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.

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A216906 Smallest palindromic number of length 12 in two bases differing by n.

Original entry on oeis.org

248876637484140, 1013960892216384, 1093158208940760, 101975569769446364, 16699985993414520, 1310809863419914830, 331113952227684640
Offset: 2

Views

Author

James G. Merickel, Sep 19 2012

Keywords

Comments

See comments in A216840, A216841. Smaller of pair of bases are 17, 20, 19, 28, 23. A216*** cross-references are other sequences in a collection that includes this one, while A171*** sequences are various record multi-base palindromes.

Examples

			The first term is 747B6336B747 in base 17 and 22B4I88I4B22 in base 19.

Crossrefs

Cf. A171701, A171702, A171703, A171704, A171705, A171706, A171740, A171741, A171742, A171775, A216840, A216841, A216843, A216899, A216900, A216901, A216902, A216903, A216904, A216905, A216907, A216908, A216909, A216910.

Extensions

a(7)-a(8) from Chai Wah Wu, Aug 19 2015

A216907 Smallest palindromic number of length 13 in two bases differing by n.

Original entry on oeis.org

318713056300, 101252880671531, 778061547945037, 8047782883062742, 37866346844745025, 247644110790746909, 828823170233185174, 10556938000726441969, 10175119610486126649, 34575397981277771773, 91506808775956056797
Offset: 1

Views

Author

James G. Merickel, Sep 19 2012

Keywords

Comments

See comments sections of A216840 and A216841. Smallest of pair of bases are 8, 12, 14, 17, 19, 22, 24, 30, 32. For the remarkable case a(10), not only does the number have four 7s in a row in base 10, but it also has prime reversal there and in its 2nd base of 13-palindromicity, 42, no digit larger than 20 appears. A216*** cross-references are others in a collection of sequences that includes this one, while A171*** ones are various record multi-base palindromes.

Examples

			The first term is 4506461646054 in base 8 and 1113578753111 in base 9.

Crossrefs

Cf. A171701, A171702, A171703, A171704, A171705, A171706, A171740, A171741, A171742, A171775, A216840, A216841, A216843, A216899, A216900, A216901, A216902, A216903, A216904, A216905, A216906, A216908, A216909, A216910.

Extensions

a(10) added by James G. Merickel, Oct 17 2012
a(11) from Chai Wah Wu, Aug 20 2015

A216908 Smallest palindromic number of length 14 in two bases differing by n.

Original entry on oeis.org

2544221971606336, 10205276352438060, 2034000581064892726, 10716579270481431300
Offset: 2

Views

Author

James G. Merickel, Sep 19 2012

Keywords

Comments

See comments to A216840 and A216841. Smaller of pair of bases are 13, 14, 21, 24. A216*** sequences in cross-reference are others in a collection including this one, while A171***ones are various record multi-base palindromes.

Examples

			The first term is 85284144148258 in base 13 and 149210BB012941 in base 15.

Crossrefs

Cf. A171701, A171702, A171703, A171704, A171705, A171706, A171740, A171741, A171742, A171775, A216840, A216841, A216843, A216899, A216900, A216901, A216902, A216903, A216904, A216905, A216906, A216907, A216909, A216910.

A216909 Smallest palindromic number of length 15 in two bases differing by n.

Original entry on oeis.org

4712469842177, 14192598568359611, 389035682624908873, 11713091929744317938, 110566060844648951845
Offset: 1

Views

Author

James G. Merickel, Sep 19 2012

Keywords

Comments

See comments to A216840 and A216841. Smaller of pair of bases are 7, 12, 15, 19, 22. A216*** sequences in cross-reference are others in a collection including this one, while A171*** ones are various multi-base palindrome records.

Examples

			The first term is 664315252513466 in base 7 and 104446505644401 in base 8.

Crossrefs

Cf. A171701, A171702, A171703, A171704, A171705, A171706, A171740, A171741, A171742, A171775, A216840, A216841, A216843, A216899, A216900, A216901, A216902, A216903, A216904, A216905, A216906, A216907, A216908, A216910.

A216910 Smallest palindromic number of length 17 in two bases differing by n.

Original entry on oeis.org

1939137135947326, 8535888851411555386, 670814726929288547601
Offset: 1

Views

Author

James G. Merickel, Sep 19 2012

Keywords

Comments

Smallest of pair of bases are 8, 13, 17. The 16-palindrome case for this collection of sequences only has the n=2 example--bases are 13 and 15--530386561769238496 as of the time of this submission (hence, no sequence for that length). A 1st term for 19-palindromes is also available. It is the even semiprime 6411682614162861146, which reads as 11759A6746476A95711 in base 11. See A216840 and A216841 for comments pertaining to entire collection of sequences.

Examples

			The second term, 8535888851411555386 in decimal, is CA9BB36AAA63BB9AC in base 13 and is 1475E0C2C2C0E5741 in base 15 (employing the notation standard for hexadecimal).

Crossrefs

Multi-base palindrome records: A171701, A171702, A171703, A171704, A171705, A171706, A171740, A171741, A171742, A171775.
Others in this collection: A216840, A216841, A216843, A216899, A216900, A216901, A216902, A216903, A216904, A216905, A216906, A216907, A216908, A216909.

A216861 a(n) = 13*a(n-1) - 65*a(n-2) + 156*a(n-3) - 182*a(n-4) + 91*a(n-5) - 13*a(n-6), with initial terms 0, -2, -9, -44, -215, -1001.

Original entry on oeis.org

0, -2, -9, -44, -215, -1001, -4446, -19058, -79677, -327418, -1329601, -5355272, -21446945, -85548138, -340268656, -1350664731, -5353389340, -21195056584, -83846301409, -331483318257, -1309872510973, -5174049465897, -20431456722794, -80660347594658
Offset: 1

Views

Author

Roman Witula, Sep 18 2012

Keywords

Comments

a(n) is equal to the rational part (with respect of the field Q(sqrt(13))) of the product sqrt(2*(13 + 3*sqrt(13)))*X(2*n-1)/13, where X(n) = sqrt((13-3*sqrt(13))/2)*X(n-1) + sqrt(13)*X(n-2) - sqrt((13+3*sqrt(13))/2)*X(n-3), with X(0)=3, X(1)=sqrt((13-3*sqrt(13))/2), and X(2)=-(13+sqrt(13))/2.
The sequence X(n) is defined in almost the same way as sequence Y(n) from the comments to A161905. The only difference is in the initial condition X(2) = -Y(2).

Examples

			We have a(3)-5*a(2)=a(4)-5a(3)=1, a(5)-5*a(4)=5, and 19000 + a(8) = a(4) + 2*a(3) - 2*a(2).

References

  • Roman Witula, On some applications of formulas for sums of the unimodular complex numbers, Wyd. Pracowni Komputerowej Jacka Skalmierskiego, Gliwice 2011 (in Polish).

Links

Crossrefs

Cf. A216905, A216801, A216540.

Programs

  • Mathematica
    LinearRecurrence[{13, -65, 156, -182, 91, -13}, {0, -2, -9, -44, -215, -1001}, 25] (* Paolo Xausa, Feb 23 2024 *)

Formula

G.f.: -x^2*(26*x^4-84*x^3+57*x^2-17*x+2) / (13*x^6-91*x^5+182*x^4-156*x^3+65*x^2-13*x+1). - Colin Barker, Jun 01 2013

Extensions

Name clarified by Robert C. Lyons, Feb 08 2025
Previous Showing 11-16 of 16 results.

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