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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A041060 Numerators of continued fraction convergents to sqrt(37).

Original entry on oeis.org

6, 73, 882, 10657, 128766, 1555849, 18798954, 227143297, 2744518518, 33161365513, 400680904674, 4841332221601, 58496667563886, 706801342988233, 8540112783422682, 103188154744060417, 1246797969712147686
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A010491, A041061.
Cf. A089926. - R. J. Mathar, Sep 09 2008

Programs

  • Mathematica
    Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[37],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 21 2011 *)
CoefficientList[Series[(6 + x)/(1 - 12  x - x^2), {x, 0, 30}], x]  (* Vincenzo Librandi_, Oct 28 2013 *)

Formula

From Philippe Deléham, Nov 21 2008: (Start)
a(n) = 12*a(n-1) + a(n-2), n > 1; a(0)=6, a(1)=73.
G.f.: (6+x)/(1-12*x-x^2). (End)

A082975 Denominators of continued fraction convergents to (sqrt(37)-4)/3.

Original entry on oeis.org

1, 1, 3, 10, 13, 36, 121, 157, 435, 1462, 1897, 5256, 17665, 22921, 63507, 213442, 276949, 767340, 2578969, 3346309, 9271587, 31161070, 40432657, 112026384, 376511809, 488538193, 1353588195, 4549302778, 5902890973, 16355084724
Offset: 0

Views

Author

Gary W. Adamson, May 27 2003

Keywords

Links

Crossrefs

Numerators are A082962.
Cf. A041061.

Programs

  • Mathematica
    Denominator[Convergents [(Sqrt[37]-4)/3, 40]] (* Vincenzo Librandi, Feb 01 2014 *)
a[ n_] := If[ n < -1, SeriesCoefficient[ (1 - 2*x + 3*x^2 + x^3 + x^4) / (1 + 12*x^3 - x^6), {x, 0, -2 - n}], SeriesCoefficient[ (1 + x + 3*x^2 - 2*x^3 + x^4) / (1 - 12*x^3 - x^6), {x, 0, n}]]; (* Michael Somos, Dec 07 2017 *)
  • PARI
    {a(n) = if( n<-1, n = -2-n; polcoeff( (1 - 2*x + 3*x^2 + x^3 + x^4) / (1 + 12*x^3 - x^6) + x * O(x^n), n), polcoeff( (1 + x + 3*x^2 - 2*x^3 + x^4) / (1 - 12*x^3 - x^6) + x * O(x^n), n))}; /* Michael Somos, Dec 07 2017 */

Formula

Empirical g.f.: 1 + x*(1+3*x+10*x^2+x^3+x^5)/(1-12*x^3-x^6). - Colin Barker, Jan 04 2012
From Michael Somos, Dec 07 2017: (Start)
G.f.: (1 + x + 3*x^2 - 2*x^3 + x^4) / (1 - 12*x^3 - x^6).
Let s := sqrt(37), c := 6 + s, b(0) = 1/2 + 2/s, b(1) = 1/2 + 7/2/s, b(2) := 3/2 + 9/s, then a(3*n + k) ~ b(k) * c^n for k=0, 1, 2.
0 = a(n)*(+3*a(n+1) +5*a(n+2) -3*a(n+3)) +a(n+1)*(+7*a(n+1) +10*a(n+2) -7*a(n+3)) +a(n+2)*(-5*a(n+2) +3*a(n+3)) for all n in Z.
a(n+3) = 12*a(n) + a(n-3), a(n+1) = a(n-1) + (1+mod(n, 3))*a(n) for all n in Z.
a(3*n - 1) = -(-1)^n * a(-3*n - 1) = 3 * A041061(n-1) for all n in Z. (End)

Extensions

Edited by Don Reble, Nov 04 2005
Prepended a(0)=1 from Vincenzo Librandi, Feb 01 2014

A305534 Index of the smallest prime in the n-Fibonacci sequence, or the Lucas U(n,-1) sequence.

Original entry on oeis.org

3, 2, 2, 3, 2, 3, 2, 5, 29, 3, 2, 5, 2, 3, 23, 3, 2, 7, 2, 3, 29, 19, 2, 3, 83, 3, 53, 19, 2, 5, 2, 5, 5, 5479, 71, 3, 2, 17, 11, 3, 2, 37, 2, 31, 5, 11, 2, 5
Offset: 1

Views

Author

Eric Chen, Jun 04 2018

Keywords

Comments

Smallest k such that the k-th Fibonacci polynomial evaluated at x=n is prime. (The first few Fibonacci polynomials are 1, x, x^2 + 1, x^3 + 2*x, x^4 + 3*x^2 + 1, x^5 + 4*x^3 + 3*x, ...)
All terms are primes, since if a divides b, then the a-th term of the n-Fibonacci sequence also divides the b-th term of the n-Fibonacci sequence.
Corresponding primes are 2, 2, 3, 17, 5, 37, 7, 4289, 726120289954448054047428229, 101, 11, 21169, 13, 197, 82088569942721142820383601, 257, 17, 34539049, 19, 401, ...
a(n) = 2 if and only if n is prime.
a(n) = 3 if and only if n^2 + 1 is prime (A005574), except n=2 (since 2 is the only prime p such that p^2 + 1 is also prime).
a(34) > 1024, does a(n) exist for all n >= 1? (However, 17 is the only prime in the first 1024 terms of the 4-Fibonacci sequence, and it seems that 17 is the only prime in the 4-Fibonacci sequence.)
a(35)..a(48) = 71, 3, 2, 17, 11, 3, 2, 37, 2, 31, 5, 11, 2, 5, a(50)..a(54) = 11, 11, 23, 2, 3, a(56) = 3, a(58)..a(75) = 5, 2, 47, 2, 5, 311, 13, 233, 3, 2, 5, 11, 5, 2, 7, 2, 3, 5. Unknown terms a(34), a(49), a(55), a(57), exceed 1024, if they exist.
a(49) > 20000, if it exists. - Giovanni Resta, Jun 06 2018

Crossrefs

Cf. A001605, A096650, A209493, which are the indices of the primes in the n-Fibonacci sequence for n = 1, 2, 3.
Cf. A005478, A086383, A201001, which are the primes in the n-Fibonacci sequence for n = 1, 2, 3.
Cf. A000045, A000129, A006190, A001076, A052918, A005668, A054413, A041025, A099371, A041041, A049666, A041061 (the n-Fibonacci sequence for n = 1 to 12).
Cf. A302990 (for n-step Fibonacci sequence instead of n-Fibonacci sequence).

Programs

  • PARI
    b(n,k)=([n,1;1,0]^k)[1,2]
a(n)=for(k=1,2^12,if(ispseudoprime(b(n,k)),return(k)))

Extensions

a(34)-a(48) from Giovanni Resta, Jun 06 2018
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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