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.

Showing 1-10 of 35 results. Next

A296354 Official position where binary expansion of n starts in the list of binary numbers in the binary Champernowne sequence A076478.

Original entry on oeis.org

0, 1, 6, 8, 22, 25, 28, 31, 66, 70, 74, 78, 82, 86, 90, 94, 178, 183, 188, 193, 198, 203, 208, 213, 218, 223, 228, 233, 238, 243, 248, 253, 450, 456, 462, 468, 474, 480, 486, 492, 498, 504, 510, 516, 522, 528, 534, 540, 546, 552, 558, 564, 570, 576, 582, 588
Offset: 0

Views

Author

N. J. A. Sloane, Dec 14 2017

Keywords

Comments

a(n) is the official position where the binary expansion of n appears. The binary expansion of n may also appear earlier, by accident, see A296355 and A296356.

Examples

			Here is the list A076478 broken up to show the successive binary numbers (the indexing starts at 0):
0,
1,
0,0,
0,1,
1,0,
1,1,
0,0,0,
0,0,1,
0,1,0,
0,1,1,
1,0,0,
1,0,1,
...
2 = 1,0 starts at position 6, so a(2) = 6.
4 = 1,0,0 starts at position 22, so a(4) = 22.

Links

Crossrefs

Cf. A029837, A036799, A061168, A076478, A296355, A296356.

Programs

  • PARI
    a(n) = my (w=#binary(n)); return (2 + 2^w*(w-2) + w*n) \\ Rémy Sigrist, Dec 15 2017

Formula

a(n) = A036799(A029837(n + 1) - 1) + A029837(n + 1) * n. - Rémy Sigrist, Dec 15 2017

Extensions

More terms from Rémy Sigrist, Dec 15 2017

A346303 Positions of words in A076478 that start with 0 and end with 0.

Original entry on oeis.org

1, 3, 7, 9, 15, 17, 19, 21, 31, 33, 35, 37, 39, 41, 43, 45, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 127, 129, 131, 133, 135, 137, 139, 141, 143, 145, 147, 149, 151, 153, 155, 157, 159, 161, 163, 165, 167, 169, 171, 173, 175, 177, 179
Offset: 1

Views

Author

Clark Kimberling, Jul 21 2021

Keywords

Comments

The sequences A346303, A171757, A346304, A346305 partition the positive integers. See A076478 for a guide to related sequences.
The position of the words 0, 00, 000, 0000, ... (all-zero) in A076478 is given by A000225. So this sequence is given by the subsequence A000225 and filling the gaps halfway up (until the leading 0 toggles) with odd numbers, from 7 to 9, from 15 to 21, from 31 to 45 etc. So the first differences are 2, 4, 2, 6, 2, 2, 2, 10, 2, 2, 2, 2, 2, 2, 2, 18, 2, 2, ... which is A052548 with 1, 3, 7, 15,... (A000225) intermediate 2's. - R. J. Mathar, Sep 08 2021

Examples

			The first fourteen words w(n) are 0, 1, 00, 01, 10, 11, 000, 001, 010, 011, 100, 101, 110, 111, so that a(3) = 7.

Crossrefs

Cf. A007931, A076478, A171757, A346304, A346305.

Programs

A346304 Positions of words in A076478 that start with 1 and end with 0.

Original entry on oeis.org

5, 11, 13, 23, 25, 27, 29, 47, 49, 51, 53, 55, 57, 59, 61, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 191, 193, 195, 197, 199, 201, 203, 205, 207, 209, 211, 213, 215, 217, 219, 221, 223, 225, 227, 229, 231, 233, 235, 237
Offset: 1

Views

Author

Clark Kimberling, Jul 21 2021

Keywords

Comments

The sequences A346303, A171757, A346304, A346305 partition the positive integers. See A076478 for a guide to related sequences.

Examples

			The first fourteen words w(n) are 0, 1, 00, 01, 10, 11, 000, 001, 010, 011, 100, 101, 110, 111, so that a(1) = 5.

Crossrefs

Cf. A007931, A076478, A171757, A346303, A346305.

Programs

A296355 True position where binary expansion of n starts in the list of binary numbers in the binary Champernowne sequence A076478.

Original entry on oeis.org

0, 1, 1, 5, 1, 6, 5, 20, 1, 17, 15, 6, 8, 5, 20, 63, 9, 1, 22, 17, 15, 55, 6, 25, 8, 21, 48, 5, 20, 27, 63, 174, 9, 111, 51, 1, 41, 22, 70, 17, 49, 15, 74, 55, 6, 154, 25, 78, 8, 65, 21, 59, 48, 73, 5, 28, 31, 20, 135, 27, 63, 89, 174, 445, 33, 9, 120, 111, 66
Offset: 0

Views

Author

N. J. A. Sloane, Dec 14 2017; corrected and extended Dec 17 2017

Keywords

Comments

A296354(n) is the official position where the binary expansion of n appears in A076478, but the binary expansion of n may also appear earlier, by accident, and it is that starting position that is listed here.
In fact every number > 1 appears earlier - see A296356 for the proof.

Examples

			Here is the list A076478 broken up to show the successive binary numbers (the indexing starts at 0):
0,
1,
0,0,
0,1,
1,0,
1,1,
0,0,0,
0,0,1,
0,1,0,
0,1,1,
1,0,0,
1,0,1,
...
2 = 1,0 officially starts at position 6, so A076478(2) = 6, but 1,0 actually can be seen starting at position 1, so a(2) = 1.
4 = 1,0,0 officially starts at position 22, so A076478(4) = 22, but 1,0,0 actually can be seen starting at position 1, so a(4) = 1.

Links

Crossrefs

Cf. A076478, A061168. A296354, A296356.

Extensions

More terms from Rémy Sigrist, Dec 19 2017

A346299 Positions of words in A076478 in which #0's < #1's.

Original entry on oeis.org

2, 6, 10, 12, 13, 14, 22, 26, 28, 29, 30, 38, 42, 44, 45, 46, 50, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 78, 86, 90, 92, 93, 94, 102, 106, 108, 109, 110, 114, 116, 117, 118, 120, 121, 122, 123, 124, 125, 126, 142, 150, 154, 156, 157, 158, 166, 170, 172, 173
Offset: 1

Views

Author

Clark Kimberling, Jul 21 2021

Keywords

Comments

The sequences A258410, A346299, A346300 partition the positive integers.
See A076478 for a guide to related sequences.

Examples

			The first fourteen words w(n) are 0, 1, 00, 01, 10, 11, 000, 001, 010, 011, 100, 101, 110, 111, so that a(1) = 2, a(2) = 6.

Crossrefs

Cf. A007931, A076478.

Programs

A346300 Positions of words in A076478 in which #0's > #1's.

Original entry on oeis.org

1, 3, 7, 8, 9, 11, 15, 16, 17, 19, 23, 31, 32, 33, 34, 35, 36, 37, 39, 40, 41, 43, 47, 48, 49, 51, 55, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 75, 79, 80, 81, 83, 87, 95, 96, 97, 99, 103, 111, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139
Offset: 1

Views

Author

Clark Kimberling, Jul 21 2021

Keywords

Comments

The sequences A258410, A346299, A346300 partition the positive integers.
See A076478 for a guide to related sequences.

Examples

			The first fourteen words w(n) are 0, 1, 00, 01, 10, 11, 000, 001, 010, 011, 100, 101, 110, 111, so that a(1) = 1, a(2) = 3.

Crossrefs

Cf. A007931, A076478.
Cf. A258410, A346299.

Programs

A346301 Positions of words in A076478 such that first digit = last digit.

Original entry on oeis.org

1, 2, 3, 6, 7, 9, 12, 14, 15, 17, 19, 21, 24, 26, 28, 30, 31, 33, 35, 37, 39, 41, 43, 45, 48, 50, 52, 54, 56, 58, 60, 62, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124
Offset: 1

Views

Author

Clark Kimberling, Jul 21 2021

Keywords

Comments

The sequences A346301 and A346302 partition the positive integers.
See A076478 for a guide to related sequences.

Examples

			The first fourteen words w(n) are 0, 1, 00, 01, 10, 11, 000, 001, 010, 011, 100, 101, 110, 111, so that a(3) = 6.

Crossrefs

Cf. A007931, A076478, A346302.

Programs

A346302 Positions of words in A076478 such that first digit != last digit.

Original entry on oeis.org

4, 5, 8, 10, 11, 13, 16, 18, 20, 22, 23, 25, 27, 29, 32, 34, 36, 38, 40, 42, 44, 46, 47, 49, 51, 53, 55, 57, 59, 61, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125
Offset: 1

Views

Author

Clark Kimberling, Jul 21 2021

Keywords

Comments

The sequences A346301 and A346302 partition the positive integers.
See A076478 for a guide to related sequences.

Examples

			The first fourteen words w(n) are 0, 1, 00, 01, 10, 11, 000, 001, 010, 011, 100, 101, 110, 111, so that a(1) = 4.

Crossrefs

Cf. A007931, A076478, A346301.

Programs

A346305 Positions of words in A076478 that start with 1 and end with 1.

Original entry on oeis.org

2, 6, 12, 14, 24, 26, 28, 30, 48, 50, 52, 54, 56, 58, 60, 62, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 192, 194, 196, 198, 200, 202, 204, 206, 208, 210, 212, 214, 216, 218, 220, 222, 224, 226, 228, 230, 232, 234, 236, 238
Offset: 1

Views

Author

Clark Kimberling, Aug 16 2021

Keywords

Comments

The sequences A346303, A171757, A346304, and this sequence partition the positive integers. See A076478 for a guide to related sequences.
Is this A079946 with a 2 added in front? - R. J. Mathar, Sep 07 2021

Examples

			The first fourteen words w(n) are 0, 1, 00, 01, 10, 11, 000, 001, 010, 011, 100, 101, 110, 111, so that a(1) = 2.

Crossrefs

Cf. A007931, A076478, A171757, A346303, A346304.

Programs

A206445 Positions of 0 in A076478; complement of A206446.

Original entry on oeis.org

1, 3, 4, 5, 8, 11, 12, 13, 14, 15, 17, 19, 20, 24, 25, 27, 31, 35, 36, 37, 38, 39, 40, 41, 43, 44, 46, 47, 48, 51, 53, 54, 55, 57, 59, 62, 63, 68, 69, 70, 72, 73, 76, 78, 80, 85, 86, 89, 94, 99, 100, 101, 102, 103, 104, 105, 106, 107, 109, 110, 111, 113, 114
Offset: 1

Views

Author

Clark Kimberling, Feb 08 2012

Keywords

Examples

			The positions of 0 in 0,1,0,0,0,1,1,0,1,1,0,0,0,0,0,1,...
are 1,3,4,5,9,11,12,13,14,15,...

Crossrefs

Cf. A076478, A206446.

Programs

  • Mathematica
    d[n_] := Rest@IntegerDigits[n + 1, 2] + 1;
t = -1 + Flatten[Array[d, 100]]
Flatten[Position[t, 0]]  (* A206445 *)
Flatten[Position[t, 1]]  (* A206446 *)
Showing 1-10 of 35 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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