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### -- OEIS
### -- Licenses: Artistic-2.0
### -- Authors: Tobias Boege
### -- File: t/data/Fibonacci-Search-all.txt
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2609,6831,4222,11054,6832,17887,
%U A118627 11055,28943,17888,46832,28944,75777,46833,122611,75778,198390,122612,321003
%N A118627 a(1) = a(2) = 1. For n >=3, a(n) = the a(n-2)th integer, among those positive integers which are missing from the first (m-1) terms of the sequence, below a(n-1) if such a positive integer exists. Otherwise, a(n) = the a(n-2)th integer, among those positive integers which are missing from the first (m-1) terms of the sequence, above a(n-1).
%e A118627 The first 8 terms of the sequence are 1,1,2,3,5,8,13,21. Those integers which are missing from the first 8 terms of the sequence form the sequence 4,6,7,9,10,11,12,14,15,16,17,18,19,20,.. Counting down from a(8)=21 a total of a(7)=13 positions in this sequence of missing terms, we land on 6. So a(9) = 6.
%e A118627 There are fewer than a(8)=21 missing positive integers below a(9)=6, so we count UP to get a(10). a(10) is what we land on when counting up from 6 a total of a(8)=21 positions, skipping over terms which occur earlier in the sequence. So a(10) = 30.
%K A118627 easy,nonn
%O A118627 1,3
%A A118627 _Leroy Quet_, May 09 2006
%E A118627 More terms from _Joshua Zucker_, Jul 27 2006

%I A120659
%S A120659 0,1,1,2,3,5,8,13,21,34,55,2483,6053383,36651957891448,
%T A120659 1343366154248994863013047386,
%U A120659 1804632624381764689171354018874685160689875147803208124
%N A120659 A determinant sum sequence of the D5 dihehral 2 X 2 representation.
%D A120659 (*http : // mathworld.wolfram.com/DihedralGroupD3.html*)
%F A120659 << MathWorld`Groups` M0 = DihedralGroupMatrices[5]; s[n_] := M0[[n]] a[0] = Table[Fibonacci[n], {n, 0, 9}]; a[1] = Table[Fibonacci[n], {n,1, 10}]; a[n_] := a[n] = {a[n - 1][[2]], a[n - 1][[3]], a[n - 1][[4]], a[n - 1][[5]], a[n - 1][[6]], a[n - 1][[7]], a[n - 1][[8]], a[n - 1][[9]], a[n - 1][[10]], Abs[Det[Sum[a[n - 1][[i]]*s[i], {i, 1, 10}]]]}
%t A120659 Clear[a, f, s] (*http : // mathworld.wolfram.com/DihedralGroupD3.html*) << MathWorld`Groups` M0 = DihedralGroupMatrices[5]; s[n_] := M0[[n]] a[0] = Table[Fibonacci[n], {n, 0, 9}]; a[1] = Table[Fibonacci[n], {n, 1, 10}]; a[n_] := a[n] = {a[n - 1][[2]], a[n - 1][[3]], a[n - 1][[4]], a[n - 1][[5]], a[n - 1][[6]], a[n - 1][[7]], a[n - 1][[8]], a[n - 1][[9]], a[n - 1][[10]], Abs[Det[Sum[a[n - 1][[i]]*s[i], {i, 1, 10}]]]} Table[Floor[a[n][[1]]], {n, 0, 16}]
%Y A120659 Cf. A120495, A120496.
%K A120659 nonn,uned
%O A120659 0,4
%A A120659 _Roger L. Bagula_, Aug 10 2006

%I A141450
%S A141450 1,1,2,1,1,4,1,1,3,7,1,1,2,4,12,1,1,2,4,8,19,1,1,2,3,6,11,30,1,1,2,3,
%T A141450 6,9,19,45,1,1,2,3,5,8,15,26,67,1,1,2,3,5,8,13,21,41,97,1,1,2,3,5,7,
%U A141450 12,18,31,56,139,1,1,2,3,5,7,12,17,28,45,83,195,1,1,2,3,5,7,11,16,25,38,63
%N A141450 Upper right triangle of the number of m's in all partitions of n.
%C A141450 The "last" column read from the bottom is A000041.
%C A141450 Mirror of triangle A066633. - _Omar E. Pol_, May 01 2012
%e A141450 A000070: 1, 2, 4, 7, 12, 19, 30, 45, 67, 97, 139, 195, 272, 373, 508, ...,
%e A141450 A024786: 0, 1, 1, 3, 4, 8, 11, 19, 26, 41, 56, 83, 112, 160, 213, ...,
%e A141450 A024787: 0, 0, 1, 1, 2, 4, 6, 9, 15, 21, 31, 45, 63, 87, 122, ...,
%e A141450 A024788: 0, 0, 0, 1, 1, 2, 3, 6, 8, 13, 18, 28, 38, 55, 74, ...,
%e A141450 A024789: 0, 0, 0, 0, 1, 1, 2, 3, 5, 8, 12, 17, 25, 35, 50, ...,
%e A141450 A024790: 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 12, 16, 24, 33, ...,
%e A141450 A024791: 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 16, 23, ...,
%e A141450 A024792: 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, ...,
%e A141450 A024793: 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, ...,
%e A141450 A024794: 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, ...
%t A141450 (* First do ) Needs["Combinatorica`"] (* then *) f[n_, m_] := Count[Flatten@ Partitions@ n, m]; Table[ f[n, m], {n, 13}, {m, n, 1, -1}]
%Y A141450 Cf. A000041, A000070, A024786, A024787, A024788, A024789, A024790, A024791, A024792, A024793, A024794.
%K A141450 nonn,tabl
%O A141450 1,3
%A A141450 _Robert G. Wilson v_, Aug 07 2008

%I A175712
%S A175712 0,1,1,2,3,5,8,13,21,33,54,79,133,176,309,365,674,709,1383,1300,2683,
%T A175712 2267,4950,3785,8735,6085,14820,9465,24285,14302,38587
%N A175712 The third column of the Lucas Fibonacci sum of binomials A175685.
%F A175712 a(n) = A175685(n,3), n>=1 .
%t A175712 Table[Sum[Binomial[n - j - 1, j], {j, Floor[(n - 1)/2] - 3, Floor[(
%t A175712     n - 1)/2]}], {n, 0, 30}]
%Y A175712 Cf. A000045, A175685, A175686.
%K A175712 nonn
%O A175712 0,4
%A A175712 _Roger L. Bagula_, Dec 04 2010

%I A177376
%S A177376 1,1,2,3,5,8,13,21,34,55,144,233,377,610,2584,4181,6765,17711,28657,
%T A177376 46368,75025,317811,832040,3524578,5702887,24157817,102334155,
%U A177376 165580141,701408733,20365011074,86267571272,225851433717,17167680177565
%N A177376 Fibonacci numbers whose decimal expansion does not contain any digit 9.
%C A177376 The probability that Fib(n) contains no 9's goes to zero as n grows to infinity. It appears that the largest term is F(188). [Corrected by _Jon E. Schoenfield_, May 08 2010]
%e A177376 a(11)=144 since it is the 11th Fibonacci containing no 9's
%t A177376 Select[Fibonacci[Range[100]],DigitCount[#,10,9]==0&] (* _Harvey P. Dale_, Jan 22 2014 *)
%Y A177376 Cf. A000045, A177194, A177195, A177231, A177245, A177246, A176253, A177247, A177272, A177372, A177374
%K A177376 nonn,base
%O A177376 1,3
%A A177376 _Carmine Suriano_, May 07 2010

%I A178526
%S A178526 1,1,0,1,1,1,1,1,2,1,1,1,2,3,2,1,1,2,3,5,3,1,1,2,3,5,8,5,1,1,2,3,5,8,
%T A178526 13,8,1,1,2,3,5,8,13,21,13,1,1,2,3,5,8,13,21,34,21,1,1,2,3,5,8,13,21,
%U A178526 34,55,34,1,1,2,3,5,8,13,21,34,55,89,55,1,1,2,3,5,8,13,21,34,55,89,144,89
%N A178526 Triangle read by rows: T(n,k) is the number of nodes of cost k in the Fibonacci tree of order n.
%C A178526 A Fibonacci tree of order n (n>=2) is a complete binary tree whose left subtree is the Fibonacci tree of order n-1 and whose right subtree is the Fibonacci tree of order n-2; each of the Fibonacci trees of order 0 and 1 is defined as a single node. In a Fibonacci tree the cost of a left (right) edge is defined to be 1 (2). The cost of a node of a Fibonacci tree is defined to be the sum of the costs of the edges that form the path from the root to this node.
%C A178526 The sum of the entries in row n is A001595(n) = 2F(n+1) - 1, where F(m)=A000045(m) (the Fibonacci numbers).
%C A178526 Sum(k*T(n,k), 0<=k<=n)=A178525(n).
%C A178526 _Daniel Forgues_, Aug 10 2012: (Start)
%C A178526 The falling diagonals are, starting from the rightmost one, with index 0:
%C A178526   d_0(i) = F(i-1), i >= 0;
%C A178526   d_j(i) = F(i+1), j >= 1, i >= 0.
%C A178526 Equivalently, as a single expression:
%C A178526   d_j(i) = F(i+1-2*0^j), j >= 0, i >= 0. (End)
%D A178526 D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, Reading, MA, 1998, p. 417.
%H A178526 Y. Horibe, <a href="http://www.fq.math.ca/Scanned/20-2/horibe.pdf">An entropy view of Fibonacci trees</a>, Fibonacci Quarterly, 20, No. 2, 1982, 168-178.
%F A178526 T(n,k)=F(k+1) if k<n; T(n,n)=F(n-1); T(n,k)=0 if k>n; here F(m)=A000045(m) (the Fibonacci numbers).
%F A178526 G.f.: (1-tz+tz^2)/[(1-z)(1-tz-t^2*z^2)].
%F A178526 The enumerating polynomials P[n] of row n are given by P[0]=1, P[n]=P[n-1]+F(n-1)*(t^{n-1}+t^n) for n>=1, where F(m)=A000045(m) (the Fibonacci numbers).
%e A178526 In the Fibonacci tree /\ of order 2 we have a node of cost 0 (the root), a node of cost 1 (the left leaf), and a node of cost 2 (the right leaf).
%e A178526 Triangle starts:
%e A178526 1;
%e A178526 1,0;
%e A178526 1,1,1;
%e A178526 1,1,2,1;
%e A178526 1,1,2,3,2;
%e A178526 1,1,2,3,5,3;
%e A178526 1,1,2,3,5,8,5;
%p A178526 with(combinat): T := proc (n, k) if k < n then fibonacci(k+1) elif k = n then fibonacci(n-1) else 0 end if end proc: for n from 0 to 12 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form
%Y A178526 Cf. A000045, A001595, A178525.
%K A178526 nonn,tabl
%O A178526 0,9
%A A178526 _Emeric Deutsch_, Jun 16 2010

%I A218032
%S A218032 1,1,1,1,2,3,5,8,13,21,35,57,94,154,254,417,687,1129,1859,3057,5032,
%T A218032 8277,13623,22412,36883,60684,99862,164312,270384,444899,732093,
%U A218032 1204629,1982228,3261701,5367131,8831505,14532200,23912499,39347839,64746320,106539481,175309363,288469809
%N A218032 G.f. A(x) satisfies A(x) = 1 + x / (1- x * A(x^2)).
%C A218032 What does this sequence count?
%o A218032 (PARI)
%o A218032 N=66;  R=O('x^N);  x='x+R;
%o A218032 F = 1 + x;
%o A218032 for (k=1,N+1, F = 1 + x / (1- x * subst(F,'x,'x^2) ) + R; );
%o A218032 Vec(F)
%K A218032 nonn
%O A218032 0,5
%A A218032 _Joerg Arndt_, Oct 19 2012

%I A280198
%S A280198 1,1,1,2,3,5,8,13,21,33,53,86,138,222,357,574,923,1484,2387,3839,6173,
%T A280198 9927,15964,25672,41284,66389,106762,171686,276091,443989,713988,
%U A280198 1148179,1846411,2969252,4774918,7678647,12348195,19857396,31933099,51352294,82580715,132799801,213558181,343427445,552272966,888121883,1428207656
%N A280198 Expansion of 1/(1 - Sum_{k>=1} mu(2*k-1)^2*x^(2*k-1)), where mu() is the Moebius function (A008683).
%C A280198 Number of compositions (ordered partitions) into odd squarefree parts (A056911).
%H A280198 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Squarefree.html">Squarefree</a>
%H A280198 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%F A280198 G.f.: 1/(1 - Sum_{k>=1} mu(2*k-1)^2*x^(2*k-1)).
%e A280198 a(4) = 3 because we have [3, 1], [1, 3] and [1, 1, 1, 1].
%t A280198 nmax = 46; CoefficientList[Series[1/(1 - Sum[MoebiusMu[2 k - 1]^2 x^(2 k - 1), {k, 1, nmax}]), {x, 0, nmax}], x]
%Y A280198 Cf. A005117, A008683, A056911, A134345, A280194.
%K A280198 nonn
%O A280198 0,4
%A A280198 _Ilya Gutkovskiy_, Dec 28 2016

%I A293865
%S A293865 0,0,0,1,1,2,3,5,8,13,21,37,56,90,144,239,374,592,948,1558,2431,3848,
%T A293865 6127,9972,15602,24658,39158,63265,99110,156505,248040,398675,625024,
%U A293865 986241,1560763,2498832,3919561,6180914,9770162,15594972,24470070,38567903,60907330
%N A293865 Number of self-intersecting walks of length n on a square lattice such that at each point the angle turns 90 degrees.
%C A293865 It is assumed that the first walk turns left and that all walks end when they intersect themselves.
%H A293865 MathStackExchange, <a href="https://math.stackexchange.com/questions/2471391/expected-number-of-steps-before-intersection">Expected Number of Steps Before Intersection</a>, Oct 2017.
%F A293865 For n>2, a(n) = 2*A189722(n-1) - A189722(n). - _Jens Randrup Rasmussen_, Oct 29 2017
%e A293865 For n = 4 we have the simplest self-intersecting walk, which is a square.
%e A293865 For n = 5 we have the walk:
%e A293865 (0,0), (0,1), (-1,1), (-1, 2), (0,2), (0,1)
%e A293865 For n = 6 we have the walks:
%e A293865 (0,0), (0,1), (-1,1), (-1, 0), (-2,0), (-2,1), (-1,1)
%e A293865 (0,0), (0,1), (-1,1), (-1, 2), (-2,2), (-2,1), (-1,1)
%o A293865 (Visual Basic for Excel)
%o A293865 Const N = 50
%o A293865 Const MaxSteps = 43
%o A293865 Dim BeenHere() As Boolean
%o A293865 Dim LoopBacks(MaxSteps) As Long
%o A293865 Dim PosX As Integer, PosY As Integer
%o A293865 Sub Macro1()
%o A293865   ReDim BeenHere(N, N)
%o A293865   PosX = N / 2: PosY = N / 2
%o A293865   BeenHere(PosX, PosY) = True
%o A293865   PosX = PosX + 1
%o A293865   BeenHere(PosX, PosY) = True
%o A293865   PosY = PosY - 1
%o A293865   BeenHere(PosX, PosY) = True
%o A293865   DoSteps 2, 3, PosX, PosY, BeenHere()
%o A293865   For i = 4 To MaxSteps
%o A293865     Cells(i - 1, 3).Value = i
%o A293865     Cells(i - 1, 4).Value = LoopBacks(i)
%o A293865   Next i
%o A293865 End Sub
%o A293865 Sub DoSteps(ByVal StepNo As Integer, Dir As Integer, X As Integer, Y As Integer, BH() As Boolean)
%o A293865 Dim BH2() As Boolean
%o A293865 Dim X1 As Integer, Y1 As Integer, X2 As Integer, Y2 As Integer
%o A293865 Dim Dir1 As Integer, Dir2 As Integer
%o A293865   BH2 = BH
%o A293865   StepNo = StepNo + 1
%o A293865   Select Case Dir
%o A293865   Case 1, 3 ' North or South
%o A293865       Dir1 = 2: X1 = X + 1: Y1 = Y
%o A293865       Dir2 = 4: X2 = X - 1: Y2 = Y
%o A293865   Case 2, 4 ' East or West
%o A293865       Dir1 = 1: Y1 = Y + 1: X1 = X
%o A293865       Dir2 = 3: Y2 = Y - 1: X2 = X
%o A293865   End Select
%o A293865   If BH2(X1, Y1) Then
%o A293865     LoopBacks(StepNo) = LoopBacks(StepNo) + 1
%o A293865   ElseIf StepNo < MaxSteps Then
%o A293865     BH2(X1, Y1) = True
%o A293865     DoSteps StepNo, Dir1, X1, Y1, BH2()
%o A293865     BH2(X1, Y1) = False
%o A293865   End If
%o A293865   If BH2(X2, Y2) Then
%o A293865     LoopBacks(StepNo) = LoopBacks(StepNo) + 1
%o A293865   ElseIf StepNo < MaxSteps Then
%o A293865     BH2(X2, Y2) = True
%o A293865     DoSteps StepNo, Dir2, X2, Y2, BH2()
%o A293865   End If
%o A293865 End Sub
%Y A293865 This sequence gives the number of self-intersecting walks while A189722 gives the number of self-avoiding walks.
%K A293865 nonn,walk
%O A293865 1,6
%A A293865 _Jens Randrup Rasmussen_, Oct 18 2017
%E A293865 The terms starting from a(11) and the program corrected by _Jens Randrup Rasmussen_, Oct 29 2017

%I A039834
%S A039834 1,1,0,1,-1,2,-3,5,-8,13,-21,34,-55,89,-144,233,-377,610,-987,1597,
%T A039834 -2584,4181,-6765,10946,-17711,28657,-46368,75025,-121393,196418,
%U A039834 -317811,514229,-832040,1346269,-2178309,3524578,-5702887,9227465,-14930352,24157817
%N A039834 a(n+2) = -a(n+1) + a(n) (signed Fibonacci numbers) with a(-2) = a(-1) = 1; or Fibonacci numbers (A000045) extended to negative indices.
%C A039834 Knuth defines the negaFibonacci numbers as follows: F(-1) = 1, F(-2) = -1, F(-3) = 2, F(-4) = -3, F(-5) = 5, ..., F(-n) = (-1)^(n-1) F(n). See A215022, A215023