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### -- Authors: Tobias Boege
### -- File: t/data/Fibonacci-Search-all.txt
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### -- Chunk 13 of 15
595,959,1546,2492,4017,6475,
%T A225396 10438,16826,27123,43722,70479,113611,183139,295217,475885,767119,
%U A225396 1236583,1993351,3213249,5179704,8349597,13459412,21696349,34974155,56377758,90880011
%N A225396 Expansion of 1/(1 - x - x^2 + x^10 - x^12).
%C A225396 Limiting ratio is 1.61198..., the largest real root of -1 + x^2 - x^10 - x^11 + x^12 = 0.
%H A225396 Vincenzo Librandi, <a href="/A225396/b225396.txt">Table of n, a(n) for n = 0..1000</a>
%H A225396 Roger L. Bagula, <a href="/A225396/a225396.txt">Demo file</a>
%H A225396 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (1, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1).
%F A225396 a(0)=1, a(1)=1, a(2)=2, a(3)=3, a(4)=5, a(5)=8, a(6)=13, a(7)=21, a(8)=34, a(9)=55, a(10)=88, a(11)=142, a(n)=a(n-1)+a(n-2)- a(n-10)+ a(n-12). - _Harvey P. Dale_, Apr 12 2014
%t A225396 CoefficientList[Series[1/(1 - x - x^2 + x^10 - x^12), {x, 0, 50}], x]
%t A225396 LinearRecurrence[{1,1,0,0,0,0,0,0,0,-1,0,1},{1,1,2,3,5,8,13,21,34,55,88,142},50] (* _Harvey P. Dale_, Apr 12 2014 *)
%Y A225396 Cf. A117791, A107293, A204631, A225393, A225394, A029826, A147660.
%K A225396 nonn,easy
%O A225396 0,3
%A A225396 _Roger L. Bagula_, May 06 2013

%I A236212
%S A236212 0,0,0,0,0,0,1,1,2,3,5,8,13,21,36,63,113,206,386,736,1433,2849,5773,
%T A236212 11919,25059,53613,116658,258032,579856,1323273,3065246,7204159,
%U A236212 17172291,41498712,101635485,252180415,633710357,1612310803,4151993262,10819115820
%N A236212 Floor of the value of Riemann's xi function at n.
%C A236212 On the interval [1, infinity), the xi function takes real values and is (strictly) increasing, so a(n) <= a(n+1) for n >= 1.
%C A236212 Same as floor of the value of the xi function at 1-n, because of the functional equation xi(1-s) = x(s).
%H A236212 J. Sondow and C. Dumitrescu, <a href="http://arxiv.org/abs/1005.1104">A monotonicity property of Riemann's xi function and a reformulation of the Riemann Hypothesis</a>, Period. Math. Hungar. 60 (2010), 37-40.
%H A236212 E. Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/Xi-Function.html">Xi Function</a>
%H A236212 Wikipedia, <a href="https://en.wikipedia.org/wiki/Riemann_Xi_function">Riemann Xi function</a>
%H A236212 <a href="/index/Z#zeta_function">Index entries for zeta function</a>
%F A236212 a(n) = [xi(n)] for n > 0.
%e A236212 xi(1) = 1/2, so a(1) = [0.5] = 0.
%e A236212 xi(8) = (4*Pi^4)/225 = 1.7317…, so a(8) = [1.7] = 1.
%t A236212 xi[ s_] := If[ s == 1, 1/2, (s/2)*(s - 1)*Pi^(-s/2)*Gamma[ s/2]*Zeta[ s]]; Table[ Floor[ xi[ n]], {n, 40}]
%Y A236212 Cf. A002410.
%K A236212 nonn
%O A236212 1,9
%A A236212 _Jonathan Sondow_, Jan 25 2014

%I A248740
%S A248740 0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,597,584,181,765,946,
%T A248740 711,657,368,25,393,418,811,229,40,269,309,578,887,465,352,817,169,
%U A248740 986,155,141,296,437,733,170,903,73,976,49,25,74,99,173,272
%N A248740 a(n) = Fibonacci(n) mod 1000.
%C A248740 The sequence is periodic with period 1500 = A001175(1000).
%C A248740 A number m of {0, 1, ..., 999} is not in the range of this sequence, iff m is congruent to 4 or 6 mod 8.
%C A248740 These numbers are the 250 = 1000 - A066853(1000) numbers of the set {4, 6, 12, 14, ..., 996, 998}. E.g., a Fibonacci number will never end in the digits '100'.
%H A248740 Alois P. Heinz, <a href="/A248740/b248740.txt">Table of n, a(n) for n = 0..3000</a>
%H A248740 <a href="/index/Rec#order_1500">Index entries for linear recurrences with constant coefficients</a>, order 1500.
%F A248740 a(n) = (a(n-1) + a(n-2)) mod 1000 for n>1, a(0) = 0, a(1) = 1.
%e A248740 a(17) = (a(16) + a(15)) mod 1000 = (987 + 610) mod 1000 = 1597 mod 1000 = 597.
%p A248740 a:= proc(n) option remember;
%p A248740       `if`(n<2, n, irem(a(n-1)+a(n-2), 1000))
%p A248740     end:
%p A248740 seq(a(n), n=0..60);  # _Alois P. Heinz_, Oct 18 2015
%o A248740 (MAGMA) [Fibonacci(n) mod 1000: n in [0..80]]; // _Vincenzo Librandi_, Oct 17 2014
%o A248740 (PARI) vector(100,n,fibonacci(n-1)%1000) \\ _Derek Orr_, Oct 17 2014
%Y A248740 Cf. A000045, A003893, A105471.
%K A248740 nonn
%O A248740 0,4
%A A248740 _Franz Vrabec_, Oct 13 2014
%E A248740 More terms from _Vincenzo Librandi_, Oct 17 2014

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

Search: seq:1,1,2,3,5,8,13,21
Showing 71-80 of 92

%I A274162
%S A274162 1,1,1,2,3,5,8,13,21,34,55,89,144,234,379,615,997,1617,2622,4252,6895,
%T A274162 11181
%N A274162 Number of real integers in n-th generation of tree T(3i) defined in Comments.
%C A274162 Let T* be the infinite tree with root 0 generated by these rules: if p is in T*, then p+1 is in T* and x*p is in T*. Let g(n) be the set of nodes in the n-th generation, so that g(0) = {0}, g(1) = {1}, g(2) = {2,x}, g(3) = {3,2x,x+1,x^2}, etc. Let T(r) be the tree obtained by substituting r for x.
%C A274162 See A274142 for a guide to related sequences.
%e A274162 If r = 3i, then g(3) = {3,2r,r+1, r^2}, in which the number of real integers is a(3) = 2.
%t A274162 z = 22; t = Join[{{0}}, Expand[NestList[DeleteDuplicates[Flatten[Map[{# + 1, x*#} &, #], 1]] &, {1}, z]]];
%t A274162 u = Table[t[[k]] /. x -> 3 I, {k, 1, z}]; Table[Count[Map[IntegerQ, u[[k]]], True], {k, 1, z}]
%Y A274162 Cf. A274142.
%K A274162 nonn,more
%O A274162 0,4
%A A274162 _Clark Kimberling_, Jun 12 2016
%E A274162 a(19)-a(21) from _Ryan Hitchman_, Sep 13 2017

%I A274163
%S A274163 1,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6766,
%T A274163 10948,17716,28667,46388,75063
%N A274163 Number of real integers in n-th generation of tree T(4i) defined in Comments.
%C A274163 Let T* be the infinite tree with root 0 generated by these rules:  if p is in T*, then p+1 is in T* and x*p is in T*.  Let g(n) be the set of nodes in the n-th generation, so that g(0) = {0}, g(1) = {1}, g(2) = {2,x}, g(3) = {3,2x,x+1,x^2}, etc.  Let T(r) be the tree obtained by substituting r for x.
%C A274163 For each integer k > 0, let s(k,n) be the number of integers in the n-th generation of T(k*i).  Conjecture:  there is a limiting sequence S(n) as k increases, and S(n) = F(n) for n >= 1, where F = A000045 (Fibonacci numbers).
%C A274163 From _Charlie Neder_, Jul 11 2018: (Start)
%C A274163 Assume for the moment that a complex number cannot be transformed back into an integer. If this is the case, then the real integers in g(n) are the real integers in g(n-1) plus 1 and the imaginary integers in g(n-1) times k*i, which are themselves k*i times the real integers in g(n-2), and so S(n) = S(n-1) + S(n-2) and S(n) = F(n).
%C A274163 However, the above assumption is false, but the earliest time such a transformation can take place is at g(k^2+5), following this path: 0 -> 1 -> k*i -> 1+k*i -> -k^2+k*i -> -(k^2-1)+k*i -> ... -> k*i -> -k^2.
%C A274163 Therefore s(k,n) matches the Fibonacci sequence for n < k^2+5 and S(n) = F(n). (End)
%C A274163 a(n) = A000045(n) only for 0 < n < 21. - _Robert G. Wilson v_, Jul 23 2018
%e A274163 If r = 4i, then g(3) = {3,2r,r+1, r^2}, in which the number of real integers is a(3) = 2.
%t A274163 z = 18; t = Join[{{0}}, Expand[NestList[DeleteDuplicates[Flatten[Map[{# + 1, x*#} &, #], 1]] &, {1}, z]]];
%t A274163 u = Table[t[[k]] /. x -> 4 I, {k, 1, z}]; Table[Count[Map[IntegerQ, u[[k]]], True], {k, 1, z}]
%Y A274163 See A274142 for a guide to related sequences.
%K A274163 nonn,more
%O A274163 0,4
%A A274163 _Clark Kimberling_, Jun 12 2016
%E A274163 a(21)-a(25) from _Robert G. Wilson v_, Jul 23 2018

%I A024466
%S A024466 1,0,0,1,1,2,3,0,0,1,1,2,3,5,8,13,21,34,55,1,1,2,3,5,8,13,21,34,55,89,
%T A024466 144,233,377,610,988,1598,2586,4184,6770,8,13,21,34,55,89,144,233,377,
%U A024466 610,987,1597,2584,4181,6765,10946,17712,28658,46370,75028,121398,196426
%N A024466 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)t(n+1-k), where k = [ (n+1)/2 ], s = (Fibonacci numbers), t = A023533.
%K A024466 nonn
%O A024466 1,6
%A A024466 _Clark Kimberling_

%I A025086
%S A025086 0,0,1,1,2,0,0,0,1,1,2,3,5,8,13,21,34,0,1,1,2,3,5,8,13,21,34,55,89,
%T A025086 144,233,377,610,988,1598,2586,4184,5,8,13,21,34,55,89,144,233,377,
%U A025086 610,987,1597,2584,4181,6765,10946,17712,28658,46370,75028,121398,196426,317824,514250
%N A025086 a(n) = s(1)t(n) + s(2)t(n-1) + ... + s(k)t(n-k+1), where k = [ n/2 ], s = (Fibonacci numbers), t = A023533.
%K A025086 nonn
%O A025086 1,5
%A A025086 _Clark Kimberling_

%I A067517
%S A067517 0,1,1,2,3,5,8,13,21,55,233,102334155
%N A067517 Fibonacci numbers which can be partitioned into group of digits which are Fibonacci numbers.
%K A067517 base,nonn
%O A067517 0,4
%A A067517 _Amarnath Murthy_, Feb 14 2002
%E A067517 102334155 from Larry Reeves (larryr(AT)acm.org), Nov 17 2003. Next term, if it exists, is > 2*10^18.

%I A069041
%S A069041 1,1,2,3,5,8,13,21,34,55,89,144,233,377,17711
%N A069041 Fibonacci numbers with at most two distinct digits.
%C A069041 Next term, if it exists, is > Fibonacci(2289000). - Lars Blomberg, May 10 2011
%t A069041 Fibonacci[ Select[Range[5*10^3], Length[Union[IntegerDigits[Fibonacci[ # ]], {}]] <= 2 &]]
%t A069041 Select[Fibonacci[Range[100]],Count[DigitCount[#],0]>7&] (* _Harvey P. Dale_, May 27 2018 *)
%K A069041 nonn,base
%O A069041 1,3
%A A069041 _Joseph L. Pe_, Apr 03 2002

%I A080787
%S A080787 1,1,2,3,5,8,13,21,24,25,29,34,43,47,50,57,57,64,71,75,76,81,87,88,95,
%T A080787 103,108,111,119,120,129,129,138,147,155,162,167,169,176,185,191,196,
%U A080787 197,203,210,213,213,216,219,225,234,239,243,252,255,257,262,269,271
%N A080787 a(1)=a(2)=1; a(n) = a(n-1) + last decimal digit of a(n-2).
%F A080787 a(n)=a(n-1)+a(n-2)(mod 10); for n>=3 a(n)-a(n-1)=A003893(n-2)=A000045(n-2)(mod 10)
%K A080787 base,nonn
%O A080787 1,3
%A A080787 _Benoit Cloitre_, Mar 12 2003

%I A093093
%S A093093 1,1,2,3,5,8,13,21,3,4,2,4,7,6,6,11,13,1,2,17,2,4,1,4,3,19,19,6,5,5,7,
%T A093093 2,2,3,8,25,11,1,0,1,2,9,4,5,11,33,3,6,1,2,1,1,3,11,13,9,1,6,4,4,3,6,
%U A093093 9,7,3,3,2,4,1,4,2,4,2,2,1,0,7,1,0,8,7,9,15,1,6,1,0,6,5,6,5,5,6,6,6,4,3,1,7
%N A093093 "Fibonacci in digits - up and down": start with a(1)=1, a(2)=1; repeatedly adjoin either the sum of the two previous terms (if that sum happens to be odd) or else adjoin digits of the sum of previous two terms (if that sum happens to be even).
%e A093093 ... a(8)=a(6)+a(7), a(9)=left digit of (a(7)+a(8)=13+21=3 4) as 34 is even, a(10)=right digit of (a(7)+a(8)=13+21=3 4) as 34 is even, a(13)=a(9)+a(10) as odd, ...
%Y A093093 Cf. A093086 to A093092.
%K A093093 nonn,base
%O A093093 1,3
%A A093093 _Bodo Zinser_, Mar 20 2004

%I A109609
%S A109609 1,1,2,3,5,8,13,21,34,55,89,144,234,378,612,990,1602,2592,4194,6786,
%T A109609 10980,17766,28746,46512,75259,121771,197030,318801,515831,834632,
%U A109609 1350463,2185095,3535558,5720653,9256211,14976864,24233076,39209940
%N A109609 Expansion of 1/((x-1)*(x+1)*(x^2+x+1)*(x^2+x-1)*(x^2-x+1)*(x^2+1)*(x^4-x^2+1)).
%C A109609 FAMP Code for s batch of sequences satisfying the recurrence relation as (a(n)): A*B with A = - .25'i - .25i' - .25'ii' + .25'jj' + .25'kk' + .25'jk' + .25'kj' - .25e, B = + 'i + i' + 'ji' + 'ki' + e. Sumtype is set to: sum[Y[15]]
%H A109609 <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, -1).
%F A109609 a(0)=1, a(1)=1, a(2)=2, a(3)=3, a(4)=5, a(5)=8, a(6)=13, a(7)=21, a(8)=34, a(9)=55, a(10)=89, a(11)=144, a(12)=234, a(13)=378, a(n)=a(n-1)+ a(n-2)+ a(n-12)-a(n-13)-a(n-14). - _Harvey P. Dale_, Sep 20 2013
%p A109609 seriestolist(series(1/((x-1)*(x+1)*(x^2+x+1)*(x^2+x-1)*(x^2-x+1)*(x^2+1)*(x^4-x^2+1)), x=0,40));
%t A109609 CoefficientList[Series[1/((x-1)(x+1)(x^2+x+1)(x^2+x-1)(x^2-x+1)(x^2+1)(x^4-x^2+1)),{x,0,40}],x] (* or *) LinearRecurrence[ {1,1,0,0,0,0,0,0,0,0,0,1,-1,-1},{1,1,2,3,5,8,13,21,34,55,89,144,234,378},40] (* _Harvey P. Dale_, Sep 20 2013 *)
%Y A109609 Cf. A000045.
%K A109609 nonn
%O A109609 0,3
%A A109609 _Creighton Dement_, Jul 31 2005

%I A117760
%S A117760 1,1,1,2,3,5,8,13,21,33,53,85,136,218,349,559,895,1433,2295,3675,5885,
%T A117760 9424,15091,24166,38698,61969,99234,158908,254467,407490,652533,
%U A117760 1044932,1673299,2679533,4290863,6871162,11003117,17619812,28215439,45182718
%N A117760 Expansion of 1/(1 - x - x^3 - x^5 - x^7).
%C A117760 Number of compositions of n into parts 1, 3, 5, and 7. - _David Neil McGrath_, Aug 18 2014
%H A117760 Milan Janjic, <a href="http://www.emis.ams.org/journals/JIS/VOL19/Janjic/janjic73.html">Binomial Coefficients and Enumeration of Restricted Words</a>, Journal of Integer Sequences, 2016, Vol 19, #16.7.3
%F A117760 a(n)=a(n-1)+a(n-3)+a(n-5)+a(n-7); a(0)=0, a(1)=1, a(2)=1, a(3)=1, a(4)=2, a(5)=3, a(6)=5.
%p A117760 a[0]:=0:a[1]:=1:a[2]:=1:a[3]:=1:a[4]:=2:a[5]:=3:a[6]:=5:for n from 7 to 45 do a[n]:=a[n-1]+a[n-3]+a[n-5]+a[n-7] od: seq(a[n],n=0..45);
%t A117760 CoefficientList[ Series[1/(1 - x - x^3 - x^5 - x^7), {x, 0, 40}], x]
%o A117760 (PARI) Vec( 1/(1-x-x^3-x^5-x^7)+O(x^66) ) \\ _Joerg Arndt_, Aug 19 2014
%K A117760 nonn
%O A117760 0,4
%A A117760 _Roger L. Bagula_, Apr 14 2006
%E A117760 Edited and extended by _N. J. A. Sloane_, Apr 20 2006

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

Search: seq:1,1,2,3,5,8,13,21
Showing 81-90 of 92

%I A118627
%S A118627 1,1,2,3,5,8,13,21,6,30,24,55,31,87,56,144,88,233,145,379,234,614,380,
%T A118627 995,615,1611,996,2608,1612,4221,