cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-17 of 17 results.

A371096 Array A read by upward antidiagonals in which the entry A(n,k) in row n and column k is defined by A(1, k) = 8*k-7, and A(n+1, k) = A371094(A(n, k)), n,k >= 1.

Original entry on oeis.org

1, 21, 9, 5461, 117, 17, 357913941, 11605, 213, 25, 1537228672809129301, 72701269, 87381, 309, 33, 28356863910078205288614550619314017621, 3752999689475413, 91625968981, 30037, 405, 41, 9649340769776349618630915417390658987772498722136713669954798667326094136661, 27043212804868893898596335048021, 100743818301219097892181, 760567125, 79189, 501, 49
Offset: 1

Views

Author

Antti Karttunen, Apr 21 2024

Keywords

Examples

			Array begins:
n\k|         1         2            3          4           5          6
---+--------------------------------------------------------------------
1  |         1,        9,          17,        25,         33,        41,
2  |        21,      117,         213,       309,        405,       501,
3  |      5461,    11605,       87381,     30037,      79189,     48469,
4  | 357913941, 72701269, 91625968981, 760567125, 1968526677, 299193685,

Crossrefs

Cf. A371094, A017077 (row 1).
Every fourth column (1, 5, 9, 13, 17, ...) of array A372282.
Cf. also arrays A257852, A371100 and A371102.

Programs

  • PARI
    up_to = 28;
A371094(n) = { my(m=1+3*n, e=valuation(m,2)); ((m*(2^e)) + (((4^e)-1)/3)); };
A371096sq(n,k) = if(1==n,8*k-7,A371094(A371096sq(n-1,k)));
A371096list(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] = A371096sq((a-(col-1)),col))); (v); };
v371096 = A371096list(up_to);
A371096(n) = v371096[n];

A371102 Array A read by upward antidiagonals in which the entry A(n,k) in row n and column k is defined by A(1, k) = 4*k-1, and A(n+1, k) = A371094(A(n, k)), n,k >= 1.

Original entry on oeis.org

3, 21, 7, 5461, 45, 11, 357913941, 1109, 69, 15, 1537228672809129301, 873813, 3413, 93, 19, 28356863910078205288614550619314017621, 1466015503701, 22369621, 2261, 117, 23, 9649340769776349618630915417390658987772498722136713669954798667326094136661, 25790417485112089060398421, 6004799503160661, 873813, 11605, 141, 27
Offset: 1

Views

Author

Antti Karttunen, Apr 21 2024

Keywords

Examples

			Array begins:
n\k|         1       2         3       4         5         6         7
---+--------------------------------------------------------------------
1  |         3,      7,       11,     15,       19,       23,       27,
2  |        21,     45,       69,     93,      117,      141,      165,
3  |      5461,   1109,     3413,   2261,    11605,     3413,     8021,
4  | 357913941, 873813, 22369621, 873813, 72701269, 22369621, 12408149,

Crossrefs

Cf. A004767 (row 1), A102603 (row 2), A371094.
Cf. also arrays A257852, A371096, A371100, A371103.

Programs

  • PARI
    up_to = 105;
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)));
A371102list(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] = A371102sq((a-(col-1)),col))); (v); };
v371102 = A371102list(up_to);
A371102(n) = v371102[n];

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

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Apr 28 2024

Keywords

Comments

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.

Examples

			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,

Links

Crossrefs

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.

Programs

  • PARI
    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];

Formula

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

Views

Author

Antti Karttunen, Apr 19 2024

Keywords

Comments

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

Examples

			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, ...
...

Crossrefs

Cf. A257852, A371092, A371100.

Programs

  • PARI
    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];

Formula

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

Views

Author

Antti Karttunen, Apr 21 2024

Keywords

Comments

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

Examples

			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,

Crossrefs

Cf. A000027 (row 1), A257852, A371092, A371102.
Cf. also arrays A371097, A371101.

Programs

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

Views

Author

Antti Karttunen, Apr 21 2024

Keywords

Comments

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

Examples

			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,

Links

Crossrefs

Cf. A000027 (row 1), A257852, A371092, A371095, A371096.
Cf. also arrays A371101, A371103.

Programs

  • PARI
    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];

Formula

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

Views

Author

Antti Karttunen, May 01 2024

Keywords

Comments

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

Links

Crossrefs

Cf. A257852, A371092, A372443, A372444.
Column 14 of A372287, column 7 of A371103.

Programs

Formula

a(n) = A371092(A372443(n)) = A371092(A372444(n)).
Previous Showing 11-17 of 17 results.

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