A022342 -id:A022342 - cp's OEIS frontend

A062879 Integers whose Zeckendorf expansion does not contain ones at even positions.

0, 2, 5, 7, 13, 15, 18, 20, 34, 36, 39, 41, 47, 49, 52, 54, 89, 91, 94, 96, 102, 104, 107, 109, 123, 125, 128, 130, 136, 138, 141, 143, 233, 235, 238, 240, 246, 248, 251, 253, 267, 269, 272, 274, 280, 282, 285, 287, 322, 324, 327, 329, 335, 337, 340, 342, 356

Offset: 1

Antti Karttunen, Jun 26 2001

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Bisection of A062877.
Subset of A022342.
Cf. A003714, A014417, A062880, A165276.

fibOddCount[n_] := Plus @@ (Reverse@IntegerDigits[n, 2])[[1 ;; -1 ;; 2]]; oddIndexed = fibOddCount /@ Select[Range[0, 10000], BitAnd[#, 2 #] == 0 &]; -1 + Position[oddIndexed, ?(# == 0 &)] // Flatten (* _Amiram Eldar, Jan 20 2020  *)

A062880(n) = A003714(a(n)).

A165276(a(n)) = 0. - Amiram Eldar, Jan 20 2020

Offset corrected by Amiram Eldar, Jan 20 2020

A095080 Fibeven primes, i.e., primes p whose Zeckendorf-expansion A014417(p) ends with zero.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 23, 29, 31, 37, 41, 47, 71, 73, 79, 83, 89, 97, 107, 109, 113, 131, 139, 149, 151, 157, 167, 173, 181, 191, 193, 199, 223, 227, 233, 241, 251, 257, 269, 277, 283, 293, 311, 317, 337, 353, 359, 367, 379, 397, 401, 409, 419, 421

Offset: 1

Antti Karttunen, Jun 01 2004

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..10000
A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Intersection of A000040 and A022342. Union of A095082 and A095087. Cf. A095060, A095081.

F:= combinat[fibonacci]:
b:= proc(n) option remember; local j;
      if n=0 then 0
    else for j from 2 while F(j+1)<=n do od;
         b(n-F(j))+2^(j-2)
      fi
    end:
a:= proc(n) option remember; local p;
      p:= `if`(n=1, 1, a(n-1));
      do p:= nextprime(p);
         if b(p)::even then break fi
      od; p
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Mar 27 2016

F = Fibonacci;
b[n_] := b[n] = Module[{j},
     If[n == 0, 0, For[j = 2, F[j + 1] <= n, j++];
     b[n - F[j]] + 2^(j - 2)]];
a[n_] := a[n] = Module[{p},
     p = If[n == 1, 1, a[n - 1]]; While[True,
     p = NextPrime[p]; If[ EvenQ[b[p]], Break[]]]; p];
Array[a, 100] (* Jean-François Alcover, Jul 01 2021, after Alois P. Heinz *)

from sympy import fibonacci, primerange
def a(n):
    k=0
    x=0
    while n>0:
        k=0
        while fibonacci(k)<=n: k+=1
        x+=10**(k - 3)
        n-=fibonacci(k - 1)
    return x
def ok(n):
    return str(a(n))[-1]=="0"
print([n for n in primerange(1, 1001) if ok(n)]) # Indranil Ghosh, Jun 07 2017

A101646 Array read by antidiagonals: T(n,k) = variant of Knuth's Fibonacci (or circle) product of n and k (A101330). Sometimes called the "arroba" product.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 4, 5, 5, 4, 5, 7, 8, 7, 5, 6, 8, 11, 11, 8, 6, 7, 10, 13, 15, 13, 10, 7, 8, 11, 16, 18, 18, 16, 11, 8, 9, 13, 18, 22, 21, 22, 18, 13, 9, 10, 15, 21, 25, 26, 26, 25, 21, 15, 10, 11, 16, 24, 29, 29, 32, 29, 29, 24, 16, 11, 12, 18, 26, 33, 34, 36, 36, 34, 33, 26, 18, 12

Offset: 1

David Applegate and N. J. A. Sloane, Jan 26 2005

Let n = Sum_{i >= 2} eps(i) Fib_i and k = Sum_{j >= 2} eps(j) Fib_j be the Zeckendorf expansions of n and k, respectively (cf. A035517, A014417). The product of n and k is defined here to be n x k = Sum_{i,j} eps(i)*eps(j) Fib_{i+j-2} (= T(n,k)). [Comment corrected by R. J. Mathar, Aug 07 2007]

Although now 1 is the multiplicative identity, in contrast to A101330, this multiplication is not associative. For example, as pointed out by Grabner et al., we have (4 x 7 ) x 9 = 25 x 9 = 198 but 4 x (7 x 9 ) = 4 x 54 = 195.

			Array begins:
  1 2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 ...
  2 3  5  7  8 10 11 13 15 16 18 20 21 23 24 26 28 29 31 ...
  3 5  8 11 13 16 18 21 24 26 29 32 34 37 39 42 45 47 50 ...
  4 7 11 15 18 22 25 29 33 36 40 44 47 51 54 58 62 65 69 ...
  5 8 13 18 21 26 29 34 39 42 47 52 55 60 63 68 73 76 81 ...
...

P. Grabner et al., Associativity of recurrence multiplication, Appl. Math. Lett. 7 (1994), 85-90.
D. E. Knuth, Fibonacci multiplication, Appl. Math. Lett. 1 (1988), 57-60.
W. F. Lunnon, Proof of formula

Cf. A101330, A101385, A035517, A014417. Main diagonal is A101711.

First 4 rows give A000027, A022342, A026274, A101741.

T[n_, k_] := With[{phi2 = GoldenRatio^2}, n k - Floor[(k + 1)/phi2] Floor[ (n + 1)/phi2]];
Table[T[n - k + 1, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 31 2020 *)

T(n, k) = my(phi2 = ((1+sqrt(5))/2)^2); n*k - floor((k+1)/phi2)*floor((n+1)/phi2); \\ Michel Marcus, Mar 29 2016

T(n, k) = n*k - [(k+1)/phi^2] [(n+1)/phi^2]. For proof see link. - Fred Lunnon, May 24 2008

A227351 Permutation of nonnegative integers: map each number by lengths of runs of zeros in its Zeckendorf expansion shifted once left to the number which has the same lengths of runs (in the same order, but alternatively of runs of 0's and 1's) in its binary representation.

Original entry on oeis.org

0, 1, 3, 7, 2, 15, 6, 4, 31, 14, 12, 8, 5, 63, 30, 28, 24, 13, 16, 9, 11, 127, 62, 60, 56, 29, 48, 25, 27, 32, 17, 19, 23, 10, 255, 126, 124, 120, 61, 112, 57, 59, 96, 49, 51, 55, 26, 64, 33, 35, 39, 18, 47, 22, 20, 511, 254, 252, 248, 125, 240, 121, 123, 224

Offset: 0

Antti Karttunen, Jul 08 2013

This permutation is based on the fact that by appending one extra zero to the right of Fibonacci number representation of n (aka "Zeckendorf expansion") and then counting the lengths of blocks of consecutive (nonleading) zeros we get bijective correspondence with compositions, and thus also with the binary representation of a unique n. See the chart below:
   n   A014417(n) A014417(A022342(n+1)) Runs of    Binary number   In dec.
                  [shifted once left]   zeros      with same runs  = a(n)
  ......0              ......0    []         .....0           0
  ......1              .....10    [1]        .....1           1
  .....10              ....100    [2]        ....11           3
  ....100              ...1000    [3]        ...111           7
  ....101              ...1010    [1,1]      ....10           2
  ...1000              ..10000    [4]        ..1111          15
  ...1001              ..10010    [2,1]      ...110           6
  ...1010              ..10100    [1,2]      ...100           4
  ..10000              .100000    [5]        .11111          31
  ..10001              .100010    [3,1]      ..1110          14
  ..10010              .100100    [2,2]      ..1100          12
  ..10100              .101000    [1,3]      ..1000           8
  ..10101              .101010    [1,1,1]    ...101           5
  .100000              1000000    [6]        111111          63
Are there any other fixed points after 0, 1, 6, 803, 407483 ?

Antti Karttunen, Table of n, a(n) for n = 0..10946
OEIS Wiki, Run-length encoding
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A227352. Cf. also A003714, A014417, A006068, A048679.

Could be further composed with A075157 or A075159, also A129594.

(define (A227351 n) (A006068 (A048679 n)))

(define (A227351v2 n) (A006068 (A106151 (* 2 (A003714 n)))))

a(n) = A006068(A048679(n)) = A006068(A106151(2*A003714(n))).
This permutation effects following correspondences:
a(A000045(n)) = A000225(n-1).
a(A027941(n)) = A000975(n).
For n >=3, a(A000204(n)) = A000079(n-2).

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

A335999 a(1) = 1; for n >= 2, a(n) = least positive integer not in {a(1),..., a(n-1), b(1),...,b(n-1)}, where for n >=1, b(n) = n + 2 + least positive integer not in {a(1),..., a(n-1), a(n), b(1),...,b(n-1)}.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 11, 13, 14, 16, 17, 19, 21, 22, 24, 26, 27, 29, 31, 32, 34, 35, 37, 39, 40, 42, 43, 45, 47, 48, 50, 51, 53, 55, 56, 58, 60, 61, 63, 64, 66, 68, 69, 71, 73, 74, 76, 77, 79, 81, 82, 84, 86, 87, 89, 90, 92, 94, 95, 97, 98, 100, 102, 103

Offset: 1

Clark Kimberling, Jul 16 2020

nonn

In general, let u(1) = 1, and let k be a positive integer. Define u(n) = least positive integer not in {u(1),..., u(n-1), v(1),...,v(n-1)} and v(n) = n - 1 + k + least positive integer not in {u(1),..., u(n-1), u(n), v(1),...,v(n-1)}. As sets, (u(n)) and (v(n)) are disjoint. If k >= -1, let a(n) = u(n) and b(n) = v(n) for all n >= 1, but if k <= -2, let a(n) = u(n) - k + 1 and b(n) = v(n) - k - 1 for all n >= 1. Then every positive integer is in exactly one of the sequences (a(n)) and (b(n)). The difference sequence of (a(n)) consists of 1's and 2's; the difference sequence of (b(n)) consists of 2's and 3's.
****
Guide to related sequences:
   k      sequences (a(n)) and (b(n))
   0      A000201 and A001950 (lower and upper Wythoff sequences)
   1      A026351 and A026352
   2      A022342 and A003622
   3      A335999 and A336008

			a(1) = 1; b(1) = 1+2+2 = 5
a(2) = 2; b(2) = 2+2+3 = 7
a(3) = 3; b(3) = 3+2+4 = 9
a(4) = 4; b(4) = 4+2+6 = 12

Cf. A336008.

mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
{a, b} = {{1}, {}}; k = 3;
Do[AppendTo[b, Length[b] + k + mex[Flatten[{a, b}], Last[a]]];
AppendTo[a, mex[Flatten[{a, b}], Last[a]]], {150}]
a  (* A335999 *)
b  (* A336008 *)
(* Peter J. C. Moses, Jul 13 2020 *)

A352538 Primes whose position in the Wythoff array is immediately followed by another prime in the next column.

Original entry on oeis.org

2, 3, 7, 19, 23, 29, 67, 97, 103, 107, 149, 181, 227, 271, 311, 353, 379, 433, 449, 563, 631, 719, 761, 883, 919, 941, 971, 997, 1049, 1087, 1223, 1291, 1297, 1427, 1447, 1453, 1531, 1601, 1627, 1699, 1753, 1831, 1861, 1877, 2039, 2207, 2213, 2239, 2269, 2281, 2287

Offset: 1

Michel Marcus, Mar 20 2022

nonn

			The Wythoff array begins:
   1    2    3    5    8 ...
   4    7   11   18   29 ...
   6   10   16   26   42 ...
   ...
So 2, 3 and 7 are terms, since they are horizontally followed by 3, 5 and 11.

Cf. A003603, A022342, A035612, A035513 (Wythoff array).

Cf. A352537 (next row and column), A352539 (next row).

T(n,k) = (n+sqrtint(5*n^2))\2*fibonacci(k+1) + (n-1)*fibonacci(k); \\ A035513
cell(n) = for (r=1, oo, for (c=1, oo, if (T(r,c) == n, return([r, c])); if (T(r,c) > n, break);););
isokh(m) = {my(pos = cell(prime(m))); isprime (T(pos[1], pos[2]+1))};
lista(nn) = for (n=1, nn, if (isokh(n), print1(prime(n), ", ")));

right(n) = n++; (sqrtint(5*n^2)+n-2)\2; \\ see A022342
isokh(n) = isprime(right(n));
lista(nn) = for (n=1, nn, my(p=prime(n)); if (isokh(p), print1(p, ", ")));

A063732 Numbers whose Lucas representation excludes L_0 = 2.

Original entry on oeis.org

0, 1, 3, 4, 5, 7, 8, 10, 11, 12, 14, 15, 16, 18, 19, 21, 22, 23, 25, 26, 28, 29, 30, 32, 33, 34, 36, 37, 39, 40, 41, 43, 44, 45, 47, 48, 50, 51, 52, 54, 55, 57, 58, 59, 61, 62, 63, 65, 66, 68, 69, 70, 72, 73, 75, 76, 77, 79, 80, 81, 83, 84, 86, 87, 88, 90

Offset: 1

Fred Lunnon, Aug 25 2001

nonn

From Michel Dekking, Aug 26 2019: (Start)
This sequence is a generalized Beatty sequence.  We know that A054770, the sequence of numbers whose Lucas representation includes L_0=2, is equal to A054770(n) = A000201(n) + 2*n - 1 = floor((phi+2)*n) - 1.
One also easily checks that the numbers 3-phi and phi+2 form a Beatty pair. This implies that the sequence with terms floor((3-phi)*n)-1 is the complement of A054770 in the natural numbers 0,1,2,...
It follows that a(n) = 3*n - floor(n*phi) - 2.
(End)

Cf. A003622, A022342. Complement of A054770.
Partial sums of A003842.
Cf. A130310 (Lucas representation).

a(n) = floor((3-phi)*n)-1, where phi is the golden mean. - Michel Dekking, Aug 26 2019

A239003 Number of partitions of n into distinct Fibonacci numbers that are all greater than 2.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 0, 2, 0, 0, 1, 0, 2, 0, 0, 2, 0, 1, 0, 0, 3, 0, 0, 2, 0, 2, 0, 0, 3, 0, 0, 1, 0, 3, 0, 0, 3, 0, 2, 0, 0, 4, 0, 0, 2, 0, 3, 0, 0, 3, 0, 1, 0, 0, 4, 0, 0, 3, 0, 3, 0, 0, 5, 0, 0, 2, 0, 4, 0, 0, 4, 0, 2, 0, 0, 5, 0, 0, 3, 0, 3, 0, 0, 4, 0, 0

Offset: 0

Clark Kimberling, Mar 08 2014

a(n) > 0 if n+1 is a term of A003622; a(n) = 0 if n+1 is a term of A022342.

			There is one partition for n=0, the empty partition.  All parts are distinct, which means that there are no two parts that are equal. So a(0)=1.

Alois P. Heinz, Table of n, a(n) for n = 0..10946

Cf. A000201, A001950, A000045, A000119, A239002, A000009.

F:= combinat[fibonacci]:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<4, 0,
       b(n, i-1)+`if`(F(i)>n, 0, b(n-F(i), i-1))))
    end:
a:= proc(n) local j; for j from ilog[(1+sqrt(5))/2](n+1)
       while F(j+1)<=n do od; b(n, j)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 15 2014

f = Table[Fibonacci[n], {n, 4, 75}];  b[n_] := SeriesCoefficient[Product[1 + x^f[[k]], {k, n}], {x, 0, n}]; u = Table[b[n], {n, 0, 60}]  (* A239003 *)
Flatten[Position[u, 0]]  (* A022342 *)

G.f.: product(1 + x^F(j), j=4..infinity). - Wolfdieter Lang, Mar 15 2014

A372501 The 2-Zeckendorf array of the second kind, read by upward antidiagonals.

Original entry on oeis.org

0, 2, 1, 5, 4, 3, 7, 9, 8, 6, 10, 12, 16, 14, 11, 13, 17, 21, 27, 24, 19, 15, 22, 29, 35, 45, 40, 32, 18, 25, 37, 48, 58, 74, 66, 53, 20, 30, 42, 61, 79, 95, 121, 108, 87, 23, 33, 50, 69, 100, 129, 155, 197, 176, 142, 26, 38, 55, 82, 113, 163, 210, 252, 320, 286, 231

Offset: 1

A.H.M. Smeets, May 03 2024

The 2-Zeckendorf array of the second kind is based on the dual Zeckendorf representation of numbers (see A104326).
Column k contains the numbers whose dual Zeckendorf expansion ends "... 0 1^(k-1)" where ^ denotes repetition.
Rows satisfy this recurrence: T(n,k+1) = T(n,k) + T(n,k-1) + 2 for all n > 0 and k > 1.
As a sequence, the array is a permutation of the nonnegative integers.
As an array, T is an interspersion (hence also a dispersion). This holds as well for all Zeckendorf arrays of the second kind.
In general, for the m-Zeckendorf array of the second kind, the row recursion is given by T(n,k) = T(n,k-1) + T(n,k-m) + m, and the first column represent the "even" numbers.

			Array begins:
       k=1    2    3    4    5    6    7
      +---------------------------------
  n=1 |  0    1    3    6   11   19   32
  n=2 |  2    4    8   14   24   40   66
  n=3 |  5    9   16   27   45   74  121
  n=4 |  7   12   21   35   58   95  155
  n=5 | 10   17   29   48   79  129  210
  n=6 | 13   22   37   61  100  163  265
  n=7 | 15   25   42   69  113  184  299
The same in dual Zeckendorf form shows the pattern of digit suffixes, for example column k=3 is all numbers ending 011:
          k=1      2       3        4
      +------------------------------
  n=1 |     0      1      11      111
  n=2 |    10    101    1011    10111
  n=3 |   110   1101   11011   110111
  n=4 |  1010  10101  101011  1010111
  n=5 |  1110  11101  111011  1110111

Cf. A104326.
Rows n=1..3: A001911, A019274, A014739.
Columns k=1..3: A090909, A276885, A188012.
Cf. k-th prepended column: A022342 (k=1), A023444 (k=2).

T(n,1) = A090909(n+1).
T(1,k) = A001911(k-1).
T(2,k) = A019274(k-2).
T(3,k) = A014739(k-1).
T(n,1) = floor((n-1)*phi^2) and T(n,k+1) = floor((T(n,k)+1)*phi) for k > 0, where phi = (1+sqrt(5))/2. This can be considered as an alternative way to define the array.

cp's OEIS Frontend

A062879 Integers whose Zeckendorf expansion does not contain ones at even positions.

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A095080 Fibeven primes, i.e., primes p whose Zeckendorf-expansion A014417(p) ends with zero.

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A101646 Array read by antidiagonals: T(n,k) = variant of Knuth's Fibonacci (or circle) product of n and k (A101330). Sometimes called the "arroba" product.

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A227351 Permutation of nonnegative integers: map each number by lengths of runs of zeros in its Zeckendorf expansion shifted once left to the number which has the same lengths of runs (in the same order, but alternatively of runs of 0's and 1's) in its binary representation.

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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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A335999 a(1) = 1; for n >= 2, a(n) = least positive integer not in {a(1),..., a(n-1), b(1),...,b(n-1)}, where for n >=1, b(n) = n + 2 + least positive integer not in {a(1),..., a(n-1), a(n), b(1),...,b(n-1)}.

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A352538 Primes whose position in the Wythoff array is immediately followed by another prime in the next column.

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PARI

A063732 Numbers whose Lucas representation excludes L_0 = 2.

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