A371092 -id:A371092 - cp's OEIS frontend

A372287 Array read by upward antidiagonals: A(n, k) = A371092(A372283(n, k)), n,k >= 1.

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

Offset: 1

Antti Karttunen, Apr 28 2024

A(n, k) gives the column index of A372282(n, k) [or equally, of A372283(n, k)] in array A257852.

Collatz conjecture is equal to the claim that in each column 1 will eventually appear.

			Array begins:
n\k| 1  2  3  4  5  6  7  8  9 10 11 12 13  14 15  16 17 18 19  20
---+---------------------------------------------------------------
1  | 1, 1, 1, 2, 2, 3, 1, 4, 3, 5, 1, 6, 4,  7, 2,  8, 5, 9, 2, 10,
2  | 1, 1, 1, 3, 2, 3, 1, 6, 1, 2, 1, 9, 5,  6, 3, 12, 4, 1, 2, 15,
3  | 1, 1, 1, 3, 3, 1, 1, 9, 1, 3, 1, 1, 2,  8, 3, 18, 5, 1, 3, 12,
4  | 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 3, 12, 1, 27, 2, 1, 3, 17,
5  | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 18, 1, 21, 3, 1, 1,  4,
6  | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 27, 1, 16, 3, 1, 1,  5,
7  | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 21, 1, 23, 1, 1, 1,  2,
8  | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 16, 1, 18, 1, 1, 1,  3,
9  | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 23, 1, 26, 1, 1, 1,  3,
10 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 18, 1, 39, 1, 1, 1,  1,
11 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 26, 1, 30, 1, 1, 1,  1,
12 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 39, 1, 44, 1, 1, 1,  1,
13 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 30, 1, 66, 1, 1, 1,  1,
14 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 44, 1, 99, 1, 1, 1,  1,
15 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 66, 1, 75, 1, 1, 1,  1,
16 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 99, 1, 28, 1, 1, 1,  1,
17 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 75, 1, 42, 1, 1, 1,  1,
18 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 28, 1, 63, 1, 1, 1,  1,
19 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 42, 1, 48, 1, 1, 1,  1,
20 | 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 63, 1, 71, 1, 1, 1,  1,

Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A257852, A371092, A371094, A372282, A372283, A372288.

Cf. also A371097 (array giving every fourth column, 1, 5, 9, ...), A371103 (array giving every even numbered column), also array A371101.

up_to = 105;
A000265(n) = (n>>valuation(n,2));
A371092(n) = floor((A000265(1+(3*n))+5)/6);
R(n) = { n = 1+3*n; n>>valuation(n, 2); };
A372283sq(n,k) = if(1==n,2*k-1,R(A372283sq(n-1,k)));
A372287sq(n,k) = A371092(A372283sq(n,k));
A372287list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A372287sq((a-(col-1)),col))); (v); };
v372287 = A372287list(up_to);
A372287(n) = v372287[n];

A(n, k) = A371092(A372282(n,k)) = A371092(A372283(n,k)).

A371101 Array A read by upward antidiagonals in which the entry A(n,k) = A371092(A371100(n, k)), n,k >= 1.

Original entry on oeis.org

1, 1, 3, 1, 2, 3, 1, 3, 1, 6, 1, 2, 3, 5, 2, 1, 3, 1, 6, 4, 9, 1, 2, 3, 5, 2, 8, 6, 1, 3, 1, 6, 4, 9, 2, 12, 1, 2, 3, 5, 2, 8, 6, 11, 1, 1, 3, 1, 6, 4, 9, 2, 12, 7, 15, 1, 2, 3, 5, 2, 8, 6, 11, 1, 14, 9, 1, 3, 1, 6, 4, 9, 2, 12, 7, 15, 4, 18, 1, 2, 3, 5, 2, 8, 6, 11, 1, 14, 9, 17, 5, 1, 3, 1, 6, 4, 9, 2, 12, 7, 15, 4, 18, 10, 21

Offset: 1

Antti Karttunen, Apr 19 2024

A(n, k) gives the column index of A371100(n, k) in array A257852.

			The array begins:
n\k|  1  2  3  4  5  6  7   8  9  10 11  12  13  14  15  16  17  18
---+--------------------------------------------------------------------
1  |  1, 3, 3, 6, 2, 9, 6, 12, 1, 15, 9, 18,  5, 21, 12, 24,  4, 27, ...
2  |  1, 2, 1, 5, 4, 8, 2, 11, 7, 14, 4, 17, 10, 20,  1, 23, 13, 26, ...
3  |  1, 3, 3, 6, 2, 9, 6, 12, 1, 15, 9, 18,  5, 21, 12, 24,  4, 27, ...
4  |  1, 2, 1, 5, 4, 8, 2, 11, 7, 14, 4, 17, 10, 20,  1, 23, 13, 26, ...
5  |  1, 3, 3, 6, 2, 9, 6, 12, 1, 15, 9, 18,  5, 21, 12, 24,  4, 27, ...
6  |  1, 2, 1, 5, 4, 8, 2, 11, 7, 14, 4, 17, 10, 20,  1, 23, 13, 26, ...
7  |  1, 3, 3, 6, 2, 9, 6, 12, 1, 15, 9, 18,  5, 21, 12, 24,  4, 27, ...
8  |  1, 2, 1, 5, 4, 8, 2, 11, 7, 14, 4, 17, 10, 20,  1, 23, 13, 26, ...
...

Cf. A257852, A371092, A371100.

up_to = 105;
A000265(n) = (n>>valuation(n,2));
A371092(n) = floor((A000265(1+(3*n))+5)/6);
A371100sq(n,k) = 4^n*(6*k - 3 - 2*(-1)^n) + (4^n - 1)/3;
A371101sq(n,k) = A371092(A371100sq(n,k));
A371101list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A371101sq((a-(col-1)),col))); (v); };
v371101 = A371101list(up_to);
A371101(n) = v371101[n];

A(n, k) = A371092(A371100(n, k)).

A(n, k) = A(n+2, k).

A371103 Array A read by upward antidiagonals in which the entry A(n,k) in row n and column k is defined by A(n, k) = A371092(A371102(n, k)), n,k >= 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 3, 3, 4, 1, 1, 1, 6, 5, 1, 1, 1, 9, 2, 6, 1, 1, 1, 1, 3, 9, 7, 1, 1, 1, 1, 3, 1, 6, 8, 1, 1, 1, 1, 1, 1, 8, 12, 9, 1, 1, 1, 1, 1, 1, 12, 18, 1, 10, 1, 1, 1, 1, 1, 1, 18, 27, 1, 15, 11, 1, 1, 1, 1, 1, 1, 27, 21, 1, 12, 9, 12, 1, 1, 1, 1, 1, 1, 21, 16, 1, 17, 7, 18, 13, 1, 1, 1, 1, 1, 1, 16, 23, 1, 4, 2, 27, 5, 14

Offset: 1

Antti Karttunen, Apr 21 2024

A(n, k) gives the column index of A371102(n, k) in array A257852.

			Array begins:
n\k| 1  2   3   ...
---+--------------------------------------------------------------------
1  | 1, 2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15, 16, 17,
2  | 1, 3,  3,  6,  2,  9,  6, 12,  1, 15,  9, 18,  5, 21, 12, 24,  4,
3  | 1, 3,  1,  9,  3,  1,  8, 18,  1, 12,  7, 27,  2,  8, 17, 36,  5,
4  | 1, 1,  1,  1,  3,  1, 12, 27,  1, 17,  2, 21,  3, 12,  4, 54,  2,
5  | 1, 1,  1,  1,  1,  1, 18, 21,  1,  4,  2, 16,  3, 18,  5, 81,  3,
6  | 1, 1,  1,  1,  1,  1, 27, 16,  1,  5,  3, 23,  1, 27,  2, 16,  3,
7  | 1, 1,  1,  1,  1,  1, 21, 23,  1,  2,  3, 18,  1, 21,  3, 23,  1,
8  | 1, 1,  1,  1,  1,  1, 16, 18,  1,  3,  1, 26,  1, 16,  3, 18,  1,
9  | 1, 1,  1,  1,  1,  1, 23, 26,  1,  3,  1, 39,  1, 23,  1, 26,  1,
10 | 1, 1,  1,  1,  1,  1, 18, 39,  1,  1,  1, 30,  1, 18,  1, 39,  1,
11 | 1, 1,  1,  1,  1,  1, 26, 30,  1,  1,  1, 44,  1, 26,  1, 30,  1,
12 | 1, 1,  1,  1,  1,  1, 39, 44,  1,  1,  1, 66,  1, 39,  1, 44,  1,
13 | 1, 1,  1,  1,  1,  1, 30, 66,  1,  1,  1, 99,  1, 30,  1, 66,  1,
14 | 1, 1,  1,  1,  1,  1, 44, 99,  1,  1,  1, 75,  1, 44,  1, 99,  1,
15 | 1, 1,  1,  1,  1,  1, 66, 75,  1,  1,  1, 28,  1, 66,  1, 75,  1,
16 | 1, 1,  1,  1,  1,  1, 99, 28,  1,  1,  1, 42,  1, 99,  1, 28,  1,
17 | 1, 1,  1,  1,  1,  1, 75, 42,  1,  1,  1, 63,  1, 75,  1, 42,  1,
18 | 1, 1,  1,  1,  1,  1, 28, 63,  1,  1,  1, 48,  1, 28,  1, 63,  1,
19 | 1, 1,  1,  1,  1,  1, 42, 48,  1,  1,  1, 71,  1, 42,  1, 48,  1,
20 | 1, 1,  1,  1,  1,  1, 63, 71,  1,  1,  1, 54,  1, 63,  1, 71,  1,
21 | 1, 1,  1,  1,  1,  1, 48, 54,  1,  1,  1, 80,  1, 48,  1, 54,  1,

Cf. A000027 (row 1), A257852, A371092, A371102.

Cf. also arrays A371097, A371101.

up_to = 105;
A000265(n) = (n>>valuation(n,2));
A371092(n) = floor((A000265(1+(3*n))+5)/6);
A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };
A371102sq(n,k) = if(1==n,4*k-1,A371094(A371102sq(n-1,k)));
A371103sq(n,k) = A371092(A371102sq(n,k));
A371103list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A371103sq((a-(col-1)),col))); (v); };
v371103 = A371103list(up_to);
A371103(n) = v371103[n];

A371097 Array A read by upward antidiagonals in which the entry A(n,k) in row n and column k is defined by A(n, k) = A371092(A371095(n, k)), n,k >= 1.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 3, 1, 4, 1, 3, 1, 5, 5, 1, 1, 1, 2, 4, 6, 1, 1, 1, 3, 5, 8, 7, 1, 1, 1, 3, 2, 12, 2, 8, 1, 1, 1, 1, 3, 18, 2, 11, 9, 1, 1, 1, 1, 3, 27, 3, 9, 7, 10, 1, 1, 1, 1, 1, 21, 3, 7, 2, 14, 11, 1, 1, 1, 1, 1, 16, 1, 2, 2, 21, 4, 12, 1, 1, 1, 1, 1, 23, 1, 2, 3, 8, 6, 17, 13, 1, 1, 1, 1, 1, 18, 1, 3, 3, 12, 9, 4, 10, 14

Offset: 1

Antti Karttunen, Apr 21 2024

A(n, k) gives the column index of A371095(n, k) [or equally, of A371096(n, k)] in array A257852.

			Array begins:
n\k| 1  2  3  ...
---+--------------------------------------------------------------
1  | 1, 2, 3, 4, 5,  6, 7,  8, 9, 10, 11, 12, 13, 14, 15,  16, 17,
2  | 1, 2, 1, 5, 4,  8, 2, 11, 7, 14,  4, 17, 10, 20,  1,  23, 13,
3  | 1, 3, 1, 2, 5, 12, 2,  9, 2, 21,  6,  4, 14, 30,  1,  18, 10,
4  | 1, 3, 1, 3, 2, 18, 3,  7, 2,  8,  9,  5, 21, 45,  1,  26, 14,
5  | 1, 1, 1, 3, 3, 27, 3,  2, 3, 12,  1,  2,  8, 17,  1,  39, 21,
6  | 1, 1, 1, 1, 3, 21, 1,  2, 3, 18,  1,  3, 12,  4,  1,  30,  8,
7  | 1, 1, 1, 1, 1, 16, 1,  3, 1, 27,  1,  3, 18,  5,  1,  44, 12,
8  | 1, 1, 1, 1, 1, 23, 1,  3, 1, 21,  1,  1, 27,  2,  1,  66, 18,
9  | 1, 1, 1, 1, 1, 18, 1,  1, 1, 16,  1,  1, 21,  3,  1,  99, 27,
10 | 1, 1, 1, 1, 1, 26, 1,  1, 1, 23,  1,  1, 16,  3,  1,  75, 21,
11 | 1, 1, 1, 1, 1, 39, 1,  1, 1, 18,  1,  1, 23,  1,  1,  28, 16,
12 | 1, 1, 1, 1, 1, 30, 1,  1, 1, 26,  1,  1, 18,  1,  1,  42, 23,
13 | 1, 1, 1, 1, 1, 44, 1,  1, 1, 39,  1,  1, 26,  1,  1,  63, 18,
14 | 1, 1, 1, 1, 1, 66, 1,  1, 1, 30,  1,  1, 39,  1,  1,  48, 26,
15 | 1, 1, 1, 1, 1, 99, 1,  1, 1, 44,  1,  1, 30,  1,  1,  71, 39,
16 | 1, 1, 1, 1, 1, 75, 1,  1, 1, 66,  1,  1, 44,  1,  1,  54, 30,
17 | 1, 1, 1, 1, 1, 28, 1,  1, 1, 99,  1,  1, 66,  1,  1,  80, 44,
18 | 1, 1, 1, 1, 1, 42, 1,  1, 1, 75,  1,  1, 99,  1,  1, 120, 66,
19 | 1, 1, 1, 1, 1, 63, 1,  1, 1, 28,  1,  1, 75,  1,  1, 180, 99,
20 | 1, 1, 1, 1, 1, 48, 1,  1, 1, 42,  1,  1, 28,  1,  1, 270, 75,
21 | 1, 1, 1, 1, 1, 71, 1,  1, 1, 63,  1,  1, 42,  1,  1, 405, 28,

Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A000027 (row 1), A257852, A371092, A371095, A371096.

Cf. also arrays A371101, A371103.

up_to = 105;
A000265(n) = (n>>valuation(n,2));
A371092(n) = floor((A000265(1+(3*n))+5)/6);
R(n) = { n = 1+3*n; n>>valuation(n, 2); };
A371095sq(n,k) = if(1==n,8*k-7,R(A371095sq(n-1,k)));
A371097sq(n,k) = A371092(A371095sq(n,k));
A371097list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A371097sq((a-(col-1)),col))); (v); };
v371097 = A371097list(up_to);
A371097(n) = v371097[n];

A(n, k) = A371092(A371095(n, k)) = A371092(A371096(n, k)).

A372445 a(n) = A371092(A372443(n)).

Original entry on oeis.org

7, 6, 8, 12, 18, 27, 21, 16, 23, 18, 26, 39, 30, 44, 66, 99, 75, 28, 42, 63, 48, 71, 54, 80, 120, 180, 270, 405, 152, 228, 342, 513, 97, 73, 55, 11, 4, 6, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Antti Karttunen, May 01 2024

nonn

a(n) gives the column index of A372443(n), or equally, of A372444(n) in array A257852.

Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A257852, A371092, A372443, A372444.

Column 14 of A372287, column 7 of A371103.

A000265(n) = (n>>valuation(n,2));
A371092(n) = floor((A000265(1+(3*n))+5)/6);
A372443(n) = { my(x=27); while(n, x=A000265(3*x+1); n--); (x); };
A372445(n) = A371092(A372443(n));

a(n) = A371092(A372443(n)) = A371092(A372444(n)).

A257852 Array A read by upward antidiagonals in which the entry A(n,k) in row n and column k is defined by A(n,k) = (2^n(6k - 3 - 2*(-1)^n) - 1)/3, n,k >= 1.

Original entry on oeis.org

3, 1, 7, 13, 9, 11, 5, 29, 17, 15, 53, 37, 45, 25, 19, 21, 117, 69, 61, 33, 23, 213, 149, 181, 101, 77, 41, 27, 85, 469, 277, 245, 133, 93, 49, 31, 853, 597, 725, 405, 309, 165, 109, 57, 35, 341, 1877, 1109, 981, 533, 373, 197, 125, 65, 39

Offset: 1

L. Edson Jeffery, Jul 12 2015

Sequence is a permutation of the odd natural numbers.

Let N_1 denote the set of odd natural numbers, and let |y|_2 denote 2-adic valuation of y. Define the map F : N_1 -> N_1 by F(x) = (3*x + 1)/2^|3*x+1|_2 (cf. A075677). Then row n of A is the set of all x in N_1 for which |3*x + 1|_2 = A371093(x) = n. Hence F(A(n,k)) = 6*k - 3 - 2*(-1)^n.

			From _Ruud H.G. van Tol_, Oct 17 2023, corrected and extended by _Antti Karttunen_, Apr 18 2024: (Start)
Array A begins:
n\k|   1|   2|   3|   4|   5|   6|   7|   8| ...
---+---------------------------------------------
1  |   3,   7,  11,  15,  19,  23,  27,  31, ...
2  |   1,   9,  17,  25,  33,  41,  49,  57, ...
3  |  13,  29,  45,  61,  77,  93, 109, 125, ...
4  |   5,  37,  69, 101, 133, 165, 197, 229, ...
5  |  53, 117, 181, 245, 309, 373, 437, 501, ...
6  |  21, 149, 277, 405, 533, 661, 789, 917, ...
... (End)

Cf. A006370, A075677, A096773 (after its initial 0, column 1 of this array).
Cf. A004767, A017077, A082285, A238477 (rows 1-4).
Cf. A371092, A371093 (column and row indices for odd numbers).
Cf. also arrays A371095, A371096, A371097, A371100, A371101, A371102, A371103.

(* Array: *)
Grid[Table[(2^n*(6*k - 3 - 2*(-1)^n) - 1)/3, {n, 10}, {k, 10}]]
(* Array antidiagonals flattened: *)
Flatten[Table[(2^(n - k + 1)*(6*k - 3 - 2*(-1)^(n - k + 1)) - 1)/ 3, {n, 10}, {k, n}]]

up_to = 105;
A257852sq(n,k) = ((2^n * (6*k - 3 - 2*(-1)^n) - 1)/3);
A257852list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A257852sq((a-(col-1)),col))); (v); };
v257852 = A257852list(up_to);
A257852(n) = v257852[n]; \\ Antti Karttunen, Apr 18 2024

From Ruud H.G. van Tol, Oct 17 2023: (Start)
A(n,k+1) = A(n,k) + 2^(n+1).
A(n+2,k) = A(n,k)*4 + 1.
A(1,k) = A004767(k-1).
A(2,k) = A017077(k-1).
A(3,k) = A082285(k-1).
A(4,k) = A238477(k). (End)
For all odd positive numbers n, A(A371093(n), A371092(n)) = n. - Antti Karttunen, Apr 24 2024

A371093 a(n) is the 2-adic valuation of 3n+1.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Apr 19 2024

When a(n) is applied to square array A257852 we obtain square array A004736, or in other words, a(n) applied to any odd number gives the index of the row where it is located in array A257852.
See further comments in A087230.
The asymptotic density of the occurrences of k = 0, 1, 2, ... is 1/2^(k+1). The asymptotic mean of this sequence is 1. - Amiram Eldar, May 28 2024

Antti Karttunen, Table of n, a(n) for n = 0..65537

Bisections: A000004, A087230.
Cf. A004736, A007814, A016777, A067745, A257852.
Cf. also A371092.

Table[IntegerExponent[3*n+1, 2], {n, 0, 105}] (* James C. McMahon, Apr 21 2024 *)

PARI
```
A371093(n) = valuation(1+3*n,2);
```

def A371093(n): return ((m:=3*n) & ~(m+1)).bit_length() # Chai Wah Wu, Apr 20 2024

a(n) = A007814(A016777(n)).
For all n >= 0, A067745(1+n) = A016777(n) / 2^a(n).
G.f.: Sum_{k>=1} k*x^(-1/3 + (-2)^(k + 1)/3 + 2^k)/(1 - x^(2^(k + 1))). - Miles Wilson, Sep 30 2024

cp's OEIS Frontend

A372287 Array read by upward antidiagonals: A(n, k) = A371092(A372283(n, k)), n,k >= 1.

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A371101 Array A read by upward antidiagonals in which the entry A(n,k) = A371092(A371100(n, k)), n,k >= 1.

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A371103 Array A read by upward antidiagonals in which the entry A(n,k) in row n and column k is defined by A(n, k) = A371092(A371102(n, k)), n,k >= 1.

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A371097 Array A read by upward antidiagonals in which the entry A(n,k) in row n and column k is defined by A(n, k) = A371092(A371095(n, k)), n,k >= 1.

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A372445 a(n) = A371092(A372443(n)).

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A257852 Array A read by upward antidiagonals in which the entry A(n,k) in row n and column k is defined by A(n,k) = (2^n*(6*k - 3 - 2*(-1)^n) - 1)/3, n,k >= 1.

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A371093 a(n) is the 2-adic valuation of 3n+1.

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A257852 Array A read by upward antidiagonals in which the entry A(n,k) in row n and column k is defined by A(n,k) = (2^n(6k - 3 - 2*(-1)^n) - 1)/3, n,k >= 1.