### ----------------------------------------------------
### -- OEIS
### -- Licenses: Artistic-2.0
### -- Authors: Tobias Boege
### -- File: t/data/Fibonacci-Search-all.txt
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### -- Chunk 12 of 15
%e A324969 The a(1) = 1 through a(7) = 8 trees:
%e A324969   o  (o)  ((o))  (o(o))   ((o(o)))   (o(o(o)))    ((o(o(o))))
%e A324969                  (((o)))  (o((o)))   (((o(o))))   (o((o(o))))
%e A324969                           ((((o))))  ((o((o))))   (o(o((o))))
%e A324969                                      (o(((o))))   ((((o(o)))))
%e A324969                                      (((((o)))))  (((o((o)))))
%e A324969                                                   ((o(((o)))))
%e A324969                                                   (o((((o)))))
%e A324969                                                   ((((((o))))))
%t A324969 durtid[n_]:=Join@@Table[Select[Union[Sort/@Tuples[durtid/@ptn]],UnsameQ@@#&&UnsameQ@@Cases[#,{__},{0,Infinity}]&],{ptn,IntegerPartitions[n-1]}];
%t A324969 Table[Length[durtid[n]],{n,15}]
%Y A324969 The Matula-Goebel numbers of these trees are given by A324968.
%Y A324969 Cf. A000081, A004111, A276625, A290689, A317713, A324923, A324936, A324971, A324978.
%K A324969 nonn,more
%O A324969 1,4
%A A324969 _Gus Wiseman_, Mar 21 2019

%I A005170 M0694
%S A005170 1,0,1,1,2,3,5,8,13,21,35,55,93,149,248,403,670,1082
%N A005170 Erroneous version of A226999.
%D A005170 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005170 R. K. Guy, <a href="/A005169/a005169_6.pdf">Letter to N. J. A. Sloane</a>, Sep 25 1986.
%H A005170 R. K. Guy, <a href="/A005728/a005728.pdf">Letter to N. J. A. Sloane, 1987</a>
%H A005170 R. K. Guy, <a href="http://www.jstor.org/stable/2322249">The strong law of small numbers</a>. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
%H A005170 R. K. Guy, <a href="/A005165/a005165.pdf">The strong law of small numbers</a>. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
%K A005170 dead
%O A005170 1,5

%I A013986
%S A013986 1,0,1,1,2,3,5,8,13,21,33,54,86,139,223,359,577,928,1492,2399,3858,
%T A013986 6203,9975,16039,25791,41471,66685,107228,172421,277250,445813,716860,
%U A013986 1152698,1853519,2980426,4792474,7706215
%N A013986 Expansion of 1/(1-x^2-x^3-x^4-x^5-x^6-x^7-x^8-x^9).
%C A013986 Number of compositions of n into parts p where 2 <= p < = 9. [_Joerg Arndt_, Jun 24 2013]
%H A013986 Vincenzo Librandi, <a href="/A013986/b013986.txt">Table of n, a(n) for n = 0..1000</a>
%H A013986 R. Mullen, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Mullen/mullen2.html">On Determining Paint by Numbers Puzzles with Nonunique Solutions</a>, JIS 12 (2009) 09.6.5
%H A013986 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (0, 1, 1, 1, 1, 1, 1, 1, 1).
%F A013986 a(0)=1, a(1)=0, a(2)=1, a(3)=1, a(4)=2, a(5)=3, a(6)=5, a(7)=8, a(8)=13, a(n)=a(n-2)+a(n-3)+a(n-4)+a(n-5)+a(n-6)+a(n-7)+a(n-8)+a(n-9). - _Harvey P. Dale_, Dec 17 2013
%t A013986 CoefficientList[Series[1 / (1 - x^2 - x^3 - x^4 - x^5 - x^6 - x^7 - x^8 - x^9), {x, 0, 40}], x] (* _Vincenzo Librandi_, Jun 24 2013 *)
%t A013986 CoefficientList[Series[1/(1-Total[x^Range[2,9]]),{x,0,40}],x] (* or *) LinearRecurrence[{0,1,1,1,1,1,1,1,1},{1,0,1,1,2,3,5,8,13},40] (* _Harvey P. Dale_, Dec 17 2013 *)
%o A013986 (MAGMA) m:=40; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-x^2-x^3-x^4-x^5-x^6-x^7-x^8-x^9))); // _Vincenzo Librandi_, Jun 24 2013
%Y A013986 See A000045 for the Fibonacci numbers.
%K A013986 nonn,easy
%O A013986 0,5
%A A013986 _N. J. A. Sloane_.

%I A013987
%S A013987 1,0,1,1,2,3,5,8,13,21,34,54,88,141,228,367,592,954,1538,2479,3996,
%T A013987 6441,10383,16736,26978,43486,70097,112991,182134,293587,473242,
%U A013987 762833,1229634,1982084,3194982,5150088,8301584,13381575,21570168,34769609,56046190
%N A013987 Expansion of 1/(1-x^2-x^3-x^4-x^5-x^6-x^7-x^8-x^9-x^10).
%C A013987 Number of compositions of n into parts p where 2 <= p < = 10. [_Joerg Arndt_, Jun 24 2013]
%H A013987 Vincenzo Librandi, <a href="/A013987/b013987.txt">Table of n, a(n) for n = 0..1000</a>
%H A013987 R. Mullen, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Mullen/mullen2.html">On Determining Paint by Numbers Puzzles with Nonunique Solutions</a>, JIS 12 (2009) 09.6.5
%t A013987 CoefficientList[Series[1 / (1 - x^2 - x^3 - x^4 - x^5 - x^6 - x^7 - x^8 - x^9 - x^10), {x, 0, 40}], x] (* _Vincenzo Librandi_, Jun 24 2013 *)
%o A013987 (MAGMA) m:=40; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-x^2-x^3-x^4-x^5-x^6-x^7-x^8-x^9-x^10))); // _Vincenzo Librandi_, Jun 24 2013
%Y A013987 See A000045 for the Fibonacci numbers.
%K A013987 nonn,easy
%O A013987 0,5
%A A013987 _N. J. A. Sloane_.

%I A023441
%S A023441 0,1,1,2,3,5,8,13,21,34,55,89,143,231,372,600,967,1559,2513,4051,6530,
%T A023441 10526,16967,27350,44086,71064,114550,184647,297638,479772,773359,
%U A023441 1246601,2009434,3239068,5221152,8416134
%N A023441 Dying rabbits: a(n) = a(n-1) + a(n-2) - a(n-11).
%H A023441 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 A023441 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.
%F A023441 G.f.: x/((x-1)*(1+x)*(x^9+x^7+x^5+x^3+x-1)). [_R. J. Mathar_, Jul 27 2009]
%o A023441 (PARI) concat(0, Vec(x/((x-1)*(1+x)*(x^9+x^7+x^5+x^3+x-1)) + O(x^50))) \\ _Michel Marcus_, Sep 06 2017
%Y A023441 See A000045 for the Fibonacci numbers.
%K A023441 nonn
%O A023441 0,4
%A A023441 _N. J. A. Sloane_.

%I A023442
%S A023442 0,1,1,2,3,5,8,13,21,34,55,89,144,232,375,605,977,1577,2546,4110,6635,
%T A023442 10711,17291,27913,45060,72741,117426,189562,306011,493996,797461,
%U A023442 1287347,2078173,3354809,5415691,8742587
%N A023442 Dying rabbits: a(n) = a(n-1) + a(n-2) - a(n-12).
%H A023442 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 A023442 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.
%F A023442 G.f.: x/ ( (x-1)*(x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2-1) ). - _R. J. Mathar_, Nov 29 2011
%o A023442 (PARI) concat(0, Vec(x/ ( (x-1)*(x^11+x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2-1) ) + O(x^50))) \\ _Michel Marcus_, Sep 06 2017
%Y A023442 See A000045 for the Fibonacci numbers.
%K A023442 nonn
%O A023442 0,4
%A A023442 _N. J. A. Sloane_.

%I A053412
%S A053412 1,1,2,3,5,8,13,21,17711,317811
%N A053412 n-th nonzero Fibonacci numbers arising in A053408.
%Y A053412 See A000045 for the Fibonacci numbers.
%K A053412 nonn
%O A053412 0,3
%A A053412 _G. L. Honaker, Jr._, Jan 09 2000

# 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 61-70 of 92

%I A055806
%S A055806 1,1,1,2,3,5,8,13,21,33,53,79,125,176,273,365,554,709,1053,1300,1891,
%T A055806 2267,3234,3785,5303,6085,8385,9465,12845,14302,19139,21065,27828,
%U A055806 30329,39593,42790,55251,59281,75772
%N A055806 T(n,n-6), array T as in A055801.
%K A055806 nonn
%O A055806 6,4
%A A055806 _Clark Kimberling_, May 28 2000

%I A117502
%S A117502 1,1,2,1,1,3,1,1,2,5,1,1,2,3,8,1,1,2,3,5,13,1,1,2,3,5,8,21,1,1,2,3,5,
%T A117502 8,13,34,1,1,2,3,5,8,13,21,55,1,1,2,3,5,8,13,21,34,89
%N A117502 Triangle, row sums = A001595.
%C A117502 Row sums = A001595 starting (1, 3, 9, 15, 25, 41...)
%F A117502 n-th row = first n Fibonacci terms, with a deletion of F(n). Columns of the triangle are difference terms of the array in A117501.
%e A117502 Row 5 of the triangle = (1, 1, 2, 3, 8); the first 5 Fibonacci terms with a deletion of F(5) = 5.
%e A117502 First few rows of the triangle are:
%e A117502 1;
%e A117502 1, 2;
%e A117502 1, 1, 3;
%e A117502 1, 1, 2, 5;
%e A117502 1, 1, 2, 3, 8;
%e A117502 ...
%Y A117502 Cf. A117501, A001595.
%K A117502 nonn,tabl
%O A117502 1,3
%A A117502 _Gary W. Adamson_, Mar 23 2006

%I A134404
%S A134404 0,0,1,0,0,1,1,1,0,0,1,1,2,1,1,0,0,1,1,2,3,2,1,1,0,0,1,1,2,3,5,3,2,1,
%T A134404 1,0,0,1,1,2,3,5,8,5,3,2,1,1,0,0,1,1,2,3,5,8,13,8,5,3,2,1,1,0,0,1,1,2,
%U A134404 3,5,8,13,21,13,8,5,3,2,1,1,0,0,1,1,2,3,5,8,13,21,34,21,13,8,5
%N A134404 Triangle read by rows in which row n contains Fib(0), ..., Fib(n-1), Fib(n), Fib(n-1), ..., Fib(0).
%e A134404 Triangle begins:
%e A134404 0
%e A134404 0 1 0
%e A134404 0 1 1 1 0
%e A134404 0 1 1 2 1 1 0
%e A134404 0 1 1 2 3 2 1 1 0
%e A134404 0 1 1 2 3 5 3 2 1 1 0
%e A134404 0 1 1 2 3 5 8 5 3 2 1 1 0
%e A134404 0 1 1 2 3 5 8 13 8 5 3 2 1 1 0
%e A134404 0 1 1 2 3 5 8 13 21 13 8 5 3 2 1 1 0
%e A134404 0 1 1 2 3 5 8 13 21 34 21 13 8 5 3 2 1 1 0
%e A134404 0 1 1 2 3 5 8 13 21 34 55 34 21 13 8 5 3 2 1 1 0
%e A134404 0 1 1 2 3 5 8 13 21 34 55 89 55 34 21 13 8 5 3 2 1 1 0
%Y A134404 Cf. A094102.
%K A134404 nonn,tabf
%O A134404 0,13
%A A134404 _N. J. A. Sloane_, Apr 07 2008

%I A137290
%S A137290 1,1,2,3,5,8,13,21,4,25,29,24,23,17,10,27,7,4,11,15,26,11,7,18,25,13,
%T A137290 8,21,29,20,19,9,28,7,5,12,17,29,16,15,1,16,17,3,20,23,13,6,19,25,14,
%U A137290 9,23,2,25,27,22,19,11,0,11,11,22,3,25,28,23,21,14,5,19,24,13,7,20,27,17,14
%N A137290 Fibonacci(n) mod 30.
%C A137290 Has period 120.
%H A137290 Michel Marcus, <a href="/A137290/b137290.txt">Table of n, a(n) for n = 1..130</a>
%H A137290 <a href="/index/Rec#order_120">Index entries for linear recurrences with constant coefficients</a>, order 120.
%o A137290 (PARI) a(n) = fibonacci(n) % 30 \\ _Michel Marcus_, Jun 12 2013
%Y A137290 Cf. A000045, A003893, A007887, A079345, A105471.
%K A137290 nonn
%O A137290 1,3
%A A137290 Aaron M. Churchill (churchil(AT)math.udel.edu), Mar 15 2008

%I A147659
%S A147659 1,1,2,3,5,8,13,21,34,55,90,146,237,385,625,1015,1648,2676,4345,7055,
%T A147659 11456,18602,30205,49046,79639,129315,209977,340953,553627,898959,
%U A147659 1459697,2370203,3848649,6249296,10147379,16476944,26754661,43443243
%N A147659 Expansion of 1/(1-x-x^2-x^10+x^12).
%C A147659 Coefficient expansion of toral of inverse of low ratio (1.6237635378007012) Pisot Polynomial: a(n)=Coefficient_Expansion(1/( 1 - x^2 - x^10 - x^11 + x^12)). [Original definition]
%C A147659 The 1 + x^2 - x^10 - x^11 + x^12, is not Pisot, so x^22 doubling is the limit that sequence of polynomials below the Golden mean ratio.
%H A147659 Vincenzo Librandi, <a href="/A147659/b147659.txt">Table of n, a(n) for n = 0..1000</a>
%H A147659 <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).
%t A147659 f[x_] = 1 - x^2 - x^10 - x^11 + x^12; g[x] = ExpandAll[x^12*f[1/x]]; a = Table[SeriesCoefficient[Series[1/g[x], {x, 0, 50}], n], {n, 0, 50}]
%t A147659 CoefficientList[Series[1/(1-x-x^2-x^10+x^12),{x,0,40}],x] (* _Harvey P. Dale_, May 31 2012 *)
%K A147659 nonn
%O A147659 0,3
%A A147659 _Roger L. Bagula_, Nov 09 2008

%I A147660
%S A147660 1,1,2,3,5,8,13,21,34,54,87,140,225,362,582,936,1505,2420,3892,6259,
%T A147660 10065,16186,26029,41858,67313,108248,174077,279938,450176,723941,
%U A147660 1164190,1872167,3010685,4841568,7785863,12520667,20134840,32379408
%N A147660 Coefficient expansion of toral of inverse of low ratio (1.6081283851873882) Pisot Polynomial: a(n)=Coefficient_Expansion(1/( -1 + x^2 - x^9 - x^10 + x^11)).
%C A147660 The next 1 + x^2 - x^10 - x^11 + x^12, is not Pisot, so x^11 is the limit that sequence of polynomials below the Golden mean ratio.
%F A147660 a(n)=Coefficient_Expansion(1/( -1 + x^2 - x^9 - x^10 + x^11)).
%t A147660 f[x_] = -1 + x^2 - x^9 - x^10 + x^11; g[x] = ExpandAll[x^11*f[1/x]]; a = Table[SeriesCoefficient[Series[1/g[x], {x, 0, 50}], n], {n, 0, 50}]
%K A147660 nonn
%O A147660 0,3
%A A147660 _Roger L. Bagula_, Nov 09 2008

%I A191869
%S A191869 0,0,1,1,2,3,5,8,13,21,34,55,88,143,231,373,603,974,1574,2543,4109,
%T A191869 6639,10727,17332,28004,45248,73109,118126,190862,308385,498273,
%U A191869 805084,1300814,2101789,3395964,5487026,8865658,14324680,23145090,37396661,60423625
%N A191869 First differences of the dying rabbits sequence A000044.
%F A191869 G.f.: x^3(1 + x + x^2 + x^3 + x^4)(1 - x + x^2 - x^3 + x^4)/(1 - x - x^3 - x^5 - x^7 - x^9 - x^11). - _Charles R Greathouse IV_, Jun 19 2011
%t A191869 A000044 = CoefficientList[Series[1/(1 - z - z^3 - z^5 - z^7 - z^9 - z^11), {z, 0, 200}], z]; GetDiff[seq_List] := Drop[seq, 1] - Drop[seq, -1]; A191869 = GetDiff[A000044]
%o A191869 (PARI) A191869_list=Vec((-x^11-x^9-x^7-x^5-x^3)/(x^11+x^9+x^7+x^5+x^3+x-1)+O(x^99)) /* returns a list of the first 96 nonzero terms, a(3)...a(99) */
%o A191869 (PARI) A191869(n)=polcoeff((1+x^2+x^4+x^6+x^8)/(1-x-x^3-x^5-x^7-x^9-x^11+O(x^max(1,n-2))),n-3)  \\ _M. F. Hasler_, Jun 19 2011
%Y A191869 Cf. A000044.
%K A191869 nonn,easy
%O A191869 1,5
%A A191869 _Vladimir Joseph Stephan Orlovsky_, Jun 18 2011

%I A225396
%S A225396 1,1,2,3,5,8,13,21,34,55,88,142,229,369,