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 1-4 of 4 results.

A004170 Reversals of Fibonacci numbers (sorted).

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 12, 16, 31, 43, 55, 98, 332, 441, 773, 789, 1814, 4852, 5676, 7951, 11771, 40238, 52057, 64901, 75682, 86364, 118713, 393121, 814691, 922415, 5647229, 7882075, 8754253, 9038712, 9626431
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

The smallest Fibonacci number with 1, 2, 3,... trailing zeros is F(15), F(150), F(750), F(7500), F(75000),.... This provides an idea of how many digits may be "lost" by reversal. - R. J. Mathar, Mar 11 2013

Links

Crossrefs

Cf. A000045, A004091.
Cf. A214855, A004086.

Programs

  • Haskell
    import Data.Set (fromList, deleteFindMin, insert)
a004170 n = a004170_list !! n
a004170_list = 0 : 1 : f (fromList us) vs where
   f s (x:xs) = m : f (insert x s') xs
     where (m,s') = deleteFindMin s
   (us,vs) = splitAt 120 $ drop 2 a004091_list
-- Reinhard Zumkeller, Mar 09 2013
  • Mathematica
    Sort[FromDigits[Reverse[IntegerDigits[#]]]&/@Fibonacci[Range[0,40]]] (* Harvey P. Dale, Jun 17 2011 *)
IntegerReverse[Fibonacci[Range[0,40]]]//Sort (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 02 2019 *)

A072912 Number of Fibonacci numbers F(k) <= 10^n which end in 0.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 24, 25, 25, 25
Offset: 0

Views

Author

Shyam Sunder Gupta and Benoit Cloitre, Aug 15 2002

Keywords

Examples

			a(2)=6 because there are 6 Fibonacci numbers F(k) <= 10^2 which end in 0.

Crossrefs

Different from A002280.
Cf. A008597, A214855.

Programs

  • PARI
    a(n) = (sum(i=0,ceil(n*log(10)/log((1+sqrt(5))/2)),if(fibonacci(i)%10+1+sign(fibonacci(i)-10^n),0,1)))

Formula

a(n) = ceiling(n*log(10)/(15*log(phi))) +0 or +1.

A215649 Reversals of tribonacci numbers (sorted).

Original entry on oeis.org

0, 0, 1, 1, 2, 4, 7, 18, 31, 42, 44, 405, 472, 729, 941, 5071, 6313, 8675, 9853, 21066, 31591, 90601, 447014, 514121, 674557, 713322, 770074, 4606468, 7359831, 7575552, 8089735, 19520951, 52494292, 69005989, 106799181, 474396516, 777547433, 2586342311, 3016782802
Offset: 0

Views

Author

Jonathan Vos Post, Mar 09 2013

Keywords

Comments

This is to A004170 as tribonacci numbers A000073 are to Fibonacci numbers A000045. Note that tribonacci(20) = 35890 is, upon reversal, 09853, then the leading 0 is truncated, making the 5-digit number into the 4-digit number 9853. Similarly, R(4700770) = 770074.

Examples

			A000073(n) for n = 7, 8, 9 = 13, 24, 44, 81. Reversed, those are 31, 42, 44, 18, and sorted that is 18, 31, 42, 44.

Links

Crossrefs

Cf. A000045, A000073, A004086, A004170, A004091, A140463, A214855, A004086.

Programs

  • Mathematica
    Sort[FromDigits[Reverse[IntegerDigits[#]]]&/@LinearRecurrence[{1,1,1},{0,0,1,1},40]] (* Harvey P. Dale, Nov 22 2015 *)

A269500 a(n) = Fibonacci(10*n).

Original entry on oeis.org

0, 55, 6765, 832040, 102334155, 12586269025, 1548008755920, 190392490709135, 23416728348467685, 2880067194370816120, 354224848179261915075, 43566776258854844738105, 5358359254990966640871840, 659034621587630041982498215, 81055900096023504197206408605
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 03 2016

Keywords

Comments

More generally, the ordinary generating function for the Fibonacci(k*n) is F(k)*x/(1 - L(k)*x + (-1)^k*x^2), where F(k) is the k-th Fibonacci number (A000045), L(k) is the k-th Lucas number (A000032), or (phi^k - (-1/phi)^k)*x/(sqrt(5)*(1 - (phi^k + (-1/phi)^k)*x + (-1)^k*x^2)), where phi is the golden ratio (A001622).

Links

Crossrefs

Cf. similar sequences of the form Fibonacci(k*n): A000045 (k = 1), A001906 (k = 2), A014445 (k = 3), A033888 (k = 4), A102312 (k = 5), A134492 (k = 6), A134498 (k = 7), A138473 (k = 8), A138590 (k = 9), this sequence (k = 10), A167398 (k = 11), A214855 (k = 15).
Cf. A000032 (Lucas numbers), A001622 (golden ratio).

Programs

  • Mathematica
    Fibonacci[10Range[0, 14]]
FullSimplify[Table[(((1 + Sqrt[5])/2)^(10 n) - (2/(1 + Sqrt[5]))^(10 n))/Sqrt[5], {n, 0, 12}]]
LinearRecurrence[{123, -1}, {0, 55}, 15]
  • PARI
    a(n) = fibonacci(10*n); \\ Michel Marcus, Mar 03 2016
  • PARI
    concat(0, Vec(55*x/(1-123*x+x^2) + O(x^100))) \\ Altug Alkan, Mar 04 2016

Formula

G.f.: 55*x/(1 - 123*x + x^2).
a(n) = 123*a(n-1) - a(n-2).
a(n) = A000045(10*n).
Lim_{n -> infinity} a(n + 1)/a(n) = phi^10 = 122.9918693812442…
Showing 1-4 of 4 results.

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

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