read a(n) = A202342(n) + n-1]

a(n) = A136496(n) - 1. - Jeffrey Shallit, Mar 08 2025

Extensions

More terms from Naohiro Nomoto, Sep 17 2001