A035513 -id:A035513 - cp's OEIS frontend

A228041 Decimal expansion of sum of reciprocals, row 3 of Wythoff array, W = A035513.

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

Offset: 0

Clark Kimberling, Aug 05 2013

Let c be the constant given by A079586, that is, the sum of reciprocals of the Fibonacci numbers F(k) for k>=1. The number c-1, the sum of reciprocals of row 1 of W, is known to be irrational (see A079586). Conjecture: the same is true for all the other rows of W.

Let h be the constant given at A153387 and s(n) the sum of reciprocals of numbers in row n of W. Then h < 1 + s(n)*floor(n*tau) < c. Thus, s(n) -> 0 as n -> oo.

			1/6 + 1/10 + 1/16 + ...  = 0.4299428331215887765860056514594635898444...

Cf. A035513, A079586, A228040, A228042, A228043.

f[n_] := f[n] = Fibonacci[n]; g = GoldenRatio; w[n_, k_] := w[n, k] = f[k + 1]*Floor[n*g] + f[k]*(n - 1);
n = 3; Table[w[n, k], {n, 1, 5}, {k, 1, 5}]
r = N[Sum[1/w[n, k], {k, 1, 2000}], 120]
RealDigits[r, 10]

Equals A079586/2 - 5/4. - Amiram Eldar, May 22 2021

A228042 Decimal expansion of sum of reciprocals, row 4 of Wythoff array, W = A035513.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Aug 05 2013

Let c be the constant given by A079586, that is, the sum of reciprocals of the Fibonacci numbers F(k) for k>=1. The number c-1, the sum of reciprocals of row 1 of W, is known to be irrational (see A079586). Conjecture: the same is true for all the other rows of W.

Let h be the constant given at A153387 and s(n) the sum of reciprocals of numbers in row n of W. Then h < 1 + s(n)*floor(n*tau) < c. Thus, s(n) -> 0 as n -> oo.

			1/9 + 1/15 + 1/24 + ... = 0.28662855541439251772400376763964239322963...

Cf. A035513, A079586, A228040, A228041, A228043.

f[n_] := f[n] = Fibonacci[n]; g = GoldenRatio; w[n_, k_] := w[n, k] = f[k + 1]*Floor[n*g] + f[k]*(n - 1);
n = 4; Table[w[n, k], {n, 1, 5}, {k, 1, 5}]
r = N[Sum[1/w[n, k], {k, 1, 2000}], 120]
RealDigits[r, 10]

Equals A079586/3 - 5/6. - Amiram Eldar, May 22 2021

A228043 Decimal expansion of sum of reciprocals, row 5 of Wythoff array, W = A035513.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Aug 05 2013

Let c be the constant given by A079586, that is, the sum of reciprocals of the Fibonacci numbers F(k) for k>=1. The number c-1, the sum of reciprocals of row 1 of W, is known to be irrational (see A079586). Conjecture: the same is true for all the other rows of W.

Let h be the constant given at A153387 and s(n) the sum of reciprocals of numbers in row n of W. Then h < 1 + s(n)*floor(n*tau) < c. Thus, s(n) -> 0 as n -> oo.

			1/12 + 1/20 + 1/32 + ... = 0.21497141656079438829300282572973179492222...

Cf. A035513, A079586, A228040, A228041, A228042.

f[n_] := f[n] = Fibonacci[n]; g = GoldenRatio; w[n_, k_] := w[n, k] = f[k + 1]*Floor[n*g] + f[k]*(n - 1);
n = 5; Table[w[n, k], {n, 1, 5}, {k, 1, 5}]
r = N[Sum[1/w[n, k], {k, 1, 2000}], 120]
RealDigits[r, 10]

Equals A079586/4 - 5/8. - Amiram Eldar, May 22 2021

A255671 Number of the column of the Wythoff array (A035513) that contains U(n), where U = A001950, the upper Wythoff sequence.

Original entry on oeis.org

2, 4, 2, 2, 6, 2, 4, 2, 2, 4, 2, 2, 8, 2, 4, 2, 2, 6, 2, 4, 2, 2, 4, 2, 2, 6, 2, 4, 2, 2, 4, 2, 2, 10, 2, 4, 2, 2, 6, 2, 4, 2, 2, 4, 2, 2, 8, 2, 4, 2, 2, 6, 2, 4, 2, 2, 4, 2, 2, 6, 2, 4, 2, 2, 4, 2, 2, 8, 2, 4, 2, 2, 6, 2, 4, 2, 2, 4, 2, 2, 6, 2, 4, 2, 2, 4

Offset: 1

Clark Kimberling, Mar 03 2015

All the terms are even, and every even positive integer occurs infinitely many times.
From Michel Dekking, Dec 09 2024 and Ad van Loon: (Start)
This sequence has a self-similarity property:
      a(U(n)) = a(n) + 2 for all n.
Proof: it is known that the columns C_h of the Wythoff array are compound Wythoff sequences. For example: C_1 = L^2, C_2 = UL.
In general column C_h is equal to LU^{(h-1)/2} if h is odd, and to U^{h/2}L if h is even (see Theorem 10 in Kimberling’s 2008 paper in JIS).
Now if h is odd then the elements of column C_h are a subsequence of L, so no U(m) can occur in such a column.
If h is even then the elements of column C_h form a subsequence of U, and so many U(m) occur. Suppose that a(m) = h. Then U(U(m)) is an element of column UU^{h/2}L = U^{(h+2)/2}L. This implies a(U(m)) = a(m) +2. (End)

			Corner of the Wythoff array:
  1   2   3   5   8   13
  4   7   11  18  29  47
  6   10  16  26  42  68
  9   15  24  39  63  102
L = (1,3,4,6,8,9,11,...); U = (2,5,7,10,13,15,18,...), so that
A255670 = (1,3,1,1,5,...) and A255671 = (2,4,2,2,6,...).

Ad van Loon, The structure of the expansions, See Section 5.

Cf. A255670, A035612, A000201, A001950.

z = 13; r = GoldenRatio; f[1] = {1}; f[2] = {1, 2};
f[n_] := f[n] = Join[f[n - 1], Most[f[n - 2]], {n}]; f[z];
g[n_] := g[n] = f[z][[n]]; Table[g[n], {n, 1, 100}]  (* A035612 *)
Table[g[Floor[n*r]], {n, 1, (1/r) Length[f[z]]}]     (* A255670 *)
Table[g[Floor[n*r^2]], {n, 1, (1/r^2) Length[f[z]]}] (* A255671 *)

a(n) = 2 if and only if n = L(j) for some j; otherwise, n = U(k) for some k.

a(n) = A255670(n) + 1 = A035612(A001950(n)).

A269729 a(n) = row number of extended Wythoff array (see A035513) which contains the sequence obtained by reading the n-th row backwards (and adjusting signs).

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 10, 5, 8, 11, 6, 9, 12, 20, 28, 15, 23, 31, 18, 26, 13, 21, 29, 16, 24, 32, 19, 27, 14, 22, 30, 17, 25, 33, 54, 75, 41, 62, 83, 49, 70, 36, 57, 78, 44, 65, 86, 52, 73, 39, 60, 81, 47, 68, 34, 55, 76, 42, 63, 84, 50, 71, 37, 58, 79, 45, 66, 87, 53, 74, 40

Offset: 0

N. J. A. Sloane, Mar 08 2016

Conjecture: sequence is its own inverse. - R. J. Mathar, May 08 2019

			Take n=5: reading row 5 of A035513 backwards gives ... 23, 14, 9, 5, 4, 1, 3, -2, 5, -7, 12, -19, ..., which after adjusting the signs is row 7, so a(5) = 7.

J. H. Conway, Postings to Math Fun Mailing List, Nov 25 1996 and Dec 02 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, A269725, A269726, A269729.

See A269733 for first differences.

A035513 := proc(r::integer, c::integer)
    option remember;
    if c = 1 then
        A003622(r) ;
    elif c > 1 then
        A022342(1+procname(r, c-1)) ;
    elif c < 1 then
        procname(r,c+2)-procname(r,c+1) ;
    end if;
end proc:
# search in A035513 for row with consecutive w1,w2
A035513inv := proc(w1::integer,w2::integer)
    local r,c,W1,W2 ;
    for r from 1 do
        if A035513(r,1) > w2 then
            return -1 ;
        end if;
        for c from 1 do
            W1 := A035513(r,c) ;
            W2 := A035513(r,c+1) ;
            if W1=w1 and W2=w2 then
                return r-1 ;
            elif W2 > w2 then
                break;
            end if;
        end do:
    end do:
end proc:
A269729 := proc(n)
    option remember;
    local c,W1,W2,r,n35513;
    n35513 := n+1 ;
    for c from 1 by -1 do
        W1 := A035513(n35513,c) ;
        W2 := A035513(n35513,c-1) ;
        if W1 < 0 and abs(W2) > abs(W1) then
            r :=  A035513inv(abs(W1),abs(W2)) ;
            if r >= 0 then
                return r;
            end if;
        end if;
    end do:
end proc:
seq(A269729(n),n=0..120) ; # R. J. Mathar, May 08 2019

W[n_, k_] := W[n, k] = Fibonacci[k+1] Floor[n*GoldenRatio] + (n-1)* Fibonacci[k];
Winv[w1_, w2_] := Winv[w1, w2] = Module[{r, c, W1, W2}, For[r = 1, True, r++, If[W[r, 1] > w2, Return[-1]]; For[c = 1, True, c++, W1 = W[r, c]; W2 = W[r, c+1]; If[W1 == w1 && W2 == w2, Return[r-1], If[W2 > w2, Break[]]]]]];
a[n_] := a[n] = Module[{c, W1, W2, r, nw}, nw = n+1; For[c = 1, True, c--, W1 = W[nw, c]; W2 = W[nw, c-1]; If[W1 < 0 && Abs[W2] > Abs[W1], r = Winv[Abs[W1], Abs[W2]]; If[r >= 0, Return[r]]]]];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 120}] (* Jean-François Alcover, Aug 09 2023, after R. J. Mathar *)

Terms from a(18) on by R. J. Mathar, May 08 2019

A274286 Numbers that are a product of distinct numbers in row 2 of the Wythoff array, A035513.

Original entry on oeis.org

4, 7, 11, 18, 28, 29, 44, 47, 72, 76, 77, 116, 123, 126, 188, 198, 199, 203, 304, 308, 319, 322, 329, 492, 504, 517, 521, 522, 532, 792, 796, 812, 836, 843, 846, 861, 1276, 1288, 1316, 1353, 1363, 1364, 1368, 1386, 1393, 2068, 2084, 2088, 2128, 2189, 2204

Offset: 1

Clark Kimberling, Jun 17 2016

See the Comment on distinct-product sequences in A160009.

			28 = 4*7, 308 = 4*7*11.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A160009, A274280.

f[1] = 4; f[2] = 7; z = 33; f[n_] := f[n - 1] + f[n - 2]; f = Table[f[n], {n, 1, z}]; f
s = {1}; Do[s = Union[s, Select[s*f[[i]], # <= f[[z]] &]], {i, z}]; s1 = Rest[s]

A274287 Numbers that are a product of distinct numbers in row 3 of the Wythoff array, A035513.

Original entry on oeis.org

6, 10, 16, 26, 42, 60, 68, 96, 110, 156, 160, 178, 252, 260, 288, 408, 416, 420, 466, 660, 672, 680, 754, 960, 1068, 1088, 1092, 1100, 1220, 1560, 1728, 1760, 1768, 1780, 1974, 2496, 2520, 2796, 2848, 2856, 2860, 2880, 3194, 4032, 4080, 4160, 4524, 4608

Offset: 1

Clark Kimberling, Jun 17 2016

See the Comment on distinct-product sequences in A160009.

			60 = 6*10, 960 = 6*10*16.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A160009.

f[1] = 6; f[2] = 10; z = 33; f[n_] := f[n - 1] + f[n - 2]; f = Table[f[n], {n, 1, z}]; f
s = {1}; Do[s = Union[s, Select[s*f[[i]], # <= f[[z]] &]], {i, z}]; Rest[s]

A328695 Rectangular array R read by descending antidiagonals: divide to each even term of the Wythoff array (A035513) by 2, and delete all others.

Original entry on oeis.org

1, 4, 2, 17, 9, 3, 72, 38, 5, 12, 305, 161, 8, 51, 6, 1292, 682, 13, 216, 10, 7, 5473, 2889, 21, 915, 16, 30, 14, 23184, 12238, 34, 3876, 26, 127, 59, 25, 98209, 51841, 55, 16419, 42, 538, 250, 106, 11, 416020, 219602, 89, 69552, 68, 2279, 1059, 449, 18, 33

Offset: 1

Clark Kimberling, Oct 26 2019

Every positive integer occurs exactly once in R, and every row of R is a linear recurrence sequence.  The appearance of a sequence s(r) below means that corresponding row of R is the same as s(r) except possibly for one or more initial terms of s(r).
Row 1 of R: A001076
Row 2 of R: A001077
Row 3 of R: A000045
Row 4 of R: A115179
Row 5 of R: A006355
Row 6 of R: A097924
Row 8 of R: A048875
Row 9 of R: A000032

			Row 1 of the Wythoff array is (1,2,3,5,8,13,21,34,55,89,144,...), so that row 1 of R is (1,4,17,72,...).
_______________
Northwest corner of R:
   1   4   17   72  305  1292   5473
   2   9   38  161  682  2889  12238
   3   5    8   13   21    34     55
  12  51  216  915 3876 16419  69552
   6  10   16   26   42    68    110
   7  30  127  538 2279  9654  40895

Cf. A035513, A001076, A001077, A000045, A115179, A006355, A097924, A048875, A000032, A328696, A328697.

w[n_, k_] := Fibonacci[k + 1] Floor[n*GoldenRatio] + (n - 1) Fibonacci[k];
Table[w[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten;
q[n_, k_] := If[Mod[w[n, k], 2] == 0, w[n, k]/2, 0];
t[n_] := Union[Table[q[n, k], {k, 1, 50}]];
u[n_] := If[First[t[n]] == 0, Rest[t[n]], t[n]]
Table[u[n], {n, 1, 10}] (* A328695 array *)
v[n_, k_] := u[n][[k]];
Table[v[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* A328695 sequence *)

A328696 Rectangular array R read by descending antidiagonals: apply x -> (x+1)/2 to each odd term of the Wythoff array (A035513), and delete all others.

Original entry on oeis.org

1, 2, 4, 3, 6, 5, 7, 15, 8, 12, 11, 24, 20, 19, 9, 28, 62, 32, 49, 23, 10, 45, 100, 83, 79, 37, 16, 13, 117, 261, 134, 206, 96, 41, 21, 14, 189, 422, 350, 333, 155, 66, 54, 36, 25, 494, 1104, 566, 871, 405, 172, 87, 58, 40, 17, 799, 1786, 1481, 1409, 655

Offset: 1

Clark Kimberling, Oct 26 2019

Every positive integer occurs exactly once in R, and every row of R is a linear recurrence sequence.

			Row 1 of the Wythoff array is (1,2,3,5,8,13,21,34,55,89,144,...), so that row 1 of R is (1,2,3,7,11,...) = A107857 (essentially).
_______________
Northwest corner of R:
   1,  2,  3,  7,  11,  28,  45,  117,  189,  494,   799
   4,  6, 15, 24,  62, 100, 261,  422, 1104, 1786,  4675
   5,  8, 20, 32,  83, 134, 350,  566, 1481, 2396,  6272
  12, 19, 49, 79, 206, 333, 871, 1409, 3688, 5967, 15621
   9, 23, 37, 96, 155, 405, 655, 1714, 2773, 7259, 11745
  10, 16, 41, 66, 172, 278, 727, 1176, 3078, 4980, 13037
  13, 21, 54, 87, 227, 367, 960, 1553, 4065, 6577, 17218

Cf. A035513, A107857, A328695, A328697.

w[n_, k_] := Fibonacci[k + 1] Floor[n*GoldenRatio] + (n - 1) Fibonacci[k];
Table[w[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten;
q[n_, k_] := If[Mod[w[n, k], 2] == 1, (1 + w[n, k])/2, 0];
t[n_] := Union[Table[q[n, k], {k, 1, 50}]];
u[n_] := If[First[t[n]] == 0, Rest[t[n]], t[n]]
s = Select[Range[40], ! u[#] == {} &]; u1[n_] := u[s[[n]]];
Column[Table[u1[n], {n, 1, 10}]] (* A328696 array *)
v[n_, k_] := u1[n][[k]];
Table[v[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* A328696 sequence *)

A360377 a(n) = number of the row of the Wythoff array (A035513) that includes prime(n).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 7, 8, 6, 2, 8, 6, 10, 17, 2, 21, 23, 24, 26, 11, 7, 12, 20, 1, 6, 39, 40, 10, 26, 17, 49, 8, 53, 21, 14, 36, 6, 63, 40, 10, 69, 27, 7, 46, 76, 2, 81, 33, 54, 88, 1, 92, 14, 23, 38, 64, 66, 42, 27, 74, 18, 80, 84, 53, 54, 90, 94, 59, 60, 24

Offset: 1

Clark Kimberling, Feb 04 2023

Conjecture: every primitive row number, as defined in A332938, occurs infinitely many times in this sequence.

			The 10th prime is 29, which occurs in row 7, so a(10) = 2.

Cf. A000040, A035513, A332938, A360378, A360379, A360380.

W[n_, k_] := Fibonacci[k + 1] Floor[n*GoldenRatio] + (n - 1) Fibonacci[k];
t = Table[W[n, k], {n, 100}, {k, 1, 20}];
a[n_] := Select[Range[100], MemberQ[t[[#]], Prime[n]] &]
Flatten[Table[a[n], {n, 1, 100}]]

Every prime p has a unique representation p = p(m,k) = F(k+1)*[m*tau] + (m-1)*F(k), where F(h) = A000045(h) = h-th Fibonacci number, [ ] = floor, and tau = (1+sqrt(5))/2 = golden ratio, as in A001622. Here, a(n) is the number m such that prime(n) = p(m,k) for some k.

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A228041 Decimal expansion of sum of reciprocals, row 3 of Wythoff array, W = A035513.

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A228042 Decimal expansion of sum of reciprocals, row 4 of Wythoff array, W = A035513.

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A228043 Decimal expansion of sum of reciprocals, row 5 of Wythoff array, W = A035513.

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A255671 Number of the column of the Wythoff array (A035513) that contains U(n), where U = A001950, the upper Wythoff sequence.

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A269729 a(n) = row number of extended Wythoff array (see A035513) which contains the sequence obtained by reading the n-th row backwards (and adjusting signs).

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A274286 Numbers that are a product of distinct numbers in row 2 of the Wythoff array, A035513.

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A274287 Numbers that are a product of distinct numbers in row 3 of the Wythoff array, A035513.

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A328695 Rectangular array R read by descending antidiagonals: divide to each even term of the Wythoff array (A035513) by 2, and delete all others.

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