A295989 -id:A295989 - cp's OEIS frontend

A352909 Pairs (i,j) of nonnegative integers with disjoint binary expansions sorted first by i+j then by i.

0, 0, 0, 1, 1, 0, 0, 2, 2, 0, 0, 3, 1, 2, 2, 1, 3, 0, 0, 4, 4, 0, 0, 5, 1, 4, 4, 1, 5, 0, 0, 6, 2, 4, 4, 2, 6, 0, 0, 7, 1, 6, 2, 5, 3, 4, 4, 3, 5, 2, 6, 1, 7, 0, 0, 8, 8, 0, 0, 9, 1, 8, 8, 1, 9, 0, 0, 10, 2, 8, 8, 2, 10, 0, 0, 11, 1, 10, 2, 9, 3, 8, 8, 3, 9, 2, 10, 1, 11, 0, 0, 12, 4, 8, 8, 4, 12, 0, 0, 13, 1, 12, 4, 9, 5, 8, 8, 5, 9, 4, 12, 1, 13, 0

Offset: 1

N. J. A. Sloane, Apr 09 2022

Pairs (i,j) with AND(i,j) = 0.

Allan C. Wechsler points out that when these points are plotted on a two-dimensional grid they form a rotated version of the Sierpinski Gasket (A047999).

			The first few pairs are [0, 0], [0, 1], [1, 0], [0, 2], [2, 0], [0, 3], [1, 2], [2, 1], [3, 0], [0, 4], [4, 0], [0, 5], [1, 4], [4, 1], [5, 0], [0, 6], [2, 4], [4, 2], [6, 0], [0, 7], [1, 6], [2, 5], [3, 4], [4, 3], [5, 2], [6, 1], [7, 0], ...

N. J. A. Sloane, Table of n, a(n) for n = 1..4374
N. J. A. Sloane, List of the 2187 pairs (i,j) with i+j <= 127. [Note this is not a b-file.]

Cf. A047999, A295989 (i values), A352910 (j values).

with(Bits);
M:=16; Nlis:=[];
for s from 0 to M do for i from 0 to s do j:=s-i;
if And(i,j)=0 then Nlis:=[op(Nlis),[i,j]]; fi;
od: od:
Nlis;

A352909list[ij_] := Select[Array[{#, ij-#} &, ij+1, 0], BitAnd @@ # == 0 &];
Flatten[Array[A352909list, 15, 0]] (* Paolo Xausa, Feb 24 2024 *)

A353174 Irregular table T(n, k), n >= 0, k = 0..A352502(n)-1; the n-th row lists in ascending order the numbers k in 0..n such that k and n-k can be added without carries in balanced ternary.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 1, 2, 3, 0, 1, 3, 4, 0, 5, 0, 1, 5, 6, 0, 1, 6, 7, 0, 2, 3, 5, 6, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 3, 4, 6, 7, 9, 10, 0, 2, 3, 8, 9, 11, 0, 1, 2, 3, 4, 8, 9, 10, 11, 12, 0, 1, 3, 4, 9, 10, 12, 13, 0, 14, 0, 1, 14, 15, 0, 1, 15, 16, 0, 2, 3, 14, 15, 17

Offset: 0

Rémy Sigrist, Apr 28 2022

Two integers can be added without carries in balanced ternary if they have no equal nonzero digit at the same position.
This sequence has connections with A295989; here we work with balanced ternary, there with binary.
The set of points {(n, T(n, k))} has interesting fractal features, with voids in the form of Koch snowflakes (see illustration in Links section).

			Irregular table T(n, k) begins:
    0: [0]
    1: [0, 1]
    2: [0, 2]
    3: [0, 1, 2, 3]
    4: [0, 1, 3, 4]
    5: [0, 5]
    6: [0, 1, 5, 6]
    7: [0, 1, 6, 7]
    8: [0, 2, 3, 5, 6, 8]
    9: [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]

Rémy Sigrist, Table of n, a(n) for n = 0..8768 (rows for n = 0..3^5 flattened)
Rémy Sigrist, Scatterplot of (n, T(n, k)) for n <= 3^6 on a hexagonal lattice
Wikipedia, Balanced ternary
Wikipedia, Hexaflake.

Cf. A059095, A295989, A352502.

ok(u,v) = { while (u && v, my (uu=[0,+1,-1][1+u%3], vv=[0,+1,-1][1+v%3]); if (abs(uu+vv)>1, return (0)); u=(u-uu)/3; v=(v-vv)/3); return (1) }
row(n) = select(k -> ok(n-k, k), [0..n])

T(n, 0) = 0.

T(n, A352502(n)-1) = n.

Index correction from Rémy Sigrist, Jan 18 2025

A374354 Irregular table T(n, k), n >= 0, 0 <= k < A277561(n), read by rows; the n-th row lists the fibbinary numbers f <= n such that n - f is also a fibbinary number whose binary expansion has no common 1's with that of f (where fibbinary numbers correspond to A003714).

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 2, 0, 4, 0, 1, 4, 5, 2, 4, 2, 5, 0, 8, 0, 1, 8, 9, 0, 2, 8, 10, 1, 2, 9, 10, 4, 8, 4, 5, 8, 9, 4, 10, 5, 10, 0, 16, 0, 1, 16, 17, 0, 2, 16, 18, 1, 2, 17, 18, 0, 4, 16, 20, 0, 1, 4, 5, 16, 17, 20, 21, 2, 4, 18, 20, 2, 5, 18, 21, 8, 16, 8, 9, 16, 17

Offset: 0

Rémy Sigrist, Jul 06 2024

In other words, we partition n into pairs of fibbinary numbers whose binary expansions have no common 1's and list the corresponding fibbinary numbers to get the n-th row.

			Triangle T(n, k) begins:
  n   n-th row
  --  -----------
   0  0
   1  0, 1
   2  0, 2
   3  1, 2
   4  0, 4
   5  0, 1, 4, 5
   6  2, 4
   7  2, 5
   8  0, 8
   9  0, 1, 8, 9
  10  0, 2, 8, 10
  11  1, 2, 9, 10
  12  4, 8
  13  4, 5, 8, 9
  14  4, 10
  15  5, 10
  16  0, 16

Rémy Sigrist, Table of n, a(n) for n = 0..8118 (rows for n = 0..1023 flattened)
Index entries for sequences related to binary expansion of n

See A295989 and A374361 for similar sequences.

Cf. A003714, A037800, A277561, A374355, A374356.

row(n) = { my (r = [0], e, x, y, b); while (n, x = y = 0; e = valuation(n, 2); for (k = 0, oo, if (bittest(n, e+k), n -= b = 2^(e+k); [x, y] = [y + b, x], r = concat([v + y | v <- r], [v + x | v <- r]); break;););); return (r); }

T(n, 0) = 0 iff n is a fibbinary number.
T(n, k) + T(n, A277561(n)-1-k) = n.
T(n, 0) = A374355(n).
T(n, A277561(n)-1) = A374356(n).
Sum_{k = 0..A277561(n)-1} T(n, k) = n * 2^A037800(n).

A380123 Irregular table T(n, k), n >= 0, k = 1..A380122(n), read by rows; the n-th row lists the integers m (possibly negative) such that the nonzero digits in the nonadjacent form for m appear in the nonadjacent form for n.

Original entry on oeis.org

0, 0, 1, 0, 2, -1, 0, 3, 4, 0, 4, 0, 1, 4, 5, -2, 0, 6, 8, -1, 0, 7, 8, 0, 8, 0, 1, 8, 9, 0, 2, 8, 10, -5, -4, -1, 0, 11, 12, 15, 16, -4, 0, 12, 16, -4, -3, 0, 1, 12, 13, 16, 17, -2, 0, 14, 16, -1, 0, 15, 16, 0, 16, 0, 1, 16, 17, 0, 2, 16, 18, -1, 0, 3, 4, 15, 16, 19, 20

Offset: 0

Rémy Sigrist, Jan 12 2025

			Irregular table T(n, k) begins:
  n   n-th row
  --  -----------------------------
   0  0
   1  0, 1
   2  0, 2
   3  -1, 0, 3, 4
   4  0, 4
   5  0, 1, 4, 5
   6  -2, 0, 6, 8
   7  -1, 0, 7, 8
   8  0, 8
   9  0, 1, 8, 9
  10  0, 2, 8, 10
  11  -5, -4, -1, 0, 11, 12, 15, 16
  12  -4, 0, 12, 16
  13  -4, -3, 0, 1, 12, 13, 16, 17
  14  -2, 0, 14, 16
  15  -1, 0, 15, 16
  16  0, 16
.
Irregular table T(n, k) begins in nonadjacent form:
  n      n-th row
  -----  ------------------------------------------
      0  0
      1  0, 1
     10  0, 10
    10T  T, 0, 10T, 100
    100  0, 100
    101  0, 1, 100, 101
   10T0  T0, 0, 10T0, 1000
   100T  T, 0, 100T, 1000
   1000  0, 1000
   1001  0, 1, 1000, 1001
   1010  0, 10, 1000, 1010
  10T0T  T0T, T00, T, 0, 10T0T, 10T00, 1000T, 10000
  10T00  T00, 0, 10T00, 10000

Rémy Sigrist, Table of n, a(n) for n = 0..6724 (rows for n = 0..2^9 flattened)
Joerg Arndt, Matters Computational (The Fxtbook), pages 61-62.
Wikipedia, Non-adjacent form

See A295989 for a similar sequence.

Cf. A184616, A380122.

row(n) = { my (r = [0], d, b = 1); while (n, if (n % 2, n -= d = 2 - (n%4); r = concat(r, [v + d*b | v <- r]);); n /= 2; b *= 2;); vecsort(r); }

T(n, 1) = A184616(n).

T(n, A380122(n)) = n.

A347204 a(n) = a(f(n)/2) + a(floor((n+f(n))/2)) for n > 0 with a(0) = 1 where f(n) = A129760(n).

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 10, 15, 5, 9, 13, 20, 17, 27, 37, 52, 6, 11, 16, 25, 21, 34, 47, 67, 26, 43, 60, 87, 77, 114, 151, 203, 7, 13, 19, 30, 25, 41, 57, 82, 31, 52, 73, 107, 94, 141, 188, 255, 37, 63, 89, 132, 115, 175, 235, 322, 141, 218, 295, 409, 372, 523, 674

Offset: 0

Mikhail Kurkov, Aug 23 2021 [verification needed]

Modulo 2 binomial transform of A243499(n).

Antti Karttunen, Table of n, a(n) for n = 0..8191
Kevin Ryde, PARI/GP Code

Cf. A000110, A000120, A002720, A005493, A005494, A007814, A008277, A035009, A045379, A053645, A129760, A143494, A143495, A196834, A265649, A295989.

Similar recurrences: A124758, A243499, A284005, A329369, A341392.

function a = A347204(max_n)
    a(1) = 1;
    a(2) = 2;
    for nloop = 3:max_n
        n = nloop-1;
        s = 0;
        for k = 0:floor(log2(n))-1
            s = s + a(1+A053645(n)-2^k*(mod(floor(n/(2^k)),2)));
        end
        a(nloop) = 2*a(A053645(n)+1) + s;
    end
end
function a_n = A053645(n)
    a_n = n - 2^floor(log2(n));
end % Thomas Scheuerle, Oct 25 2021

f[n_] := BitAnd[n, n - 1]; a[0] = 1; a[n_] := a[n] = a[f[n]/2] + a[Floor[(n + f[n])/2]]; Array[a, 100, 0] (* Amiram Eldar, Nov 19 2021 *)

f(n) = bitand(n, n-1); \\ A129760
a(n) = if (n<=1, n+1, if (n%2, a(n\2)+a(n-1), a(f(n/2)) + a(n/2+f(n/2)))); \\ Michel Marcus, Oct 25 2021

PARI
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\\ Also see links.
```

A129760(n) = bitand(n, n-1);
memoA347204 = Map();
A347204(n) = if (n<=1, n+1, my(v); if(mapisdefined(memoA347204,n,&v), v, v = if(n%2, A347204(n\2)+A347204(n-1), A347204(A129760(n/2)) + A347204(n/2+A129760(n/2))); mapput(memoA347204,n,v); (v))); \\ (Memoized version of Michel Marcus's program given above) - Antti Karttunen, Nov 20 2021

a(n) = a(n - 2^f(n)) + (1 + f(n))*a((n - 2^f(n))/2) for n > 0 with a(0) = 1 where f(n) = A007814(n).
a(2n+1) = a(n) + a(2n) for n >= 0.
a(2n) = a(n - 2^f(n)) + a(2n - 2^f(n)) for n > 0 with a(0) = 1 where f(n) = A007814(n).
a(n) = 2*a(f(n)) + Sum_{k=0..floor(log_2(n))-1} a(f(n) - 2^k*T(n,k)) for n > 1 with a(0) = 1, a(1) = 2, and where f(n) = A053645(n), T(n,k) = floor(n/2^k) mod 2.
Sum_{k=0..2^n - 1} a(k) = A035009(n+1) for n >= 0.
a((4^n - 1)/3) = A002720(n) for n >= 0.
a(2^n - 1) = A000110(n+1),
a(2*(2^n - 1)) = A005493(n),
a(2^2*(2^n - 1)) = A005494(n),
a(2^3*(2^n - 1)) = A045379(n),
a(2^4*(2^n - 1)) = A196834(n),
a(2^m*(2^n-1)) = T(n,m+1) is the n-th (m+1)-Bell number for n >= 0, m >= 0 where T(n,m) = m*T(n-1,m) + Sum_{k=0..n-1} binomial(n-1,k)*T(k,m) with T(0,m) = 1.
a(n) = Sum_{j=0..2^A000120(n)-1} A243499(A295989(n,j)) for n >= 0. Also A243499(n) = Sum_{j=0..2^f(n)-1} (-1)^(f(n)-f(j)) a(A295989(n,j)) for n >= 0 where f(n) = A000120(n). In other words, a(n) = Sum_{j=0..n} (binomial(n,j) mod 2)*A243499(j) and A243499(n) = Sum_{j=0..n} (-1)^(f(n)-f(j))*(binomial(n,j) mod 2)*a(j) for n >= 0 where f(n) = A000120(n).
Generalization:
b(n, x) = (1/x)*b((n - 2^f(n))/2, x) + (-1)^n*b(floor((2n - 2^f(n))/2), x) for n > 0 with b(0, x) = 1 where f(n) = A007814(n).
Sum_{k=0..2^n - 1} b(k, x) = (1/x)^n for n >= 0.
b((4^n - 1)/3, x) = (1/x)^n*n!*L_{n}(x) for n >= 0 where L_{n}(x) is the n-th Laguerre polynomial.
b((8^n - 1)/7, x) = (1/x)^n*Sum_{k=0..n} (-x)^k*A265649(n, k) for n >= 0.
b(2^n - 1, x) = (1/x)^n*Sum_{k=0..n} (-x)^k*A008277(n+1, k+1),
b(2*(2^n - 1), x) = (1/x)^n*Sum_{k=0..n} (-x)^k*A143494(n+2, k+2),
b(2^2*(2^n - 1), x) = (1/x)^n*Sum_{k=0..n} (-x)^k*A143495(n+3, k+3),
b(2^m*(2^n - 1), x) = (1/x)^n*Sum_{k=0..n} (-x)^k*T(n+m+1, k+m+1, m+1) for n >= 0, m >= 0 where T(n,k,m) is m-Stirling numbers of the second kind.

A361755 Irregular triangle T(n, k), n >= 0, k = 1..2^A007895(n), read by rows; the n-th row lists the numbers k such that the Fibonacci numbers that appear in the Zeckendorf representation of k also appear in that of n.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 3, 0, 1, 3, 4, 0, 5, 0, 1, 5, 6, 0, 2, 5, 7, 0, 8, 0, 1, 8, 9, 0, 2, 8, 10, 0, 3, 8, 11, 0, 1, 3, 4, 8, 9, 11, 12, 0, 13, 0, 1, 13, 14, 0, 2, 13, 15, 0, 3, 13, 16, 0, 1, 3, 4, 13, 14, 16, 17, 0, 5, 13, 18, 0, 1, 5, 6, 13, 14, 18, 19, 0, 2, 5, 7, 13, 15, 18, 20

Offset: 0

Rémy Sigrist, Mar 23 2023

In other words, the n-th row lists the numbers k such that A003714(n) AND A003714(k) = A003714(k) (where AND denotes the bitwise AND operator).

The Zeckendorf representation is also known as the greedy Fibonacci representation (see A356771 for further details).

			Triangle T(n, k) begins:
  n   n-th row
  --  ------------------------
   0  0
   1  0, 1
   2  0, 2
   3  0, 3
   4  0, 1, 3, 4
   5  0, 5
   6  0, 1, 5, 6
   7  0, 2, 5, 7
   8  0, 8
   9  0, 1, 8, 9
  10  0, 2, 8, 10
  11  0, 3, 8, 11
  12  0, 1, 3, 4, 8, 9, 11, 12

Rémy Sigrist, Table of n, a(n) for n = 0..10924 (rows for n = 0..610 flattened)
Rémy Sigrist, PARI program
Index entries for sequences related to Zeckendorf expansion of n

See A361756 for a similar sequence.

Cf. A003714, A007895, A139764, A295989, A356771.

PARI
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See Links section.
```

T(n, 1) = 0.
T(n, 2) = A139764(n) for any n > 0.
T(n, 2^A007895(n)) = n.

A331533 Irregular table read by rows; for n >= 0, the n-th row corresponds to the nonnegative integers k such that (n^2) AND (k^2) = k^2, in ascending order (where AND denotes the bitwise AND operator).

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 1, 3, 0, 4, 0, 1, 3, 4, 5, 0, 2, 6, 0, 1, 4, 7, 0, 8, 0, 1, 4, 8, 9, 0, 2, 6, 8, 10, 0, 1, 3, 4, 5, 7, 8, 9, 11, 0, 4, 12, 0, 1, 3, 13, 0, 2, 8, 14, 0, 1, 8, 15, 0, 16, 0, 1, 16, 17, 0, 2, 8, 16, 18, 0, 1, 3, 8, 16, 17, 19, 0, 4, 12, 16, 20

Offset: 0

Rémy Sigrist, Jan 19 2020

The n-th row has A331532(n) terms, leading term 0 and last term n.

			Table begins:
    0;
    0, 1;
    0, 2;
    0, 1, 3;
    0, 4;
    0, 1, 3, 4, 5;
    0, 2, 6;
    0, 1, 4, 7;
    0, 8;
    ...

Rémy Sigrist, Table of n, a(n) for n = 0..7118 (rows for n = 0..512)

Cf. A295989, A331532 (row lengths).

row(n) = select(k -> bitand(n^2, k^2)==k^2, [0..n])

A343757 Irregular table read by rows; the n-th row contains the sums of distinct terms of the n-th row of table A343835, in ascending order.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 3, 0, 4, 0, 1, 4, 5, 0, 6, 0, 7, 0, 8, 0, 1, 8, 9, 0, 2, 8, 10, 0, 3, 8, 11, 0, 12, 0, 1, 12, 13, 0, 14, 0, 15, 0, 16, 0, 1, 16, 17, 0, 2, 16, 18, 0, 3, 16, 19, 0, 4, 16, 20, 0, 1, 4, 5, 16, 17, 20, 21, 0, 6, 16, 22, 0, 7, 16, 23, 0, 24

Offset: 0

Rémy Sigrist, May 01 2021

In other words, the n-th row contains the numbers k whose runs of 1's in the binary expansion also appear in that of n.
The n-th row has 2^A069010(n) terms.
This sequence has similarities with A295989.

			Table begins:
     0:    [0]
     1:    [0, 1]
     2:    [0, 2]
     3:    [0, 3]
     4:    [0, 4]
     5:    [0, 1, 4, 5]
     6:    [0, 6]
     7:    [0, 7]
     8:    [0, 8]
     9:    [0, 1, 8, 9]
    10:    [0, 2, 8, 10]
    11:    [0, 3, 8, 11]
    12:    [0, 12]
    13:    [0, 1, 12, 13]
    14:    [0, 14]
    15:    [0, 15]
Table begins in binary:
       0:   [0]
       1:   [0, 1]
      10:   [0, 10]
      11:   [0, 11]
     100:   [0, 100]
     101:   [0, 1, 100, 101]
     110:   [0, 110]
     111:   [0, 111]
    1000:   [0, 1000]
    1001:   [0, 1, 1000, 1001]
    1010:   [0, 10, 1000, 1010]
    1011:   [0, 11, 1000, 1011]
    1100:   [0, 1100]
    1101:   [0, 1, 1100, 1101]
    1110:   [0, 1110]
    1111:   [0, 1111]

Rémy Sigrist, Table of n, a(n) for n = 0..8120
Rémy Sigrist, Scatterplot of (n, T(n, k)) for n < 2^10
Index entries for sequences related to binary expansion of n

Cf. A069010, A295989, A343835.

row(n) = { my (rr=[]); while (n, my (z=valuation(n, 2), o=valuation(n/2^z+1, 2), r=(2^o-1)*2^z); n-=r; rr = concat(rr, r)); vector(2^#rr, k, vecsum(vecextract(rr, k-1))) }

T(n, 0) = 0.
T(n, 1) = A342410(n) for any n > 0.
T(n, 2^A069010(n)-1) = n.

A352910 The j-values of pairs (i,j) listed in A352909.

Original entry on oeis.org

0, 1, 0, 2, 0, 3, 2, 1, 0, 4, 0, 5, 4, 1, 0, 6, 4, 2, 0, 7, 6, 5, 4, 3, 2, 1, 0, 8, 0, 9, 8, 1, 0, 10, 8, 2, 0, 11, 10, 9, 8, 3, 2, 1, 0, 12, 8, 4, 0, 13, 12, 9, 8, 5, 4, 1, 0, 14, 12, 10, 8, 6, 4, 2, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 0

Offset: 1

N. J. A. Sloane, Apr 09 2022

See A295989 for the i-values.

Rémy Sigrist, Table of n, a(n) for n = 1..6563 (values for i+j <= 256)

Cf. A352909, A295989.

A352910list[ij_] := Select[Range[ij, 0, -1], BitAnd[#, ij-#] == 0 &];
Flatten[Array[A352910list, 25, 0]] (* Paolo Xausa, Feb 24 2024 *)

A370727 Lexicographically earliest sequence of distinct positive integers such that for any n > 0, prime(n) AND a(n) = a(n) (where prime(n) denotes the n-th prime number and AND denotes the bitwise AND operator).

Original entry on oeis.org

2, 1, 4, 3, 8, 5, 16, 17, 6, 9, 7, 32, 33, 10, 11, 20, 18, 12, 64, 65, 72, 13, 19, 24, 96, 36, 34, 35, 37, 48, 14, 128, 129, 130, 21, 22, 25, 131, 38, 40, 49, 52, 15, 192, 68, 66, 67, 23, 97, 69, 41, 39, 80, 26, 256, 257, 260, 258, 261, 264, 27, 288, 50, 51

Offset: 1

Rémy Sigrist, Feb 28 2024

In other words, the 1's in the binary expansion of the n-th term also appear in that of the n-th prime number.

This sequence is a permutation of the positive integers with inverse A370727: for any w > 0, there are infinitely many prime numbers whose binary expansions end with w 1's, and these are all occasions for an integer < 2^w to appear in the sequence.

			The first terms, alongside the corresponding binary expansions, are:
  n   a(n)  bin(a(n))  bin(prime(n))
  --  ----  ---------  -------------
   1     2         10             10
   2     1          1             11
   3     4        100            101
   4     3         11            111
   5     8       1000           1011
   6     5        101           1101
   7    16      10000          10001
   8    17      10001          10011
   9     6        110          10111
  10     9       1001          11101

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A295609, A295989, A370728 (inverse).

PARI
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A352909 Pairs (i,j) of nonnegative integers with disjoint binary expansions sorted first by i+j then by i.

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Maple

Mathematica

A353174 Irregular table T(n, k), n >= 0, k = 0..A352502(n)-1; the n-th row lists in ascending order the numbers k in 0..n such that k and n-k can be added without carries in balanced ternary.

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PARI

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A374354 Irregular table T(n, k), n >= 0, 0 <= k < A277561(n), read by rows; the n-th row lists the fibbinary numbers f <= n such that n - f is also a fibbinary number whose binary expansion has no common 1's with that of f (where fibbinary numbers correspond to A003714).

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PARI

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A380123 Irregular table T(n, k), n >= 0, k = 1..A380122(n), read by rows; the n-th row lists the integers m (possibly negative) such that the nonzero digits in the nonadjacent form for m appear in the nonadjacent form for n.

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PARI

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A347204 a(n) = a(f(n)/2) + a(floor((n+f(n))/2)) for n > 0 with a(0) = 1 where f(n) = A129760(n).

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MATLAB

Mathematica

PARI

PARI

PARI

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A361755 Irregular triangle T(n, k), n >= 0, k = 1..2^A007895(n), read by rows; the n-th row lists the numbers k such that the Fibonacci numbers that appear in the Zeckendorf representation of k also appear in that of n.

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PARI

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A331533 Irregular table read by rows; for n >= 0, the n-th row corresponds to the nonnegative integers k such that (n^2) AND (k^2) = k^2, in ascending order (where AND denotes the bitwise AND operator).

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PARI