A220249 -id:A220249 - cp's OEIS frontend

A173027 Numbers of rows R of the Wythoff array such that R is the n-th multiple of a tail of the Fibonacci sequence.

1, 3, 4, 5, 16, 19, 22, 25, 28, 31, 34, 97, 105, 113, 121, 129, 137, 145, 153, 161, 169, 177, 185, 193, 201, 209, 217, 225, 233, 631, 652, 673, 694, 715, 736, 757, 778, 799, 820, 841, 862, 883, 904, 925, 946, 967, 988, 1009, 1030, 1051, 1072, 1093, 1114, 1135

Offset: 1

Clark Kimberling, Feb 07 2010

nonn

Row 1 of the array A173028.
Contribution from K. G. Stier, Dec 08 2012: (Start)
It appears that the numbers of this sequence form groups of m members respectively with same distance d of two consecutive values a(n) such that d is equal to even-indexed Fibonacci numbers (A001906) while m is equal to even-indexed Lucas numbers (A005248). Example: from n=1365 to 3571 d=987 and m=2207;
Also of interest are the gaps between two consecutive groups which appear to be sums of Fibonacci numbers F(2*n) plus F(4*n-2). Example: gap 5 after a(76) is 2639 = F(10) + F(18) = 55 + 2584.
Likewise, the tail (as mentioned in this sequence's name) of the Fibonacci sequence is chopped off by two initial terms at each of the gap positions. (End)

			Referring to rows of the Wythoff array (A035513),
Row 1: (1,2,3,5,...) = 1*(1,2,3,...)
Row 3: (6,10,16,...) = 2*(3,5,8,...)
Row 4: (9,15,24,...) = 3*(3,5,8,...)
Row 5: (12,20,32,...) = 4*(3,5,8,...)
Row 16: (40,65,105...) = 8*(5,13,21,...).

K. G. Stier, Table of n, a(n) for n = 1..10000

Cf. A000045, A035513, A173028, A220249.

A173028 Partition of the row numbers of the Wythoff array W: two numbers are in the same row if and only if their rows in W have (essentially) a common divisor greater than 1.

Original entry on oeis.org

1, 3, 2, 4, 9, 6, 5, 13, 29, 7, 16, 45, 43, 35, 8, 19, 56, 57, 52, 15, 10, 22, 67, 186, 181, 58, 51, 11, 25, 78, 223, 226, 77, 199, 55, 12, 28, 89, 260, 271, 96, 265, 82, 61, 14, 31, 262, 297, 316, 115, 331, 109, 91, 71, 17, 34, 291, 334, 361, 351, 397, 136, 317, 106, 87, 18

Offset: 1

Clark Kimberling, Feb 07 2010

(Row 1) = A173027, (Row 2) = A220249. Every positive integer occurs exactly once, so that, as a sequence, this is a permutation of the natural numbers.

			First four rows of R:
1...3....4....5.....16....19....22...25...28...
2...9....13...45....56....67....78...89...262..
6...29...43...57....186...223...260..297..334...
7...35...52...181...226...271...316..361..1063...
For example, row 3 begins with 6, which is the least positive
integer not in rows 1 and 2. Row 6 of W is (14,23,37,60,...)
Row 29 of W is (74,120,194,...) = 2*(37,60,97...).
Row 43 of W is (111,180,291,...) = 3*(37,60,97,...).
So row 3 of R begins with (6,29,43...) as there are no other rows
of W numbered <43 which are multiples of row 6 of W.

Cf. A000045, A035513, A173027, A220249.

Let R(n,k) be the number in row n, column k. After Row 1 (A173027),
inductively, R(n,1) is the least positive integer not in the first n-1
rows, and the rest of row n consists of the numbers of rows X of the
Wythoff array W for X a multiple of a tail of row R(n,1) of W.

Corrections (these have been made): a(31) should read 223 instead of 225, a(63) 317 instead of 314 - K. G. Stier, Dec 21 2012

A269726 a(n) = row number of extended Wythoff array (see A035513) which contains the sequence n times the Lucas numbers 1,3,4,7,11,18,... (A000204).

Original entry on oeis.org

1, 8, 12, 44, 55, 66, 77, 88, 261, 290, 319, 348, 377, 406, 435, 464, 493, 522, 551, 580, 609, 1672, 1748, 1824, 1900, 1976, 2052, 2128, 2204, 2280, 2356, 2432, 2508, 2584, 2660, 2736, 2812, 2888, 2964, 3040, 3116, 3192, 3268, 3344, 3420, 3496, 3572, 3648

Offset: 1

N. J. A. Sloane, Mar 07 2016

J. H. Conway, Posting to Math Fun Mailing List, Dec 02 1996.

Cf. A000045, A035513, A269725, A357097.

a(n) = A220249(n)-1. - R. J. Mathar, May 06 2017

a(n) = A357097(a(1), A269725(n)), where we write a(1) here to emphasize the semantics of the relationship. - Peter Munn, Aug 21 2025

cp's OEIS Frontend

A173027 Numbers of rows R of the Wythoff array such that R is the n-th multiple of a tail of the Fibonacci sequence.

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A173028 Partition of the row numbers of the Wythoff array W: two numbers are in the same row if and only if their rows in W have (essentially) a common divisor greater than 1.

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A269726 a(n) = row number of extended Wythoff array (see A035513) which contains the sequence n times the Lucas numbers 1,3,4,7,11,18,... (A000204).

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