A019586 -id:A019586 - cp's OEIS frontend

A026355 a(n) = least k such that s(k) = n+1, where s = A026354.

1, 2, 3, 5, 6, 8, 10, 11, 13, 14, 16, 18, 19, 21, 23, 24, 26, 27, 29, 31, 32, 34, 35, 37, 39, 40, 42, 44, 45, 47, 48, 50, 52, 53, 55, 57, 58, 60, 61, 63, 65, 66, 68, 69, 71, 73, 74, 76, 78, 79, 81, 82, 84, 86, 87, 89, 90, 92, 94, 95, 97, 99, 100, 102, 103, 105, 107, 108, 110

Offset: 0

Clark Kimberling

Let f(1)=1, f(2)=q and f(k+2) = f(k+1)+f(k)-n; a(n) is the smallest positive integer q such that f(k) -> infinity as k -> infinity. - Benoit Cloitre, Aug 04 2002

J. H. Conway and N. J. A. Sloane, Notes on the Para-Fibonacci and related sequences

Cf. A000201, A005614, A026351. Different from A007067.

from math import isqrt
def A026355(n): return (n-1+isqrt(5*(n-1)**2)>>1)+2 if n else 1 # Chai Wah Wu, Aug 25 2022

For n>0, a(n) = floor((n-1)*phi) + 2, where phi=(1+sqrt(5))/2.

Recurrences: a(n+1) = a(n)+(3 + sign(phi*n-a(n)))/2 for n>=0. Also a(n+1) = a(n) + 1 + A005614(n-2) for n>=2. - Benoit Cloitre, Aug 04 2002

A348853 Delete any least significant 0's from the Zeckendorf representation of n, leaving its "odd" part.

Original entry on oeis.org

1, 1, 1, 4, 1, 6, 4, 1, 9, 6, 4, 12, 1, 14, 9, 6, 17, 4, 19, 12, 1, 22, 14, 9, 25, 6, 27, 17, 4, 30, 19, 12, 33, 1, 35, 22, 14, 38, 9, 40, 25, 6, 43, 27, 17, 46, 4, 48, 30, 19, 51, 12, 53, 33, 1, 56, 35, 22, 59, 14, 61, 38, 9, 64, 40, 25, 67, 6, 69, 43, 27, 72

Offset: 1

Kevin Ryde, Nov 14 2021

Terms are odd Zeckendorfs A003622 and the fixed points are where n is odd already so that a(n) = n iff n is in A003622.
A139764(n) is the least significant "10..00" part of n so Zeckendorf multiplication n = A101646(a(n), A139764(n)).
The equivalent delete least significant 0's in binary is A000265 so that conversion to Fibbinary (A003714) and back gives a(n) = A022290(A000265(A003714(n))).
a(n) = 1 iff n is a Fibonacci number >= 1 (A000045) since they are Zeckendorf 100..00.
a(n) = 4 iff n is a Lucas number >= 4 (A000032) since they are Zeckendorf 10100..00 which reduces to 101.
In the Wythoff array A035513, a(n) is the term in column 0 of the row containing n, and hence the formula below using row number A019586 to select which of the odds (column 0) is a(n).

			n    = 81 = Zeckendorf 101001000.
a(n) = 19 = Zeckendorf 101001.

Cf. A189920 (Zeckendorf digits), A003622 (odds), A003849 (final digit), A005206, A319433 (shift down).
Cf. A000045 (Fibonacci), A000032 (Lucas).
Cf. A035513 (Wythoff array), A019586 (row number).
Cf. A003714 (Fibbinary), A022290 (its inverse).
In other bases: A000265 (binary), A004151 (decimal).
Cf. A001622, A101646, A139764.

my(phi=quadgen(5)); a(n) = my(q,r); while([q,r]=divrem(n+2,phi); r<1, n=q-1); n;

a(n) = n if A003849(n)=1, otherwise a(n) = a(A005206(n)) = a(A319433(n)).
a(n) = A003622(A019586(n) + 1).
Sum_{k=1..n} a(k) ~ n^2/(2*phi), where phi is the golden ratio (A001622). - Amiram Eldar, Feb 17 2024

A035339 5th column of Wythoff array.

Original entry on oeis.org

8, 29, 42, 63, 84, 97, 118, 131, 152, 173, 186, 207, 228, 241, 262, 275, 296, 317, 330, 351, 364, 385, 406, 419, 440, 461, 474, 495, 508, 529, 550, 563, 584, 605, 618, 639, 652, 673, 694, 707, 728, 741, 762, 783, 796, 817, 838, 851, 872, 885, 906, 927, 940, 961

Offset: 0

N. J. A. Sloane and J. H. Conway

nonn

The asymptotic density of this sequence is 1/phi^6 = A094214^6 = 0.05572809... . - Amiram Eldar, Mar 24 2025

Seiichi Manyama, Table of n, a(n) for n = 0..10000
John H. Conway and N. J. A. Sloane, Notes on the Para-Fibonacci and related sequences.
N. J. A. Sloane, Classic Sequences.

Column k of A035513: A003622 (k=1), A035336 (k=2), A035337 (k=3), A035338 (k=4), this sequence (k=5), A035340 (k=6).

Cf. A094214.

t:= (1+sqrt(5))/2: [ seq(8*floor((n+1)*t)+5*n,n=0..80) ];

a[n_] := 8 * Floor[n * GoldenRatio] + 5*(n-1); Array[a, 100] (* Amiram Eldar, Mar 24 2025 *)

A035340 6th column of Wythoff array.

Original entry on oeis.org

13, 47, 68, 102, 136, 157, 191, 212, 246, 280, 301, 335, 369, 390, 424, 445, 479, 513, 534, 568, 589, 623, 657, 678, 712, 746, 767, 801, 822, 856, 890, 911, 945, 979, 1000, 1034, 1055, 1089, 1123, 1144, 1178, 1199, 1233, 1267, 1288, 1322, 1356, 1377, 1411, 1432

Offset: 0

N. J. A. Sloane and J. H. Conway

nonn

The asymptotic density of this sequence is 1/phi^7 = A094214^7 = 0.03444185... . - Amiram Eldar, Mar 24 2025

Seiichi Manyama, Table of n, a(n) for n = 0..10000
John H. Conway and N. J. A. Sloane, Notes on the Para-Fibonacci and related sequences.
N. J. A. Sloane, Classic Sequences.

Column k of A035513: A003622 (k=1), A035336 (k=2), A035337 (k=3), A035338 (k=4), A035339 (k=5), this sequence (k=6).

Cf. A094214.

t:= (1+sqrt(5))/2: [ seq(13*floor((n+1)*t)+8*n,n=0..80) ];

a[n_] := 13 * Floor[n * GoldenRatio] + 8*(n-1); Array[a, 100] (* Amiram Eldar, Mar 24 2025 *)

A051656 Sum_{i=0..n} (C(n,i) mod 2)Fibonacci(2i).

Original entry on oeis.org

0, 1, 3, 12, 21, 77, 168, 609, 987, 3572, 7755, 28059, 47376, 171409, 372099, 1346268, 2178309, 7881197, 17108664, 61899729, 104512485, 378129724, 820851717, 2969869413, 4809706272, 17401680769, 37775923491, 136674575148

Offset: 0

Antti Karttunen, Nov 30 1999

Positions in the first column (A003622) of Wythoff array of the terms which have their Zeckendorf Expansion patterned as row[2n+1] in Pascal's Triangle computed modulo 2 (A047999)

Proof in preparation, to be published (see A048757).

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A048757, A047999, A035513, A038183, A051256. First row of A050609, First column of A050610.
a(n) = A019586[A048757[n]]. A048757[n] = SS(Athis_sequence[n])+1, where SSx means the second Fibonacci Successor of x (= x's Z.E. shifted left twice).
Cf. A001906.

a051656 = sum . zipWith (*) a001906_list . a047999_row
-- Reinhard Zumkeller, Feb 27 2011

Table[Sum[Mod[Binomial[n,i],2]Fibonacci[2i],{i,0,n}],{n,0,30}] (* Harvey P. Dale, Oct 30 2011 *)

a(n)=sum(i=0,n,if(!bitand(i,n-i),fibonacci(2*i))) \\ Charles R Greathouse IV, Jan 04 2013

a(n) = sum_{i=0..n} (C(2n, 2i) mod 2)*F(2*i) = FL(n)product_{i=0..inf} L(2^i)^bit(n, i) where L is n-th Lucas number (A000032) and FL is defined as in A048757: FL(n) = n-th fibonacci number if n even, n-th Lucas number if n odd.

A098862 a(n) = column number of Stolarsky array that contains n.

Original entry on oeis.org

0, 1, 2, 0, 3, 1, 0, 4, 0, 2, 1, 0, 5, 0, 1, 3, 0, 2, 1, 0, 6, 0, 1, 2, 0, 4, 0, 1, 3, 0, 2, 1, 0, 7, 0, 1, 2, 0, 3, 1, 0, 5, 0, 1, 2, 0, 4, 0, 1, 3, 0, 2, 1, 0, 8, 0, 1, 2, 0, 3, 1, 0, 4, 0, 2, 1, 0, 6, 0, 1, 2, 0, 3, 1, 0, 5, 0, 1, 2, 0, 4, 0, 1, 3, 0, 2, 1, 0, 9

Offset: 1

Casey Mongoven, Oct 12 2004

nonn

D. R. Morrison, A Stolarsky Array of Wythoff Pairs, A Collection of Manuscripts Related to the Fibonacci Sequence, edited by V. E. Hoggatt Jr., M. Bicknell-Johnson, published by The Fibonacci Association, (1980) pp. 134-136.

Ron Knott, General Fibonacci Series
Casey Mongoven, Sonification of multiple Fibonacci-related sequences, Annales Mathematicae et Informaticae, 41 (2013) pp. 175-192.

Cf. A035614, A019586, A098862.

The segment between the first x and the first x+1 is given by the segment before the first x-1.

A167198 Fractal sequence of the interspersion A083047.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 4, 2, 3, 5, 1, 4, 6, 2, 7, 3, 5, 8, 1, 9, 4, 6, 10, 2, 7, 11, 3, 12, 5, 8, 13, 1, 9, 14, 4, 15, 6, 10, 16, 2, 17, 7, 11, 18, 3, 12, 19, 5, 20, 8, 13, 21, 1, 22, 9, 14, 23, 4, 15, 24, 6, 25, 10, 16, 26, 2, 17, 27, 7, 28, 11, 18, 29, 3, 30, 12, 19, 31, 5, 20, 32, 8, 33, 13

Offset: 1

Clark Kimberling, Oct 30 2009

nonn

As a fractal sequence, if the first occurrence of each term is deleted, the remaining sequence is the original. In general, the interspersion of a fractal sequence is constructed by rows: row r consists of all n, such that a(n)=r; in particular, A083047 is constructed in this way from A167198.

a(n-1) gives the row number which contains n in the dual Wythoff array A126714 (beginning the row count at 1), see also A223025 and A019586. - Casey Mongoven, Mar 11 2013

			To produce row 5, first write row 4: 2,3,1, then place 4 right before 2, and then place 5 right before 1, getting 4,2,3,5,1.

Clark Kimberling, Stolarsky interspersions, Ars Combinatoria 39 (1995), 129-138.

Clark Kimberling, The first column of an interspersion, The Fibonacci Quarterly 32 (1994), 301-315.

Cf. A003603, A083047, A035513, A000045.

Following is a construction that avoids reference to A083047.
Write initial rows:
Row 1: .... 1
Row 2: .... 1
Row 3: .... 2..1
Row 4: .... 2..3..1
For n>=4, to form row n+1, let k be the least positive integer not yet used; write row n, and right before the first number that is also in row n-1, place k; right before the next number that is also in row n-1, place k+1, and continue. A167198 is the concatenation of the rows. (If "before" is replaced by "after", the resulting fractal sequence is A003603, and the associated interspersion is the Wythoff array, A035513.)

A167237 Lower trim of the Wythoff fractal sequence, A003603.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 5, 3, 2, 6, 1, 7, 4, 8, 5, 3, 9, 2, 10, 6, 1, 11, 7, 4, 12, 13, 8, 5, 14, 3, 15, 9, 2, 16, 10, 6, 17, 1, 18, 11, 7, 19, 4, 20, 12, 21, 13, 8, 22, 5, 23, 14, 3, 24, 15, 9, 25, 2, 26, 16, 10, 27, 6, 28, 17, 1, 29, 18, 11, 30, 7, 31, 19, 4, 32, 20, 12

Offset: 1

Clark Kimberling, Oct 31 2009

nonn

A fractal sequence: if you delete the first occurrence of each positive
integer, the remaining sequence is the original. This procedure is called
upper trimming, in contrast to lower trimming, which consists of
subtracting 1 from each term of the original fractal sequence and then
deleting all 0's. In general, the lower trim of a fractal sequence is a
fractal sequence; in particular, the lower trim of A003603 is A167237.

			The first 7 rows in the construction of A003603 are
1
1
1 2
1 3 2
1 4 3 2 5
1 6 4 3 7 2 8 5
1 9 6 4 10 3 11 7 2 12 8 5 13
Subtracting 1 and deleting 0's leaves
1
2 1
3 2 1 4
5 4 2 6 1 7 4
8 5 3 9 2 10 6 1 11 7 4 12

Clark Kimberling, "Fractal Sequences and Interspersions," Ars Combinatoria 45 (1997) 157-168.

Cf. A003603, A019586, A035513.

Although A167237 is closely associated with the Wythoff array (A035513)
and Fibonacci numbers (A000045), it can be constructed independently.
First, construct the fractal sequence of the Wythoff array inductively
as described at A003603; then subtract 1 from all terms and delete
all 0's.

A357097 A multiplication table for the rows of the extended Wythoff array. See comments for definition.

Original entry on oeis.org

0, 1, 1, 2, 15, 2, 3, 8, 8, 3, 4, 12, 4, 12, 4, 5, 44, 18, 18, 44, 5, 6, 19, 24, 27, 24, 19, 6, 7, 62, 28, 96, 96, 28, 62, 7, 8, 26, 34, 42, 128, 42, 34, 26, 8, 9, 30, 14, 51, 56, 56, 51, 14, 30, 9, 10, 91, 44, 57, 180, 65, 180, 57, 44, 91, 10, 11, 37, 50, 66, 76, 79, 79, 76, 66, 50, 37, 11

Offset: 0

Peter Munn, Sep 11 2022

Square array A(x,y), x >= 0, y >= 0, defined as follows:
(1) Extend the Wythoff array infinitely to the left, maintaining the Fibonacci recurrence (see A287870 examples). We denote this extended array as eW(n,m), n >= 0, m any integer, indexed such that eW(n,0) = n. From each row n, form the set of pairs S_n = {(eW(n,m+1),eW(n,m)) : integer m)}.
(2) Define addition and multiplication of pairs by (j1,k1) + (j2,k2) = (j1+j2, k1+k2) and (j1,k1) o (j2,k2) = (j1*j2 + k1*k2, j1*k2 + k1*j2 - k1*k2). (This defines a commutative ring with identity (1,0).)
(3) For nonnegative integers x and y, there is an integer z such that for every pair (j_x,k_x) in S_x and every pair (j_y,k_y) in S_y, (j_x,k_x) o (j_y,k_y) is in S_z. Define A(x,y) = z.
As a binary operation, A(.,.) is analogous to multiplication of coefficients in scientific numeric notation. The column position, m, used to define a pair in (1) above does not affect the eventual outcome, A(x,y), in (3), as no special pairs are selected from the pairs in S_x or S_y. The column position is analogous to the exponent. Notice also A(1,1) = 15 is substantially larger than A(2,2) = 4. This can be seen as analogous to 0.3 * 0.4 = 0.12 requiring more digits than 0.5 * 0.8 = 0.4.

			Calculation for A(1,2). Rows 1 and 2 of A287870 (indexed from 0) start 1, 3, ... and 2, 4, ... . So we may use the pairs (3,1) and (4,2). The defined multiplication gives (3*4 + 1*2, 3*2 + 4*1 - 1*2) = (14,8). 8, 14 , ... is in row 8 of A287870, so A(1,2) = 8.
For A(1,1), we start as above to get (3*3 + 1*1, 3*1 + 3*1 - 1*1) = (10,5). In the more general case, we form a sequence using the Fibonacci recurrence (as ..., 5, 10, ... may be in the extension leftwards of A287870). This starts 5, 10, 5+10=15, 10+15=25, 15+25=40, ... . We observe 15, 25, 40, ... is in row 15. So A(1,1) = 15.
The top left corner of the array:
  0   1   2    3    4    5    6    7    8    9
  1  15   8   12   44   19   62   26   30   91
  2   8   4   18   24   28   34   14   44   50
  3  12  18   27   96   42   51   57   66  198
  4  44  24   96  128   56  180   76   88  264
  5  19  28   42   56   65   79   33  102  116
  6  62  34   51  180   79  253  107  124  371
  7  26  14   57   76   33  107   45  138  157
  8  30  44   66   88  102  124  138  160  182
  9  91  50  198  264  116  371  157  182  544

Peter G. Anderson, More Properties of the Zeckendorf Array, Fib. Quart. 52-5 (2014), 15-21.
P. Arnoux, Some remarks about Fibonacci multiplication, Appl. Math. Lett. 2 (1989), 319-320.
Clark Kimberling, The Zeckendorf array equals the Wythoff array, Fibonacci Quarterly, Vol. 33, No. 1 (1995), pp. 3-8.

Cf. A022344 (norm), A035513, A120873, A287870, A348853.

See the formula section for the relationships with A000201, A003622, A019586, A035336, A101330.

lowerw(n) = (n+sqrtint(5*n^2))\2 ; \\ A000201
upperw(n) = (sqrtint(n^2*5)+n*3)\2; \\ A001950
compoundw(n) = (sqrtint(n^2*5)+n*3)\2 - 1; \\ A003622
wpair(p) = {my(x=p[2], y = p[1], z); while(1, my(n=1, ok=1); while(ok, my(xx = lowerw(n), yy = upperw(n)); if ((x == xx) && (y == yy), return([xx, yy])); if (xx > x, ok = 0); n++;); z = y; y += x; x = z;);}
row(p) = {my(x=p[1], y=p[2], u); while (1, my(n=1, ok=1); while(ok, my(xx = lowerw(n), yy = compoundw(n)); if ((x == xx) && (y == yy), return(n)); if (xx > x, ok = 0); n++;); u = x; x = y - u; y = u;);} \\ similar to A120873
wrow(p) = row(wpair(p));
prodpair(v1, v2) = my(j1=v1[1], j2 = v2[1], k1 = v1[2], k2 = v2[2]); [j1*j2 + k1*k2, j1*k2 + k1*j2 - k1*k2];
pair(n) = [lowerw(n+1), n];
T(n, k) = my(pn = pair(n), pk = pair(k), px = prodpair(pn, pk)); wrow(px)-1; \\ Michel Marcus, Sep 18 2022

A(x,y) = g(j1*j2 + k1*k2, j1*k2 + k1*j2 - k1*k2), where j1 = A035336(x+1), j2 = A035336(y+1), k1 = A003622(x+1), k2 = A003622(y+1) and g(j,k) = (if j = A000201(k+1) then k otherwise g(k,j-k)).
A(x,y) = A(y,x).
A(x,0) = x.
A(x, A(y,z)) = A(A(x,y), z).
A022344(A(x,y)) = A022344(x) * A022344(y).
A(A019586(x), A019586(y)) = A019586(A101330(x,y)). (conjectured)

A090247 a(n) = 26*a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 26.

Original entry on oeis.org

2, 26, 674, 17498, 454274, 11793626, 306180002, 7948886426, 206364867074, 5357537657498, 139089614227874, 3610972432267226, 93746193624720002, 2433790061810452826, 63184795413447053474, 1640370890687812937498

Offset: 0

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 24 2004

a(n+1)/a(n) converges to (13+sqrt(168)) =25.9614813... Lim a(n)/a(n+1) as n approaches infinity = 0.0385186... = 1/(13+sqrt(168)) = (13-sqrt(168)). Lim a(n+1)/a(n) as n approaches infinity = 25.9614813... = (13+sqrt(168)) = 1/(13-sqrt(168)). Lim a(n)/a(n+1) = 26 - Lim a(n+1)/a(n).

			a(4) = 454274 = 26*a(3) - a(2) = 26*17498 - 674 = (13+sqrt(168))^4 + (13-sqrt(168))^4 = 454273.9999977986 + 0.0000022013 = 454274.

Indranil Ghosh, Table of n, a(n) for n = 0..705
Tanya Khovanova, Recursive Sequences
Index entries for recurrences a(n) = k*a(n - 1) +/- a(n - 2)
Index entries for linear recurrences with constant coefficients, signature (26,-1).

Cf. A032000, A019586, A097308.

a[0] = 2; a[1] = 26; a[n_] := 26a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 15}] (* Robert G. Wilson v, Jan 30 2004 *)

[lucas_number2(n,26,1) for n in range(0,16)] # Zerinvary Lajos, Jun 27 2008

a(n) = 26a(n-1) - a(n-2), starting with a(0) = 2 and a(1) = 26. a(n) = (13+sqrt(168))^n + (13-sqrt(168))^n. (a(n))^2 =a(2n)+2.
G.f.: (2-26*x)/(1-26*x+x^2). - Philippe Deléham, Nov 02 2008
a(n) = 2*A097308(n). - R. J. Mathar, Sep 27 2014

cp's OEIS Frontend

A026355 a(n) = least k such that s(k) = n+1, where s = A026354.

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A348853 Delete any least significant 0's from the Zeckendorf representation of n, leaving its "odd" part.

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A035339 5th column of Wythoff array.

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A035340 6th column of Wythoff array.

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A051656 Sum_{i=0..n} (C(n,i) mod 2)*Fibonacci(2*i).

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A098862 a(n) = column number of Stolarsky array that contains n.

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A167198 Fractal sequence of the interspersion A083047.

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A167237 Lower trim of the Wythoff fractal sequence, A003603.

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A051656 Sum_{i=0..n} (C(n,i) mod 2)Fibonacci(2i).