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 61-68 of 68 results.

A067009 Continued fraction expansion of Product_{p prime} (1 + 1/(p+1)^2).

Original entry on oeis.org

1, 3, 1, 3, 40, 1, 4, 1, 1, 1, 3, 1, 3, 2, 4, 2, 40, 2, 2, 2, 4, 3, 1, 1, 1, 1, 1, 1, 7, 5, 1, 2, 1, 2, 3, 3, 1, 12, 1, 4, 2, 1, 1, 1, 2, 2, 1, 1, 5, 1, 2, 1, 2, 4, 1, 2, 1, 6, 2, 4, 1, 26, 4, 5, 6, 1, 17, 7, 1, 1, 7, 4, 1, 17, 7, 1, 3, 1, 1, 62, 1, 2, 3, 2, 36, 3, 1, 1, 1, 5, 1, 12, 12, 1, 4, 4, 1, 2, 22, 1
Offset: 0

Views

Author

N. J. A. Sloane, Nov 30 2002

Keywords

Examples

			1.26655850147152857161454711262964...

Links

Crossrefs

Cf. A065486 (decimal expansion).

Programs

  • PARI
    contfrac(prodeulerrat(1 + 1/(p+1)^2)) \\ Amiram Eldar, Mar 15 2021

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 01 2003
Offset changed by Andrew Howroyd, Jul 04 2024

A072801 Continued fraction expansion of Product_{p prime} (1 - 1/(p*(p^2-1))).

Original entry on oeis.org

0, 1, 3, 1, 2, 1, 1, 2, 1, 1, 1, 18, 1, 1, 6, 5, 59, 1, 2, 159, 1, 5, 1, 3, 1, 1, 73, 1, 4, 1, 3, 22, 1, 90, 9, 4, 145, 1, 32, 1, 2, 3, 1, 2, 1, 2, 4, 1, 2, 12, 1, 462, 1, 3, 3, 2, 2, 1, 1, 1, 1, 7, 2, 4, 23, 1, 6, 1, 2, 1, 3, 1, 3, 2, 6, 1, 1, 10, 2, 2, 10, 5, 24, 3, 1, 3, 1, 181, 5, 1, 1, 3, 4, 3, 1
Offset: 0

Views

Author

Rick L. Shepherd, Jul 11 2002

Keywords

Links

Crossrefs

Cf. A065470 (decimal expansion).

Programs

  • PARI
    contfrac(prodeulerrat(1 - 1/(p*(p^2-1)))) \\ Amiram Eldar, Mar 13 2021

A072802 Continued fraction expansion of Product_{p prime} (1 + 1/(p*(p^2-1))).

Original entry on oeis.org

1, 4, 3, 11, 33, 1, 2, 1, 4, 103, 1, 1, 1, 2, 2, 12, 1, 1, 3, 2, 3, 1, 1, 15, 1, 5, 11, 1, 1, 1, 5, 1, 1, 14, 1, 1, 1, 2, 20, 2, 1, 2, 2, 29, 1, 8, 1, 1, 4, 2, 2, 6, 4, 1, 1, 1, 2, 1, 12, 1, 1, 3, 2, 2, 2, 4, 2, 7, 1, 11, 2, 4, 2, 1, 1, 1, 2, 2, 1, 3, 1, 2, 7, 2, 3, 1, 11, 3, 15, 1, 1, 3, 2, 1, 3, 1, 1
Offset: 0

Views

Author

Rick L. Shepherd, Jul 11 2002

Keywords

Links

Crossrefs

Cf. A065487 (decimal expansion).

Programs

  • PARI
    \p 1002 x=1.2312911488886035627747876512720337... (cut-and-paste all 1002 digits from link) contfrac(x)
  • PARI
    contfrac(prodeulerrat(1 + 1/(p*(p^2-1)))) \\ Amiram Eldar, Jun 13 2021

A077387 Continued fraction expansion of Product_{p prime} (1 + p/((p-1)^2*(p+1))).

Original entry on oeis.org

2, 4, 1, 9, 1, 1, 2, 1, 48, 1, 12, 1, 1, 2, 1, 1, 3, 9, 3, 3, 1, 16, 3, 1, 1, 6, 12, 50, 23, 8, 1, 1, 2, 1, 1, 1, 2, 1, 11, 1, 2, 2, 1, 1, 10, 5, 7, 2, 1, 3, 1, 4, 3, 8, 1, 2, 1, 1, 4, 11, 1, 1, 1, 1, 1, 16, 1, 1, 1, 7, 1, 13, 1, 3, 1, 6, 2, 7, 1, 2, 1, 11, 2, 1, 5, 1, 9, 4, 2, 9, 26, 1, 2, 1, 20, 1, 1, 4
Offset: 0

Views

Author

N. J. A. Sloane, Nov 30 2002

Keywords

Links

Crossrefs

Cf. A065484 (decimal expansion).

Programs

  • PARI
    contfrac(prodeulerrat(1 + p/((p-1)^2*(p+1)))) \\ Amiram Eldar, Jun 13 2021

Extensions

Extended by Ray Chandler, Sep 27 2006
Offset changed by Andrew Howroyd, Jul 06 2024

A349001 The number of Lyndon words of size n from an alphabet of 5 letters and 1st and 2nd letter of the alphabet with equal frequency in the words.

Original entry on oeis.org

1, 3, 4, 14, 46, 174, 656, 2640, 10790, 45340, 193600, 839820, 3686424, 16353924, 73187456, 330052646, 1498335650, 6841899606, 31404443032, 144814450188, 670552118244, 3116578216310, 14534401932712, 67992210407514, 318969964124256, 1500268062754830
Offset: 0

Views

Author

R. J. Mathar, Nov 05 2021

Keywords

Comments

Counts a subset of the Lyndon words in A001692. Here there is no requirement of how often the 3rd to 5th letter of the alphabet are in the admitted word, only on the frequency of the 1st and 2nd letter of the alphabet.
Let T(n,k,M) be the number of words of length n drawn from an alphabet of size M where the first k letters of the alphabet appear with the same frequency f in each word. Then T(n,k,M) = Sum_{f=0..floor(n/k)} (M-k)^(n-f*k) * Product_{i=0..k-1} binomial(n-i*f,f) and T(n,2,5) = A026375(n), T(n,3,6) = A294035(n), T(n,2,6) = A081671(n). Removing the words with cycles by the inclusion-exclusion principle by a Mobius Transform gives words of length n of that type without cycles and division through n the Lyndon words of that type. - R. J. Mathar, Nov 07 2021

Examples

			Examples for the alphabet {0,1,2,3,4}:
a(0)=1 counts (), the empty word.
a(3)=14 counts (021) (031) (041) (012) (013) (223) (233) (243) (014) (224) (234) (334) (244) (344), words of length 3 where the letters 0 and the 1 occur both either not or once.
a(4)=46 counts (0011) (0221) (0321) (0421) (0231) (0331) (0431) (0241) (0341) (0441) (0212) (0312) (0412) (0122) (0132) (0142) (0213) (0313) (0413) (0123) (2223) (0133) (2233) (2333) (2433) (0143) (2243) (2343) (2443) (0214) (0314) (0414) (0124) (2224) (2324) (0134) (2234) (2334) (3334) (2434) (0144) (2244) (2344) (3344) (2444) (3444).

Links

Crossrefs

Cf. A022553 (alphabet of 2 letters), A290277 (of 3 letters), A060165 (of 4 letters), A026375.

Programs

  • PARI
    a(n) = if(n>0, sumdiv(n, d, moebius(n/d)*sum(k=0, d, binomial(d,k)*binomial(2*k,k)))/n, n==0) \\ Andrew Howroyd, Jan 14 2023

Formula

n*a(n) = Sum_{d|n} mu(d)*A026375(n/d) where mu = A008683.

Extensions

Terms a(16) and beyond from Andrew Howroyd, Jan 14 2023

A032324 Number of aperiodic necklaces with n labeled beads of 5 colors.

Original entry on oeis.org

5, 20, 240, 3600, 74880, 1857600, 56246400, 1965600000, 78744960000, 3542608742400, 177187481856000, 9744664849920000, 584718747604992000, 38006232139459584000, 2660470029606445056000
Offset: 1

Views

Author

Christian G. Bower

Keywords

Links

Formula

"CHJ" (necklace, identity, labeled) transform of 5, 0, 0, 0...
n! * A001692.

A066517 Continued fraction expansion of Artin constant of rank 2: product(1-1/(p^3-p^2), p=prime).

Original entry on oeis.org

0, 1, 2, 3, 3, 1, 2, 2, 1, 3, 1, 8, 1, 4, 1, 1, 2, 1, 2, 31, 9, 2, 1, 3, 1, 3, 13, 3, 2, 11, 3, 1, 1, 1, 2, 2, 10, 10, 2, 1, 204, 4, 2, 12, 2, 8, 1, 1, 6, 17, 5, 2, 34, 4, 2, 2, 1, 5, 1, 1, 1, 1, 4, 1, 2, 1, 54, 3, 1, 6, 1, 13, 3, 2, 12, 1, 1, 1, 2, 2, 5, 2, 2, 7, 2, 2, 2, 1, 2, 1, 10, 3, 3, 1, 8, 9, 1
Offset: 0

Views

Author

Simon Plouffe Jan 05 2002

Keywords

Examples

			0.697501358496365903284670350820922924...

Links

Crossrefs

Cf. A065414.

A066834 Continued fraction expansion of Product_{p prime} (1 - 1/(p^5 - p^4)).

Original entry on oeis.org

0, 1, 13, 1, 1, 4, 1, 1, 1, 3, 16, 16, 1, 11, 1, 1, 1, 1, 9, 13, 1, 5, 22, 4, 2, 6, 1, 1, 1, 39, 1, 1, 3, 1, 12, 4, 1, 106, 15, 19, 7, 7, 4, 1, 5, 1, 2, 1, 1, 2, 4, 3, 1, 10, 1, 1, 1, 1, 2, 3, 1, 6, 7, 8, 1, 7, 5, 1, 2, 44, 2, 1, 5, 1, 4, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 20, 8, 2, 3, 1, 1, 1, 4, 1, 1, 2
Offset: 0

Views

Author

Randall L Rathbun, Jan 16 2002

Keywords

Links

Crossrefs

Cf. A065416 (decimal expansion).

Programs

  • PARI
    contfrac(prodeulerrat(1-1/(p^5-p^4))) \\ Amiram Eldar, Jun 13 2021
Previous Showing 61-68 of 68 results.

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

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