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 11 results. Next

A309223 Bisection A164733(2*n).

Original entry on oeis.org

0, 1, 2, 2, 3, 5, 6, 8, 12, 14, 17, 21, 25, 30, 36, 43, 49, 58, 66, 75, 85, 96, 109, 121, 136, 150, 167, 184, 202, 222, 242, 265, 287, 313, 338
Offset: 1

Views

Author

N. J. A. Sloane, Aug 29 2019, following a suggestion from Manuj Mishra

Keywords

Comments

This sequence and the other bisection A309224 are initially very similar: there appear to be blocks of terms that are identical except that the initial terms differ by 1. For example, [30, 36, 43, 49, 58, 66, 75] here versus [31, 36, 43, 49, 58, 66, 75] in A309224. Is there a simple explanation? - N. J. A. Sloane, Aug 31 2019

Crossrefs

Cf. A099009, A164733, A309224.

A309224 Bisection A164733(2*n+1).

Original entry on oeis.org

1, 1, 0, 0, 2, 1, 1, 2, 2, 3, 5, 7, 8, 12, 14, 17, 21, 25, 31, 36, 43, 49, 58, 66, 75, 86, 96, 109, 121, 136, 150, 167, 185, 202, 222
Offset: 1

Views

Author

N. J. A. Sloane, Aug 29 2019, following a suggestion from Manuj Mishra

Keywords

Crossrefs

Cf. A099009, A164733, A309223.

A164731 Number of cycles of n-digit numbers (including fixed points) under the Kaprekar map A151949.

Original entry on oeis.org

1, 0, 1, 1, 3, 3, 1, 4, 3, 8, 3, 16, 5, 27, 8, 46, 9, 73, 11, 110, 16, 162, 25, 231, 37, 318, 58, 429, 88, 572, 132, 747, 192, 963, 269, 1229, 372, 1551, 500, 1939, 662, 2401, 864, 2948, 1115, 3586, 1421, 4330, 1792, 5194, 2240, 6191, 2764, 7338, 3382, 8650, 4105
Offset: 1

Views

Author

Joseph Myers, Aug 23 2009

Keywords

Links

Crossrefs

Cf. A151949, A164716, A164732, A164733, A164734, A164735, A164736.
In other bases: A004526 (base 2, adjusted to start 1, 0, 0, 1, 1, ...), A165006 (base 3), A165025 (base 4), A165045 (base 5), A165064 (base 6), A165084 (base 7), A165103 (base 8), A165123 (base 9). [Joseph Myers, Sep 05 2009]

A164732 Number of n-digit numbers in a cycle (including fixed points) under the Kaprekar map A151949.

Original entry on oeis.org

1, 0, 1, 1, 10, 9, 8, 12, 16, 22, 14, 42, 18, 73, 29, 125, 34, 199, 38, 308, 49, 462, 71, 665, 105, 920, 161, 1243, 249, 1658, 379, 2170, 555, 2806, 780, 3587, 1075, 4539, 1449, 5689, 1922, 7059, 2516, 8677, 3252, 10566, 4156, 12774, 5255, 15337, 6578, 18300
Offset: 1

Views

Author

Joseph Myers, Aug 23 2009

Keywords

Links

Crossrefs

Cf. A151949, A164716, A164731, A164733, A164734, A164735, A164736.
In other bases: A004526 (base 2, adjusted to start 1, 0, 0, 1, 1, ...), A165007 (base 3), A165026 (base 4), A165046 (base 5), A165065 (base 6), A165085 (base 7), A165104 (base 8), A165124 (base 9). [Joseph Myers, Sep 05 2009]

A165027 Number of n-digit fixed points under the base-4 Kaprekar map A165012.

Original entry on oeis.org

1, 0, 1, 1, 0, 3, 1, 3, 3, 5, 3, 8, 5, 9, 8, 12, 9, 16, 12, 18, 16, 22, 18, 27, 22, 30, 27, 35, 30, 41, 35, 45, 41, 51, 45, 58, 51, 63, 58, 70, 63, 78, 70, 84, 78, 92, 84, 101, 92, 108, 101, 117, 108, 127, 117, 135, 127, 145, 135, 156, 145, 165, 156, 176, 165, 188, 176, 198
Offset: 1

Views

Author

Joseph Myers, Sep 04 2009

Keywords

Links

Crossrefs

Cf. A165012, A165016, A165025, A165026.
In other bases: A004526 (base 2, adjusted to start 1, 0, 0, 1, 1, ...), A008615 (base 3), A008617 (base 5), A165066 (base 6), A008722 (base 7, conjecturally), A165105 (base 8), A165125 (base 9), A164733 (base 10).

Formula

Conjectures from Chai Wah Wu, Apr 13 2024: (Start)
a(n) = 2*a(n-2) + a(n-3) - a(n-4) - 2*a(n-5) + a(n-7) for n > 8.
G.f.: x*(x^7 - x^6 - 2*x^5 + x^4 + x^2 - 1)/((x - 1)^3*(x + 1)^2*(x^2 + x + 1)). (End)

A165066 Number of n-digit fixed points under the base-6 Kaprekar map A165051.

Original entry on oeis.org

1, 0, 1, 0, 1, 3, 0, 2, 1, 4, 2, 3, 2, 5, 4, 5, 3, 8, 4, 8, 6, 9, 7, 10, 8, 13, 9, 13, 10, 17, 12, 16, 14, 19, 16, 21, 16, 24, 19, 25, 21, 28, 23, 29, 26, 33, 27, 35, 29, 39, 33, 39, 35, 44, 38, 46, 40, 50, 43, 53, 46, 56, 50, 58, 53, 64, 55, 66, 59, 71, 63, 73, 66, 78, 71, 81, 73, 87
Offset: 1

Views

Author

Joseph Myers, Sep 04 2009

Keywords

Links

Crossrefs

Cf. A165051, A165055, A165064, A165065.
In other bases: A004526 (base 2, adjusted to start 1, 0, 0, 1, 1, ...), A008615 (base 3), A165027 (base 4), A008617 (base 5), A008722 (base 7, conjecturally), A165105 (base 8), A165125 (base 9), A164733 (base 10).

Formula

Conjectures from Chai Wah Wu, Apr 13 2024: (Start)
a(n) = - a(n-1) + a(n-3) + 2*a(n-4) + 2*a(n-5) + a(n-6) - a(n-7) - 2*a(n-8) - 2*a(n-9) - a(n-10) + a(n-12) + a(n-13) for n > 15.
G.f.: x*(-x^14 + x^8 - x^5 + x^4 - x^2 - x - 1)/((x - 1)^3*(x + 1)^2*(x^2 + 1)*(x^2 + x + 1)*(x^4 + x^3 + x^2 + x + 1)). (End)

A165105 Number of n-digit fixed points under the base-8 Kaprekar map A165090.

Original entry on oeis.org

1, 1, 1, 0, 0, 2, 1, 1, 1, 1, 0, 4, 0, 4, 2, 2, 2, 4, 2, 3, 6, 5, 2, 7, 2, 6, 5, 10, 5, 8, 5, 8, 7, 9, 11, 12, 7, 11, 9, 12, 9, 21, 10, 14, 12, 15, 13, 18, 20, 18, 15, 20, 15, 23, 16, 30, 20, 23, 20, 26, 21, 27, 32, 29, 23, 32, 25, 32, 28, 43
Offset: 1

Views

Author

Joseph Myers, Sep 04 2009

Keywords

Links

Crossrefs

Cf. A165090, A165094, A165103, A165104.
In other bases: A004526 (base 2, adjusted to start 1, 0, 0, 1, 1, ...), A008615 (base 3), A165027 (base 4), A008617 (base 5), A165066 (base 6), A008722 (base 7, conjecturally), A165125 (base 9), A164733 (base 10).

Formula

Conjectures from Chai Wah Wu, Apr 13 2024: (Start)
a(n) = - a(n-1) + a(n-3) + 2*a(n-4) + 2*a(n-5) + a(n-6) - a(n-7) - 2*a(n-8) - 2*a(n-9) - a(n-10) + a(n-12) + a(n-13) for n > 15.
G.f.: x*(-x^14 + x^8 - x^5 + x^4 - x^2 - x - 1)/((x - 1)^3*(x + 1)^2*(x^2 + 1)*(x^2 + x + 1)*(x^4 + x^3 + x^2 + x + 1)). (End)

A165125 Number of n-digit fixed points under the base-9 Kaprekar map A165110.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 2, 1, 1, 2, 2, 3, 2, 2, 1, 4, 2, 3, 2, 3, 3, 4, 2, 3, 4, 4, 4, 4, 4, 4, 5, 4, 4, 5, 4, 5, 5, 5, 5, 6, 5, 6, 6, 6, 5, 7, 6, 7, 6, 6, 6, 8, 6, 7, 7, 8, 8, 8, 7, 7, 9, 8, 8
Offset: 1

Views

Author

Joseph Myers, Sep 04 2009

Keywords

Links

Crossrefs

Cf. A165110, A165114, A165123, A165124.
In other bases: A004526 (base 2, adjusted to start 1, 0, 0, 1, 1, ...), A008615 (base 3), A165027 (base 4), A008617 (base 5), A165066 (base 6), A008722 (base 7, conjecturally), A165105 (base 8), A164733 (base 10).

Formula

Conjectures from Chai Wah Wu, Apr 13 2024: (Start)
a(n) = a(n-2) - a(n-3) + a(n-5) - a(n-6) + a(n-8) + a(n-15) - a(n-17) + a(n-18) - a(n-20) + a(n-21) - a(n-23) for n > 24.
G.f.: x*(x^23 + x^22 - x^21 + x^20 + 2*x^19 - x^18 + x^17 + 3*x^16 - 2*x^15 + 3*x^13 - x^12 + 2*x^10 - x^9 + 2*x^7 - x^5 + x^4 + x^3 - x^2 + 1)/(x^23 - x^21 + x^20 - x^18 + x^17 - x^15 - x^8 + x^6 - x^5 + x^3 - x^2 + 1). (End)

A164734 Number of n-digit cycles of length 2 under the Kaprekar map A151949.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 2, 1, 1, 1, 1, 1, 0, 3, 0, 3, 0, 2, 0, 2, 3, 1, 2, 1, 2, 1, 1, 4, 1, 4, 0, 3, 0, 3, 4, 2, 3, 2, 3, 1, 2, 5, 2, 5, 1, 4, 1, 4, 5, 3, 4, 3, 4, 2, 3, 7, 3, 6, 2, 5, 2, 5, 7, 4, 6, 4, 5, 3, 4, 9, 4, 8, 3, 7, 3, 6, 9, 5, 8, 5, 7, 4
Offset: 1

Views

Author

Joseph Myers, Aug 23 2009

Keywords

Links

Crossrefs

Cf. A151949, A164723, A164724, A164731, A164732, A164733, A164735, A164736.

Formula

Conjectures from Chai Wah Wu, Apr 13 2024: (Start)
a(n) = a(n-7) + a(n-9) + a(n-14) - a(n-16) - a(n-21) - a(n-23) + a(n-30) for n > 41.
G.f.: x*(-x^40 - x^38 - x^36 + x^33 - x^32 + 2*x^31 - x^30 + 2*x^29 - x^28 + x^27 + x^25 - x^24 + 2*x^23 - x^22 + 2*x^21 - 2*x^20 + x^19 - x^16 - x^15 - x^14 + x^13 - x^12 + x^11 - x^4)/(x^30 - x^23 - x^21 - x^16 + x^14 + x^9 + x^7 - 1). (End)

A164735 Number of n-digit cycles of length 3 under the Kaprekar map A151949.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 4, 0, 10, 0, 20, 0, 36, 0, 60, 1, 94, 4, 141, 10, 204, 21, 286, 39, 392, 66, 527, 105, 696, 159, 906, 231, 1164, 326, 1477, 449, 1854, 605, 2304, 801, 2836, 1044, 3462, 1341, 4194, 1701, 5044, 2133, 6027, 2646, 7158, 3252, 8452, 3963
Offset: 1

Views

Author

Joseph Myers, Aug 23 2009

Keywords

Links

Crossrefs

Cf. A151949, A164725, A164726, A164731, A164732, A164733, A164734, A164736.

Formula

Conjectures from Chai Wah Wu, Apr 13 2024: (Start)
a(n) = 4*a(n-2) - 6*a(n-4) + 5*a(n-6) - 5*a(n-8) + a(n-9) + 6*a(n-10) - 4*a(n-11) - 4*a(n-12) + 6*a(n-13) + a(n-14) - 5*a(n-15) + 5*a(n-17) - 6*a(n-19) + 4*a(n-21) - a(n-23) for n > 25.
G.f.: x*(-x^24 + x^22 + x^18 - x^16 + x^15 - x^13 + x^7)/((x - 1)^6*(x + 1)^5*(x^2 - x + 1)*(x^2 + x + 1)^2*(x^6 + x^3 + 1)). (End)
Showing 1-10 of 11 results. Next

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

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