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 1-3 of 3 results.

A090316 a(n) = 24*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 24.

Original entry on oeis.org

2, 24, 578, 13896, 334082, 8031864, 193098818, 4642403496, 111610782722, 2683301188824, 64510839314498, 1550943444736776, 37287153512997122, 896442627756667704, 21551910219673022018, 518142287899909196136, 12456966819817493729282, 299485345963519758698904
Offset: 0

Views

Author

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Jan 25 2004

Keywords

Comments

Lim_{n->infinity} a(n)/a(n+1) = 0.0415945... = 1/(12+sqrt(145)) = sqrt(145) - 12.
Lim_{n->infinity} a(n+1)/a(n) = 24.0415945... = 12+sqrt(145) = 1/(sqrt(145)-12).

Examples

			a(4) =334082 = 24a(3) + a(2) = 24*13896+ 578 = (12+sqrt(145))^4 + (12-sqrt(145))^4 = 334081.99999700672 + 0.00000299327 = 334082.

Links

Crossrefs

Cf. A041264, A058168, A056949.
Lucas polynomials Lucas(n,m): A000032 (m=1), A002203 (m=2), A006497 (m=3), A014448 (m=4), A087130 (m=5), A085447 (m=6), A086902 (m=7), A086594 (m=8), A087798 (m=9), A086927 (m=10), A001946 (m=11), A086928 (m=12), A088316 (m=13), A090300 (m=14), A090301 (m=15), A090305 (m=16), A090306 (m=17), A090307 (m=18), A090308 (m=19), A090309 (m=20), A090310 (m=21), A090313 (m=22), A090314 (m=23), this sequence (m=24), A330767 (m=25).

Programs

  • GAP
    a:=[2,24];; for n in [3..20] do a[n]:=24*a[n-1]+a[n-2]; od; a; # G. C. Greubel, Dec 29 2019
  • Magma
    I:=[2,24]; [n le 2 select I[n] else 24*Self(n-1) +Self(n-2): n in [1..20]]; // G. C. Greubel, Dec 29 2019
  • Maple
    seq(simplify(2*(-I)^n*ChebyshevT(n, 12*I)), n = 0..20); # G. C. Greubel, Dec 29 2019
  • Mathematica
    LinearRecurrence[{24,1},{2,24},20] (* Harvey P. Dale, Aug 30 2015 *)
LucasL[Range[20]-1,24] (* G. C. Greubel, Dec 29 2019 *)
  • PARI
    vector(21, n, 2*(-I)^(n-1)*polchebyshev(n-1, 1, 12*I) ) \\ G. C. Greubel, Dec 29 2019
  • Sage
    [2*(-I)^n*chebyshev_T(n, 12*I) for n in (0..20)] # G. C. Greubel, Dec 29 2019

Formula

a(n) = 24*a(n-1) + a(n-2), starting with a(0) = 2 and a(1) = 24.
a(n) = (12+sqrt(145))^n + (12-sqrt(145))^n.
(a(n))^2 = a(2n) - 2 if n=1,3,5,..., (a(n))^2 = a(2n)+2 if n=2,4,6,....
G.f.: 2*(1-12*x)/(1-24*x-x^2). - Philippe Deléham, Nov 02 2008
a(n) = 2*(-i)^n * ChebyshevT(n, 12*i) = Lucas(n, 24). - G. C. Greubel, Dec 29 2019
a(n) = 2 * A041264(n-1) for n>0. - Alois P. Heinz, Dec 29 2019

Extensions

More terms from Ray Chandler, Feb 14 2004
Corrected by T. D. Noe, Nov 07 2006

A040132 Continued fraction for sqrt(145).

Original entry on oeis.org

12, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			12 + 1/(24 + 1/(24 + 1/(24 + 1/(24 + ...)))) = sqrt(145).

Links

Crossrefs

Cf. A041264/A041265 (convergents), A176910 (decimal expansion).
Cf. A040000, A040002, A040006.

Programs

  • Maple
    Digits := 100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):
  • Mathematica
    ContinuedFraction[Sqrt[145],300] (* Vladimir Joseph Stephan Orlovsky, Mar 13 2011*)

Formula

From Elmo R. Oliveira, Feb 12 2024: (Start)
a(n) = 24 for n >= 1.
G.f.: 12*(1+x)/(1-x).
E.g.f.: 24*exp(x) - 12.
a(n) = 12*A040000(n) = 6*A040002(n) = 4*A040006(n). (End)

A041265 Denominators of continued fraction convergents to sqrt(145).

Original entry on oeis.org

1, 24, 577, 13872, 333505, 8017992, 192765313, 4634385504, 111418017409, 2678666803320, 64399421297089, 1548264777933456, 37222754091700033, 894894362978734248, 21514687465581321985, 517247393536930461888, 12435452132351912407297
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

From Michael A. Allen, May 04 2023: (Start)
Also called the 24-metallonacci sequence; the g.f. 1/(1-k*x-x^2) gives the k-metallonacci sequence.
a(n) is the number of tilings of an n-board (a board with dimensions n X 1) using unit squares and dominoes (with dimensions 2 X 1) if there are 24 kinds of squares available. (End)

Links

Crossrefs

Cf. A041264, A176910, A040132.
Row n=24 of A073133, A172236 and A352361 and column k=24 of A157103.

Programs

  • Mathematica
    Denominator[Convergents[Sqrt[145], 30]] (* Vincenzo Librandi, Dec 14 2013 *)

Formula

a(n) = F(n, 24), the n-th Fibonacci polynomial evaluated at x=24. - T. D. Noe, Jan 19 2006
From Philippe Deléham, Nov 21 2008: (Start)
a(n) = 24*a(n-1) + a(n-2) for n > 1, a(0)=1, a(1)=24.
G.f.: 1/(1-24*x-x^2). (End)

Extensions

More terms from Colin Barker, Nov 14 2013
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.