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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A376054 Denominator of sum S(n) defined in A376052.

Original entry on oeis.org

3, 15, 105, 315, 3465, 45045, 90090, 765765, 451035585, 25508568085, 411827311870584610, 525058386770138717020639964850, 528134692562568161116953143877712480332943632586669596900
Offset: 1

Views

Author

N. J. A. Sloane, Sep 14 2024

Keywords

Examples

			The initial values of S(n) are 1/3, 8/15, 71/105, 248/315, 3043/3465, 43024/45045, 89051/90090, ...
		

Crossrefs

A376059 a(n) is the denominator of the sum S(n) defined in A376058.

Original entry on oeis.org

1, 2, 6, 78, 18330, 1679962830, 22578200883132834030, 6627077016548303724729207245056971365730, 922281145448518091883798423085535218757314338662318933097843039655721026758456630
Offset: 0

Views

Author

N. J. A. Sloane, Sep 14 2024

Keywords

Examples

			The first few values of S(n) are 0, 1/2, 5/6, 77/78, 18329/18330, 1679962829/1679962830, 22578200883132834029/22578200883132834030, ...
		

Crossrefs

Programs

  • Mathematica
    RecurrenceTable[{a[n+1] == Fibonacci[n+1]*a[n]^2 + a[n], a[0] == 1}, a, {n, 0, 8}] (* Amiram Eldar, Sep 15 2024 *)

Formula

a(n+1) = Fibonacci(n+1)*a(n)^2 + a(n), with a(0) = 1.

A376184 Lexicographically earliest sequence of positive integers a(1), a(2), a(3), ... such that for any n > 0, S(n) = Sum_{k = 1..n} b(k)/a(k) < 1, where {b(k)} is the sequence b(1)=5/4, b(2*i)=3/2, b(2*i+1)=6/5 (i>0).

Original entry on oeis.org

2, 5, 17, 341, 92753, 10753782821, 92515075960384748177, 10698799099944699918936107506299150093941, 91571441744782016867976366392607084634231243149599342901251284090792487979854033
Offset: 1

Views

Author

N. J. A. Sloane, Sep 15 2024

Keywords

Comments

This sequence and A376062 were discovered by Rémy Sigrist on Sep 09 2024. The two sequences {b(1)=7/6, b(k)=5/4 for k>1} and {b(1)=5/4, b(2*k)=3/2, b(2*k+1)=6/5 for k>0} are the first sequences {b(i)} discovered with the property that the sums S(n) do not converge to numbers of the form (e_n - 1)/e_n as n-> oo.

Examples

			The initial values of S(n) are 5/8, 37/40, 677/680, 231877/231880, 21507565637/21507565640, 231287689900961870437/231287689900961870440, ...
		

Crossrefs

A376185 a(n) = denominator of the sum S(n) defined in A376062.

Original entry on oeis.org

12, 48, 624, 97968, 2399530224, 1439436326371902768, 517994234419759747473589427583418224, 67079506723028253472357256785558488997471406450171845011442457607246768
Offset: 1

Views

Author

N. J. A. Sloane, Sep 15 2024

Keywords

Examples

			The initial values of S(n) are 7/12, 43/48, 619/624, 97963/97968, 2399530219/2399530224, 1439436326371902763/1439436326371902768 ...
		

Crossrefs

Formula

1/a(n) = 1/a(n-1) - 1/(4*A376062(n)) for n >= 2.

A376186 a(n) = denominator of the sum S(n) defined in A376184.

Original entry on oeis.org

8, 40, 680, 231880, 21507565640, 231287689900961870440, 21397598199889399837872215012598300187880, 228928604361955042169940915981517711585578107873998357253128210226981219949635080
Offset: 1

Views

Author

N. J. A. Sloane, Sep 15 2024

Keywords

Examples

			The initial values of S(n) are 5/8, 37/40, 677/680, 231877/231880, 21507565637/21507565640, 231287689900961870437/231287689900961870440, ...
		

Crossrefs

A376060 Lexicographically earliest sequence of positive integers a(0), a(1), a(2), a(3), ... such that for any n > 0, S(n) = Sum_{k = 0..n-1} Catalan(k)/a(k) < 1.

Original entry on oeis.org

2, 3, 13, 391, 426973, 546916547269, 940084230410591812263433, 2872214670866692695441731060944339347071024216683
Offset: 0

Views

Author

N. J. A. Sloane, Sep 14 2024

Keywords

Crossrefs

Formula

a(n+1) = Catalan(n+1)*A376061(n) + 1.
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