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### -- Licenses: Artistic-2.0
### -- Authors: Tobias Boege
### -- File: t/data/Fibonacci-Search-all.txt
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### -- Chunk 9 of 15
,25,28,53
%N A261606 a(n) = Fibonacci(n) mod 60.
%C A261606 a(n) = A261575(n,0).
%C A261606 Periodic with period length 120. - _Ray Chandler_, Sep 23 2015
%H A261606 Reinhard Zumkeller, <a href="/A261606/b261606.txt">Table of n, a(n) for n = 0..10000</a>
%H A261606 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Sexagesimal.html">Sexagesimal</a>
%H A261606 Wikipedia, <a href="http://www.wikipedia.org/wiki/Sexagesimal">Sexagesimal</a>
%F A261606 a(0) = 0, a(1) = 1 and for n > 1: a(n) = (a(n-1) + a(n-2)) mod 60.
%o A261606 (Haskell)
%o A261606 a261606 n = a261606_list !! n
%o A261606 a261606_list = 0 : 1 : map (flip mod 60)
%o A261606                            (zipWith (+) a261606_list $ tail a261606_list)
%Y A261606 Cf. A000045, A261575, A261607, A003893.
%K A261606 nonn,base
%O A261606 0,4
%A A261606 _Reinhard Zumkeller_, Sep 09 2015

%I A261607
%S A261607 0,1,1,2,3,5,8,13,21,34,55,1,2,3,6,10,16,26,43,1,1,3,4,7,12,20,33,54,
%T A261607 1,2,3,6,10,16,26,42,1,1,3,4,7,12,20,33,54,1,2,3,6,10,16,26,42,1,1,2,
%U A261607 4,7,12,20,33,53,1,2,3,6,9,16,25,42,1,1,2,4,7,12
%N A261607 Initial digit of Fibonacci number F(n) in base 60.
%C A261607 a(n) = A261575(n,A261585(n)-1).
%H A261607 Reinhard Zumkeller, <a href="/A261607/b261607.txt">Table of n, a(n) for n = 0..1000</a>
%H A261607 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Sexagesimal.html">Sexagesimal</a>
%H A261607 Wikipedia, <a href="http://www.wikipedia.org/wiki/Sexagesimal">Sexagesimal</a>
%o A261607 (Haskell)
%o A261607 a261607 = last . a261575_row
%Y A261607 Cf. A000045, A261575, A261585, A261606, A008963.
%K A261607 nonn,base
%O A261607 0,4
%A A261607 _Reinhard Zumkeller_, Sep 09 2015

%I A023438
%S A023438 0,1,1,2,3,5,8,13,21,33,53,84,134,213,339,539,857,1363,2167,3446,5479,
%T A023438 8712,13852,22025,35020,55682,88535,140771,223827,355886,565861,
%U A023438 899722,1430563,2274603,3616631,5750463
%N A023438 Dying rabbits: a(n) = a(n-1) + a(n-2) - a(n-8).
%H A023438 J. H. E. Cohn, <a href="http://www.fq.math.ca/Scanned/2-2/cohn1.pdf">Letter to the editor</a>, Fib. Quart. 2 (1964), 108.
%H A023438 V. E. Hoggatt, Jr. and D. A. Lind, <a href="http://www.fq.math.ca/Scanned/7-5/hoggatt.pdf">The dying rabbit problem</a>, Fib. Quart. 7 (1969), 482-487.
%H A023438 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,0,0,0,0,-1).
%F A023438 G.f.: x / ( (x-1)*(x^7+x^6+x^5+x^4+x^3+x^2-1) ). - _R. J. Mathar_, Nov 29 2011
%o A023438 (PARI) concat(0, Vec(x / ( (x-1)*(x^7+x^6+x^5+x^4+x^3+x^2-1) ) + O(x^60))) \\ _Michel Marcus_, Sep 06 2017
%Y A023438 See A000045 for the Fibonacci numbers.
%K A023438 nonn
%O A023438 0,4
%A A023438 _N. J. A. Sloane_.

%I A077371
%S A077371 0,1,1,2,3,5,8,13,21,34,55,89,233,610,987
%N A077371 Fibonacci numbers whose internal digits form a Fibonacci number. Equivalently, Fibonacci numbers from which deleting the MSD and LSD leaves a Fibonacci number.
%C A077371 Conjecture: The sequence is finite.
%C A077371 No more terms < 10^6. - _Lars Blomberg_, May 20 2015
%C A077371 From _Manfred Scheucher_, Jun 02 2015 (Start)
%C A077371 No more terms < 10^10000.
%C A077371 When considering binary representations, the sequence would be 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 144, and no further terms < 2^150000 (about 10^44095).
%C A077371 When considering k-ary representations with k=2..100, each of the sequences has some small terms in the beginning (as in the 10-ary case) and no further terms <10^1000.
%C A077371 The sequence seems to be finite for any base, not just for base 10.
%C A077371 Another observation: When considering k-ary representations with k=55,144,377,... (Fibonacci numbers with even index, A001906), the number of "initial terms" (terms <10^1000) increases very fast.
%C A077371 (End)
%H A077371 Manfred Scheucher, <a href="/A077371/a077371.sage.txt">Sage Script</a>
%Y A077371 Cf. A077372, A077373, A077374, A077375.
%K A077371 base,more,nonn
%O A077371 1,4
%A A077371 _Amarnath Murthy_, Nov 06 2002

%I A077372
%S A077372 0,1,1,2,3,5,8,13,21,34,55,89,121393,1836311903,2504730781961,
%T A077372 10610209857723,10284720757613717413913,184551825793033096366333,
%U A077372 59425114757512643212875125,155576970220531065681649693
%N A077372 Fibonacci numbers whose external digits form a Fibonacci number. Or Fibonacci numbers whose MSD and LSD form a Fibonacci number.
%C A077372 Sequence can be easily generated up to >10000 terms with the appended Sage script. - _Manfred Scheucher_, Jun 02 2015
%H A077372 Manfred Scheucher, <a href="/A077372/b077372.txt">Table of n, a(n) for n = 1..100</a>
%H A077372 Manfred Scheucher, <a href="/A077372/a077372.sage.txt">Sage Script</a>
%o A077372 (Sage) F=[fibonacci(i) for i in [0..200]]
%o A077372 [x for x in F if x<10 or (x>10 and (10*x.digits()[-1]+x.digits()[0]) in F)]
%o A077372 # _Tom Edgar_, Jun 03 2015
%Y A077372 Cf. A077371, A077373.
%K A077372 base,nonn
%O A077372 1,4
%A A077372 _Amarnath Murthy_, Nov 06 2002
%E A077372 More terms from _Lior Manor_, Nov 06 2002

%I A093089
%S A093089 1,1,2,3,5,8,13,21,34,55,89,1,44,90,45,1,34,1,35,46,35,35,36,81,81,70,
%T A093089 71,1,17,1,62,1,51,1,41,72,18,18,63,63,52,52,42,1,13,90,36,81,1,26,1,
%U A093089 15,1,4,94,43,14,1,3,1,26,1,17,82,27,27,16,16,5,98,1,37,57,15,4,4,27,27,18
%N A093089 "Fibonacci in pairs": start with a(1)=1, a(2)=1; repeatedly adjoin sum of previous two terms but chopped from the right into pairs of 2 digits.
%C A093089 Do all pairs of digits appear infinitely often? The sequence is not periodic.
%e A093089 ... a(11)=a(9)+a(10), a(12)=left pair of (a(10)+a(11)=55+89=1 44), a(13)=right pair of (a(10)+a(11)=55+89=1 44), a(14)=a(11)+a(12) ...
%Y A093089 Cf. A093086, A093087, A093088.
%K A093089 nonn,base
%O A093089 1,3
%A A093089 _Bodo Zinser_, Mar 20 2004

%I A132916
%S A132916 0,1,1,1,1,1,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,
%T A132916 4181,6765,10946,21892,39603,72441,133936,245980,452357,832273,
%U A132916 1530610,2815240,5178123,9523973,17517336,32219432,59260741,108997509,200477682
%N A132916 a(0)=0; a(1)=1; a(n) = Sum_{k=1..floor(n^(1/3))} a(n-k) for n >= 2.
%C A132916 Lim_{n->infinity} a(n+1)/a(n) = 2. Contrast with Fibonacci sequence. Also a(n+1)/a(n) = 2 iff n+1 >= 8 is a cube.
%C A132916 Up to a(26) = 10946, but not beyond, the sequence consists of the Fibonacci numbers A000045(0..21). - _M. F. Hasler_, May 10 2017
%H A132916 Robert Israel, <a href="/A132916/b132916.txt">Table of n, a(n) for n = 0..3345</a>
%F A132916 a(n) = Sum_{k=1..floor(n^(1/3))} a(n-k) for n >= 2; a(0)=0; a(1)=1.
%e A132916 a(27) = a(24) + a(25) + a(26) = 4181 + 6765 + 10946 = 21892.
%p A132916 f:= proc(n) option remember;
%p A132916 add(procname(n-k),k=1..floor(n^(1/3)))
%p A132916 end proc:
%p A132916 f(0):= 0: f(1):= 1:
%p A132916 map(f, [$0..50]); # _Robert Israel_, Dec 16 2018
%t A132916 a[n_] := a[n] = If[n < 2, n, Sum[a[n - k], {k, Floor[n^(1/3)]}]]; Array[a, 43, 0] (* _Michael De Vlieger_, May 10 2017 *)
%Y A132916 Cf. A000045, A132915.
%K A132916 nonn
%O A132916 0,9
%A A132916 _Rick L. Shepherd_, Sep 04 2007
%E A132916 Incorrect g.f. and programs deleted by _Colin Barker_, Dec 17 2018

%I A141169
%S A141169 0,0,1,0,1,1,0,1,1,2,0,1,1,2,3,0,1,1,2,3,5,0,1,1,2,3,5,8,0,1,1,2,3,5,
%T A141169 8,13,0,1,1,2,3,5,8,13,21,0,1,1,2,3,5,8,13,21,34,0,1,1,2,3,5,8,13,21,
%U A141169 34,55,0,1,1,2,3,5,8,13,21,34,55,89,0,1,1,2,3,5,8,13,21,34,55,89,144,0,1,1,2,3,5,8,13,21
%N A141169 Triangle of Fibonacci numbers, read by rows: T(n,k) = A000045(k), 0<=k<=n.
%C A141169 T(n,0) = A000004(n); T(n,n) = A000045(n);
%C A141169 central terms: T(2*n,n) = A000045(n);
%C A141169 sums of rows: Sum(T(n,k): 0<=k<=n) = A000071(n+2);
%C A141169 alternating sums of rows: Sum(T(n,k)*(-1)^k: 0<=k<=n) = A119282(n);
%C A141169 T(n,k) + T(n,n-k) = A094570(n,k).
%H A141169 Reinhard Zumkeller, <a href="/A141169/b141169.txt">Rows n=0..125 of triangle, flattened</a>
%o A141169 (Haskell)
%o A141169 import Data.List (inits)
%o A141169 a141169 n k = a141169_tabl !! n !! k
%o A141169 a141169_row n = a141169_tabl !! n
%o A141169 a141169_tabl = tail $ inits a000045_list
%o A141169 a141169_list = concat $ a141169_tabl
%o A141169 -- _Reinhard Zumkeller_, Aug 24 2015, Mar 21 2011
%Y A141169 Cf. A000045, A000004, A000071, A094570, A119282.
%K A141169 nonn,tabl
%O A141169 0,10
%A A141169 _Reinhard Zumkeller_, Mar 21 2011

%I A177247
%S A177247 1,1,2,3,5,8,13,21,34,55,89,144,233,377,987,1597,2584,4181,17711,
%T A177247 75025,121393,317811,514229,832040,2178309,3524578,5702887,14930352,
%U A177247 24157817,102334155,433494437,701408733,1134903170,2971215073,7778742049
%N A177247 Fibonacci numbers Fib(n) whose decimal expansion does not contain any digit 6.
%C A177247 Probability that Fib(n) contains no 6's goes to zero as n grows to infinity. I suppose the maximum number is F(258) having 54 digits with no 6's.
%e A177247 a(7)=13 since it is the 7th Fibonacci having no 6's
%Y A177247 Cf. A000045, A177194, A177195, A177231, A177245, A177246, A176253
%K A177247 nonn,base
%O A177247 1,3
%A A177247 _Carmine Suriano_, May 06 2010

# Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE

# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/

Search: seq:1,1,2,3,5,8,13,21
Showing 31-40 of 92

%I A229194
%S A229194 1,1,1,2,3,5,8,13,21,35,58,97,163,275,466,793,1353,2315,3969,6817,
%T A229194 11726,20195,34816,60073,103724,179195,309724,535537,926275,1602515,
%U A229194 2773034,4799353,8307516,14381675,24899377,43112257,74651790,129271235,223862687,387682633,671402698,1162785755,2013837368,3487832977,6040770648,10462450355,18120829034,31385253913,54359521280,94151567435,163072632198
%N A229194 Integers nearest to (2^((n-3)/2) + 3^((n-3)/2)).
%C A229194 This sequence illustrates the second law of small numbers because it is a coincidence that the terms for 1 <= n <= 8 are the same as the Fibonacci numbers F(n) (A000045): a(n) = F(n) for 1 <= n <= 8.
%C A229194 Furthermore, the following terms are the sum of two Fibonacci numbers: a(9) = F(9) + F(2), a(10) = F(10) + F(4), a(11) = F(11) + F(6), a(14) = F(14) + F(11); or the algebraic sum of three Fibonacci numbers: a(12) = F(12) + F(8) - F(3), a(13) = F(13) + F(10) - F(7), a(14) = F(14) + F(12) - F(10), a(18) = F(19) - F(13) - F(8), a(19) = F(20) + F(10) - F(4); or the algebraic sum of four Fibonacci numbers: a(15) = F(15) + F(12) + F(9) + F(5), a(16) = F(16) + F(14) - F(6) - F(4), a(17) = F(18) - F(13) - F(9) - F(3), a(18) = F(18) + F(16) + F(14) + F(8), a(19) = F(19) + F(18) + F(10) - F(3).
%C A229194 Note that, for following values of n, a(n) > F(n+1) for n >= 20.
%C A229194 Remark as well that (2^(1/2) + 3^(1/2)) = 3.14626437... ~=  Pi (see A135611).
%D A229194 T. Koshy, Fibonacci and Lucas Numbers with Applications, New York, Wiley-Interscience, 2001
%D A229194 I. Stewart, L'univers des nombres, Belin-Pour La Science, Paris 2000.
%H A229194 Vladimir Pletser, <a href="/A229194/b229194.txt">Table of n, a(n) for n = 0..1000</a>
%H A229194 R. K. Guy, <a href="http://www.jstor.org/stable/2691503">The Second Strong Law of Small Numbers</a>, Math. Mag, 63 (1990), no. 1, 3-20.
%H A229194 I. Stewart, <a href="http://www.whydomath.org/Reading_Room_Material/ian_stewart/9505.html">Fibonacci Forgeries</a>
%H A229194 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number.</a>
%H A229194 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StrongLawofSmallNumbers.html">Strong Law of Small Numbers.</a>
%F A229194 a(n) = round(2^((n-3)/2) + 3^((n-3)/2)).
%p A229194 seq(round(2^((n-3)/2)+3^((n-3)/2)), n=0..50);
%t A229194 Table[Round[2^((n - 3)/2) + 3^((n - 3)/2)], {n, 0, 50}] (* _Vincenzo Librandi_, Sep 20 2013 *)
%o A229194 (MAGMA) [Round(2^((n-3)/2) + 3^((n-3)/2)): n in [0..50]]; // _Vincenzo Librandi_, Sep 20 2013
%Y A229194 Cf. A000045, A005181.
%K A229194 nonn
%O A229194 0,4
%A A229194 _Vladimir Pletser_, Sep 15 2013

%I A232666
%S A232666 0,1,1,2,3,5,8,13,21,34,55,89,4,93,97,190,287,477,764,1241,2005,541,
%T A232666 2546,3087,5633,8720,14353,23073,37426,60499,97925,26404,124329,
%U A232666 150733,275062,425795,700857,1126652,1827509,2954161,796945,3751106,4548051,8299157,12847208,21146365,33993573
%N A232666 6-free Fibonacci numbers.
%C A232666 The sequences of n-free Fibonacci numbers were suggested by John H. Conway.
%C A232666 a(n) is the sum of the two previous terms divided by the largest possible power of 6.
%C A232666 4-free Fibonacci numbers are A224382.
%C A232666 The sequence coincides with the Fibonacci sequence until the first multiple of 6 in the Fibonacci sequence: 144, which in this sequence is divided by 36 to produce 4.
%C A232666 7-free Fibonacci numbers is A078414.
%H A232666 B. Avila, T. Khovanova, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Avila/avila4.html">Free Fibonacci Sequences</a>, J. Int. Seq. 17 (2014) # 14.8.5.
%t A232666 sixPower[n_] := (a = Transpose[FactorInteger[n]]; a2 = Position[a[[1]], 2]; a3 = Position[a[[1]], 3]; If[Length[a2] == 0 || Length[a3] == 0 , res = 0, res = Min[a[[2]][[a2[[1]][[1]]]], a[[2]][[a3[[1]][[1]]]]]]; res); sixFree[n_] := n/6^sixPower[n]; appendNext6Free[list_] := Append[list, sixFree[list[[-1]] + list[[-2]]]]; Nest[appendNext6Free, {0, 1}, 50]
%Y A232666 Cf. A224382, A214684.
%K A232666 nonn
%O A232666 0,4
%A A232666 _Brandon Avila_ and _Tanya Khovanova_, Nov 27 2013

%I A321021
%S A321021 0,1,1,2,3,5,8,13,21,34,0,34,34,68,102,170,7,1,8,9,17,26,43,69,2,71,
%T A321021 73,1,74,75,149,4,153,157,310,467,0,467,467,934,40,974,4,978,982,1960,
%U A321021 94,2054,2148,40,21,61,82,143,5,148,153,301,5,306,3
%N A321021 a(0)=0, a(1)=1; thereafter a(n) = a(n-2)+a(n-1), keeping just the digits that appear exactly once.
%C A321021 a(n