A173027 -id:A173027 - cp's OEIS frontend

A269725 a(n) = row number of extended Wythoff array (see A035513) which contains the sequence n times the Fibonacci numbers 1,2,3,5,8,13,21,... .

0, 2, 3, 4, 15, 18, 21, 24, 27, 30, 33, 96, 104, 112, 120, 128, 136, 144, 152, 160, 168, 176, 184, 192, 200, 208, 216, 224, 232, 630, 651, 672, 693, 714, 735, 756, 777, 798, 819, 840, 861, 882, 903, 924, 945, 966, 987, 1008, 1029, 1050, 1071, 1092, 1113, 1134, 1155, 1176, 1197, 1218, 1239, 1260

Offset: 1

N. J. A. Sloane, Mar 07 2016

			Take n=5: 5 times 1,2,3,5,8,13,... gives 5,10,15,25,40,65,.., which is row 15 of the extended Wythoff array (when extended to the left), so a(5) = 15.

J. H. Conway, Posting to Math Fun Mailing List, Nov 25 1996.

J. H. Conway, Allan Wechsler, Marc LeBrun, Dan Hoey, N. J. A. Sloane, On Kimberling Sums and Para-Fibonacci Sequences, Correspondence and Postings to Math-Fun Mailing List, Nov 1996 to Jan 1997

Cf. A000045, A022413-A022423, A035513, A269726, A269729.

A269725 := proc(n)
    local f,sl,r,c,wrks ;
    f := [seq(n*combinat[fibonacci](i),i=2..30)] ;
    for sl from 0 do
        for r from 1 do
            if A035513(r,1) = op(1+sl,f) then
                wrks := true;
                for c from 2 to 5 do
                    if A035513(r,c) <> op(c+sl,f) then
                        wrks := false;
                    end if;
                end do:
                if wrks then
                    print(n,f,r) ;
                    return r-1 ;
                end if;
            elif A035513(r,1) > op(1+sl,f) then
                break ;
            end if;
        end do:
    end do:
end proc: # R. J. Mathar, May 06 2017

W[n_, k_] := Fibonacci[k+1] Floor[n*GoldenRatio] + (n-1) Fibonacci[k];
a[n_] := Module[{f, sl, r, c, wrks}, f = Table[n*Fibonacci[i], {i, 2, 30}]; For[sl = 0, True, sl++, For[r = 1, True, r++, Which[W[r, 1] == f[[1 + sl]], wrks = True; For[c = 2, c <= 5, c++, If[W[r, c] != f[[c+sl]], wrks = False]]; If[wrks, Return[r-1]], W[r, 1] > f[[1+sl]], Break[]]]]];
Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Sep 13 2022, after R. J. Mathar *)

a(n) = A173027(n)-1. - R. J. Mathar, May 06 2017

A332938 Indices of the primitive rows of the Wythoff array (A035513); see Comments.

Original entry on oeis.org

1, 2, 6, 7, 8, 10, 11, 12, 14, 17, 18, 20, 21, 23, 24, 26, 27, 30, 32, 33, 36, 37, 38, 39, 40, 42, 44, 46, 48, 49, 50, 53, 54, 59, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 79, 80, 81, 84, 85, 86, 88, 90, 92, 94, 95, 98, 100, 101, 102, 104, 107

Offset: 1

Clark Kimberling, Mar 03 2020

In a row of the Wythoff array, either every two consecutive terms are relatively prime or else no two consecutive terms are relatively prime. In the first case, we call the row primitive; otherwise, the row is an integer multiple of a tail of a preceding row. Conjectures: the maximal number of consecutive primitive rows is 5, and the limiting proportion of primitive rows exists and is approximately 0.608.

			The Wythoff array begins:
   1    2    3    5    8   13   21   34   55   89  144 ...
   4    7   11   18   29   47   76  123  199  322  521 ...
   6   10   16   26   42   68  110  178  288  466  754 ...
   9   15   24   39   63  102  165  267  432  699 1131 ...
  12   20   32   52   84  136  220  356  576  932 1508 ...
  14   23   37   60   97  157  254  411  665 1076 1741 ...
  17   28   45   73  118  191  309  500  809 1309 2118 ...
  19   31   50   81  131  212  343  555  898 1453 2351 ...
  22   36   58   94  152  246  398  644 1042 1686 2728 ...
Row 1: A000045 (Fibonacci numbers, a primitive row)
Row 2: A000032 (Lucas numbers, primitive)
Row 3: 2 times a tail of row 1
Row 4: 3 times a tail of row 1
Row 5  4 times a tail of row 1
Row 6:  essentially A000285, primitive
Row 7:  essentially A022095, primitive
Row 8:  essentially A013655, primitive
Row 9:  2 times a tail of row 2
Thus first five terms of (a(n)) are 1,2,6,7,8.

Cf. A000045, A173027, A173028, A035513, A332937.

W[n_, k_] := Fibonacci[k + 1] Floor[n*GoldenRatio] + (n - 1) Fibonacci[k]; (* A035513 *)
t = Table[GCD[W[n, 1], W[n, 2]], {n, 1, 160}]  (* A332937 *)
Flatten[Position[t, 1]]  (* A332938 *)

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

A220249 Numbers of rows R of the Wythoff array such that R is the n-th multiple of a tail of the Lucas sequence.

Original entry on oeis.org

2, 9, 13, 45, 56, 67, 78, 89, 262, 291, 320, 349, 378, 407, 436, 465, 494, 523, 552, 581, 610, 1673, 1749, 1825, 1901, 1977, 2053, 2129, 2205, 2281, 2357, 2433, 2509, 2585, 2661, 2737, 2813, 2889, 2965, 3041, 3117, 3193, 3269, 3345, 3421, 3497, 3573, 3649

Offset: 1

K. G. Stier, Dec 08 2012

nonn

This sequence is corresponding to A173027. Also Row 2 of the array A173028.
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 odd-indexed Lucas numbers (A002878) while m is equal to odd-indexed Fibonacci numbers (A001519). Example: from n=988 to 2584 d=3571 and m=1597;
Also of interest are the gaps between two consecutive groups which appear to be sums of one Lucas number L(2n+1) plus one Fibonacci number F(4n). Example: gap 5 after a(55) is 6964 = L(11) + F(20) = 199 + 6765
Likewise, the tail (as mentioned in this sequence's name) of the Lucas sequence is chopped off by two initial terms at each of the gap positions.

			Referring to rows of the Wythoff array (A035513),
Row 2: (4,7,11,18,...) = 1*(4,7,11,18,29,47,76,...)
Row 9: (22,36,58,...) = 2*(11,18,29,47,76...)
Row 13: (33,54,87,...) = 3*(11,18,29,47,76...)
Row 45: (116,188,304,...) = 4*(29,47,76...)

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

Cf. A173027, A035513, A173028, A002878, A001519.

A332937 a(n) is the greatest common divisor of the first two terms of row n of the Wythoff array (A035513).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Mar 03 2020

a(n) is also the gcd of every pair of consecutive terms of row n of the Wythoff array. Conjectures: the maximal number of consecutive 1's is 5, and the limiting proportion of 1's exists. See A332938.

If seems that for all primes p > 3, a(1+p) = 1. - Antti Karttunen, Jan 15 2025

			See A332938.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000045, A173027, A173028, A035513, A332938 (positions of 1's).

W[n_, k_] := Fibonacci[k + 1] Floor[n*GoldenRatio] + (n - 1) Fibonacci[k]; (* A035513 *)
t = Table[GCD[W[n, 1], W[n, 2]], {n, 1, 160}]  (* A332937 *)
Flatten[Position[t, 1]]  (* A332938 *)

T(n, k) = (n+sqrtint(5*n^2))\2*fibonacci(k+1) + (n-1)*fibonacci(k); \\ A035513
a(n) = gcd(T(n, 0), T(n, 1)); \\ Michel Marcus, Mar 03 2020

More terms from Antti Karttunen, Jan 15 2025

cp's OEIS Frontend

A269725 a(n) = row number of extended Wythoff array (see A035513) which contains the sequence n times the Fibonacci numbers 1,2,3,5,8,13,21,... .

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A332938 Indices of the primitive rows of the Wythoff array (A035513); see Comments.

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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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A220249 Numbers of rows R of the Wythoff array such that R is the n-th multiple of a tail of the Lucas sequence.

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A332937 a(n) is the greatest common divisor of the first two terms of row n of the Wythoff array (A035513).

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