A022086 -id:A022086 - cp's OEIS frontend

A022089 Fibonacci sequence beginning 0, 6.

0, 6, 6, 12, 18, 30, 48, 78, 126, 204, 330, 534, 864, 1398, 2262, 3660, 5922, 9582, 15504, 25086, 40590, 65676, 106266, 171942, 278208, 450150, 728358, 1178508, 1906866, 3085374, 4992240, 8077614, 13069854, 21147468, 34217322, 55364790, 89582112, 144946902

Offset: 0

N. J. A. Sloane

Starting with a(0)=1, a(1)=3, a(n) = the number of ternary length-2 squarefree words of length n.

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 15.

G. C. Greubel, Table of n, a(n) for n = 0..1000
N. H. Bong, C. Dalfó, M. À. Fiol, and D. Závacká, Some inner metric parameters of a digraph: Iterated line digraphs and integer sequences, arXiv:2409.02125 [math.CO], 2024. See p. 17.
Cristina Dalfó and Miquel Àngel Fiol, A Note on the Order of Iterated Line Digraphs, Journal of Graph Theory, Volume 85, Issue 2, June 2017, Pages 395-39, 2016; DOI: 10.1002/jgt.22068; arXiv:1607.08832 [math.CO], 2016.
Tanya Khovanova, Recursive Sequences
Christoph Richard and Uwe Grimm, On the entropy and letter frequencies of ternary squarefree words, arXiv:math/0302302 [math.CO], 2003.
Index entries for linear recurrences with constant coefficients, signature (1,1).

Cf. A000032, A000045, A119457.

Sequences of the form m*Fibonacci listed in A022086.

A022089:= func< n | 6*Fibonacci(n) >;
[A022089(n): n in [0..50]]; // G. C. Greubel, Apr 13 2025

a:= n-> 6*(<<0|1>, <1|1>>^n)[1,2]:
seq(a(n), n=0..40);  # Alois P. Heinz, Jan 18 2019

6*Fibonacci[Range[0,50]] (* G. C. Greubel, Apr 13 2025 *)
LinearRecurrence[{1,1},{0,6},50] (* Harvey P. Dale, Dec 05 2015 *)

def A022089(n): return 6*fibonacci(n)
print([A022089(n) for n in range(51)]) # G. C. Greubel, Apr 13 2025

a(n) = round( (12*phi-6)/5 * phi^n)  for n>3. - Thomas Baruchel, Sep 08 2004
a(n) = 6*F(n) = F(n+3) + F(n+1) + F(n-4), n>3, where F=A000045.
a(n) = A119457(n+4,n-1) for n>1. - Reinhard Zumkeller, May 20 2006
G.f.: 6*x/(1-x-x^2). - Philippe Deléham, Nov 20 2008
a(n) = 6 * A000045(n). - Alois P. Heinz, Jan 18 2019

A022090 Fibonacci sequence beginning 0, 7.

Original entry on oeis.org

0, 7, 7, 14, 21, 35, 56, 91, 147, 238, 385, 623, 1008, 1631, 2639, 4270, 6909, 11179, 18088, 29267, 47355, 76622, 123977, 200599, 324576, 525175, 849751, 1374926, 2224677, 3599603, 5824280, 9423883, 15248163, 24672046, 39920209, 64592255, 104512464

Offset: 0

N. J. A. Sloane

The number of heptagons in the n-th ring of the Klein Quartic. - Amiram Eldar, Nov 14 2023

Arthur T. Benjamin and Jennifer J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A., 2003, p. 15.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences.
William P. Thurston, The Eightfold Way: A Mathematical Sculpture by Helaman Ferguson, in: The Eightfold Way: The Beauty of the Klein Quartic (ed. Silvio Levy), Cambridge University Press, New York, 1999, pp. 1-7.
Eric Weisstein's World of Mathematics, Klein Quartic.
Index entries for linear recurrences with constant coefficients, signature (1,1).

Cf. A000032, A000045, A001622, A119457.
Sequences of the form Fibonacci(n+k) + Fibonacci(n-k) are listed in A280154.
Sequences of the form m*Fibonacci are listed in A022086.

A022090:= func< n | 7*Fibonacci(n) >; [A022090(n): n in [0..40]]; // G. C. Greubel, Apr 10 2025

7*Fibonacci[Range[0,40]] (* Vladimir Joseph Stephan Orlovsky, Jul 23 2008 *)

def A022090(n): return 7*fibonacci(n)
[A022090(n) for n in range(41)] # G. C. Greubel, Apr 10 2025

a(n) = round(((14*phi-7)/5) * phi^n), for n>3. - Thomas Baruchel, Sep 08 2004
a(n) = 7*Fibonacci(n) = Fibonacci(n+4) + Fibonacci(n-4) for n>3.
a(n) = A119457(n+5, n-1) for n>1. - Reinhard Zumkeller, May 20 2006
G.f.: 7*x/(1-x-x^2). - Philippe Deléham, Nov 20 2008

A249783 Smallest index of a Fibonacci-like sequence containing n, see comments.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 3, 1, 3, 2, 3, 4, 1, 4, 3, 2, 5, 3, 5, 4, 1, 6, 4, 3, 5, 2, 7, 5, 3, 6, 5, 4, 6, 1, 7, 6, 4, 7, 3, 5, 7, 2, 8, 7, 5, 8, 3, 6, 8, 5, 9, 4, 6, 9, 1, 7, 9, 6, 10, 4, 7, 10, 3, 8, 5, 7, 11, 2, 8, 11, 7, 9, 5, 8, 12, 3, 9, 6, 8, 10, 5, 9, 13, 4, 10, 6, 9, 11, 1, 10, 7, 9, 11, 6, 10, 12, 4, 11, 7, 10

Offset: 0

Allan C. Wechsler, Nov 05 2014

Any two nonnegative integers F0 and F1 generate a Fibonacci-like sequence where for n > 1, Fn = F[n-1] + F[n-2]. Call F0 + F1 the "index" of such a sequence. In this sequence, a(n) is the smallest occurring index of any Fibonacci-like sequence containing n.

			For n = 0, the trivial sequence 0, 0, 0, ... has index 0.
For n = 5, the classic Fibonacci sequence 0, 1, 1, 2, 3, 5, ... contains 5 and has index 1.
For n = 7, the Lucas sequence 2, 1, 3, 4, 7, ... contains 7, and no such sequence with a smaller index contains 7.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
Index to sequences related to the complexity of n

If n > 0 is an element of A000045, a(n) = 1. If n > 2 is twice such an element, a(n) = 2. If n > 3 is an element of A000032 or of A022086, a(n) = 3.

Haskell
```
bi x y = if (x
				
```

a[n_] := Module[{a, k, A, B}, If[n<2, Return[n]]; For[k=1, k <= n-1, k++, For[a=0, a <= k-1, a++, A=a; B=k-A; While[BJean-François Alcover, Jan 06 2017, translated from PARI *)

a(n)=if(n<2,return(n));for(k=1,n-1,for(a=0,k-1,my(A=a,B=k-A);while(BCharles R Greathouse IV, Nov 06 2014

a(n) <= A054495(n) <= n. - Charles R Greathouse IV, Nov 06 2014

Extended by Charles R Greathouse IV, Nov 06 2014

A280154 a(n) = 5*Lucas(n).

Original entry on oeis.org

10, 5, 15, 20, 35, 55, 90, 145, 235, 380, 615, 995, 1610, 2605, 4215, 6820, 11035, 17855, 28890, 46745, 75635, 122380, 198015, 320395, 518410, 838805, 1357215, 2196020, 3553235, 5749255, 9302490, 15051745, 24354235, 39405980, 63760215, 103166195, 166926410, 270092605, 437019015

Offset: 0

Bruno Berselli, Dec 27 2016

Fibonacci sequence beginning 10, 5.
After 5, the sequence provides the 3rd column of the rectangular array in A213590.
After 5, all terms belong to A191921 because a(n) = Lucas(n+4) - 3*Lucas(n-1).
From G. C. Greubel, Dec 27 2016: (Start)
{a(n) mod 3} yields (1,2,0,2,2,1,0,1), repeated, and is given as A082115.
{a(n) mod 6} yields (4,5,3,2,5,1,0,1,1,2,3,5,2,1,3,4,1,5,0,5,5,4,3,1) and is given as A082117. (End)

Bruno Berselli, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (1,1).

Subsequence of A084176.
Cf. A022088: 5*Fibonacci(n).
Cf. A022359: Lucas(n+5) + Lucas(n-5).
Cf. A000032, A000045, A191921, A213590.
Cf. sequences with formula Fibonacci(n+k) + Fibonacci(n-k): A006355 (k=0, without the initial 1), A000032 (k=1), A022086 (k=2), A022112 (k=3, with an initial 4), A022090 (k=4), this sequence (k=5), A022352 (k=6).

Magma
```
[5*Lucas(n): n in [0..40]];
```

F := n -> combinat:-fibonacci(n):
seq(F(n+5) + F(n-5), n=0..38); # Peter Luschny, Dec 29 2016

Mathematica
```
Table[5 LucasL[n], {n, 0, 40}]
```

vector(40, n, n--; fibonacci(n+5)+fibonacci(n-5))

def A280154():
    x, y = 10, 5
    while True:
        yield x
        x, y = y, x + y
a = A280154(); print([next(a) for  in range(39)]) # _Peter Luschny, Dec 29 2016

G.f.: 5*(2 - x)/(1 - x - x^2).
a(n) = a(n-1) + a(n-2) for n>1.
a(n) = Fibonacci(n+5) + Fibonacci(n-5), with Fibonacci(-k) = -(-1)^k*Fibonacci(k) for the negative indices.

A050610 Sum_{i=0..y} (C(y,i) mod 2)F(2i+x) = FL(y+x)A050613(y), where A050613(y) = Product_{i=0..floor(log_2(y+1))} L(2^i)^bit(y,i).

Original entry on oeis.org

0, 1, 1, 1, 3, 3, 2, 4, 6, 12, 3, 7, 9, 21, 21, 5, 11, 15, 33, 35, 77, 8, 18, 24, 54, 56, 126, 168, 13, 29, 39, 87, 91, 203, 273, 609, 21, 47, 63, 141, 147, 329, 441, 987, 987, 34, 76, 102, 228, 238, 532, 714, 1596, 1598, 3572, 55, 123, 165, 369, 385, 861, 1155, 2583

Offset: 0

Antti Karttunen, Oct 24 1999

Rows cut from column 2 onward form a subset of Wythoff array (A035513), where the terms of column 0 (A051656) give the positions of those rows in that array.

Transpose of A050609. First row: A000045, second row: A000032, third row: A022086.

a(n) = generic_bincoeff_fibsum_as_sum( (n-((trinv(n)*(trinv(n)-1))/2)),(((trinv(n)-1)*(((1/2)*trinv(n))+1))-n) );

A097135 a(0) = 1; for n>0, a(n) = 3*Fibonacci(n).

Original entry on oeis.org

1, 3, 3, 6, 9, 15, 24, 39, 63, 102, 165, 267, 432, 699, 1131, 1830, 2961, 4791, 7752, 12543, 20295, 32838, 53133, 85971, 139104, 225075, 364179, 589254, 953433, 1542687, 2496120, 4038807, 6534927, 10573734, 17108661, 27682395, 44791056, 72473451, 117264507

Offset: 0

Paul Barry, Jul 26 2004

Binomial transform is A097136.

Index entries for linear recurrences with constant coefficients, signature (1,1).

Cf. A000032, A000045.

Essentially the same as A022086.

Join[{1}, Table[3*Fibonacci[n], {n, 70}]] (* Vladimir Joseph Stephan Orlovsky, Feb 10 2012 *)

a(n)=if(n,3*finonacci(n),1) \\ Charles R Greathouse IV, Oct 16 2015

G.f. : (1+2*x-x^2)/(1-x-x^2).
a(n) = a(n-1)+a(n-2) for n>2.
a(2n) = A097134(n); a(2n+1) = 3*F(2n+1).

Definition rewritten by N. J. A. Sloane, Jan 24 2010

A126714 Dual Wythoff array read along antidiagonals downwards.

Original entry on oeis.org

1, 2, 4, 3, 6, 7, 5, 10, 11, 9, 8, 16, 18, 14, 12, 13, 26, 29, 23, 19, 15, 21, 42, 47, 37, 31, 24, 17, 34, 68, 76, 60, 50, 39, 27, 20, 55, 110, 123, 97, 81, 63, 44, 32, 22, 89, 178, 199, 157, 131, 102, 71, 52, 35, 25, 144, 288, 322, 254, 212, 165, 115, 84, 57, 40, 28

Offset: 1

R. J. Mathar, Feb 12 2007

The dual Wythoff array is the dispersion of the sequence w given by w(n)=2+floor(n*x), where x=(golden ratio), so that w=2+A000201(n). For a discussion of dispersions, see A191426. - Clark Kimberling, Jun 03 2011

			Array starts:
  row
   1:   1   2   3    5    8   13   21   34    55    89   144
   2:   4   6  10   16   26   42   68  110   178   288   466
   3:   7  11  18   29   47   76  123  199   322   521   843
   4:   9  14  23   37   60   97  157  254   411   665  1076
   5:  12  19  31   50   81  131  212  343   555   898  1453
   6:  15  24  39   63  102  165  267  432   699  1131  1830
   7:  17  27  44   71  115  186  301  487   788  1275  2063
   8:  20  32  52   84  136  220  356  576   932  1508  2440
   9:  22  35  57   92  149  241  390  631  1021  1652  2673
  10:  25  40  65  105  170  275  445  720  1165  1885  3050
  11:  28  45  73  118  191  309  500  809  1309  2118  3427

Clark Kimberling, "Stolarsky Interspersions," Ars Combinatoria 39 (1995) 129-138. See page 135 for the dual Wythoff array and other dual arrays. - Clark Kimberling, Oct 29 2009

P. Hegarty, U. Larsson, Permutations of the natural numbers with prescribed difference multisets, Electr. J. Combin. Numb. Theory 6 (2006) #A03.

First three rows identical to A035506. First column is A007066. First row is A000045. 2nd row is essentially A006355. 3rd row is essentially A000032. 4th row essentially A000285. 5th row essentially A013655 or A001060. 6th row essentially A022086 or A097135. 7th row essentially A022120. 8th row essentially A022087. 9th row essentially A022130. 10th row essentially A022088. 11th row essentially A022095. 12th row essentially A022089 etc.

Cf. A035513 (Wythoff array).

Tn1 := proc(T,nmax,row) local n,r,c,fnd; n := 1; while true do fnd := false; for r from 1 to row do for c from 1 to nmax do if T[r,c] = n then fnd := true; fi; od; if T[r,nmax] < n then RETURN(-1); fi; od; if fnd then n := n+1; else RETURN(n); fi; od; end; Tn2 := proc(T,nmax,row,ai1) local n,r,c,fnd; for r from 1 to row do for c from 1 to nmax do if T[r,c]+1 = ai1 then RETURN(T[r,c+1]+1); fi; od; od; RETURN(-1); end; T := proc(nmax) local a,col,row; a := array(1..nmax,1..nmax); for col from 1 to nmax do a[1,col] := combinat[fibonacci](col+1); od; for row from 2 to nmax do a[row,1] := Tn1(a,nmax,row-1); a[row,2] := Tn2(a,nmax,row-1,a[row,1]); for col from 3 to nmax do a[row,col] := a[row,col-2]+a[row,col-1]; od; od; RETURN(a); end; nmax := 12; a := T(nmax); for d from 1 to nmax do for row from 1 to d do printf("%d, ",a[row,d-row+1]); od; od;

(* program generates the dispersion array T of the complement of increasing sequence f[n] *)
r = 40; r1 = 12;  (* r=# rows of T, r1=# rows to show *)
c = 40; c1 = 12;   (* c=# cols of T, c1=# cols to show *)
x = GoldenRatio; f[n_] := Floor[n*x + 2]
(* f(n) is complement of column 1 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];  (* the array T *)
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1,10}]]
(* Dual Wythoff array, A126714 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* array as a sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011; added here by Clark Kimberling, Jun 03 2011 *)

A131326 Row sums of A131325.

Original entry on oeis.org

1, 3, 4, 9, 13, 24, 37, 63, 100, 165, 265, 432, 697, 1131, 1828, 2961, 4789, 7752, 12541, 20295, 32836, 53133, 85969, 139104, 225073, 364179, 589252, 953433, 1542685, 2496120, 4038805, 6534927, 10573732, 17108661, 27682393, 44791056, 72473449, 117264507

Offset: 0

Gary W. Adamson, Jun 28 2007

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-1).

Cf. A131325, A049310, A062114, A052952, A022086.

Vec((1+2*x-x^2) / ((x-1)*(1+x)*(x^2+x-1)) + O(x^50)) \\ Colin Barker, Jul 12 2017

a(n) = A062114(n+1)+A052952(n-1).
G.f. ( 1+2*x-x^2 ) / ( (x-1)*(1+x)*(x^2+x-1) ).  - R. J. Mathar, Aug 12 2012
a(n) = 3*Fibonacci(n+1)-2 if n even, a(n) = 3*Fibonacci(n+1) if n odd. - R. J. Mathar, Aug 12 2012
From Colin Barker, Jul 12 2017: (Start)
a(n) = (3*2^(-n-1)*((1 + sqrt(5))^(n+1) - (1 - sqrt(5))^(n+1))) / sqrt(5) - 2 for n even.
a(n) = (3*2^(-n-1)*((1 + sqrt(5))^(n+1) - (1 - sqrt(5))^(n+1))) / sqrt(5) for n odd.
a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4) for n>3.
(End)

A022092 Fibonacci sequence beginning 0, 9.

Original entry on oeis.org

0, 9, 9, 18, 27, 45, 72, 117, 189, 306, 495, 801, 1296, 2097, 3393, 5490, 8883, 14373, 23256, 37629, 60885, 98514, 159399, 257913, 417312, 675225, 1092537, 1767762, 2860299, 4628061, 7488360, 12116421, 19604781, 31721202, 51325983, 83047185, 134373168

Offset: 0

N. J. A. Sloane

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 15.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (1,1).

Cf. A000032, A000045.

Sequences of the form m*Fibonacci listed in A022086.

A022092:= func< n | 9*Fibonacci(n) >;
[A022092(n): n in [0..40]]; // G. C. Greubel, Apr 10 2025

9*Fibonacci[Range[0,40]] (* Vladimir Joseph Stephan Orlovsky, Sep 17 2008 *)

def A022092(n): return 9*fibonacci(n)
print([A022092(n) for n in range(41)]) # G. C. Greubel, Apr 10 2025

a(n) = 9*Fibonacci(n).
a(n) = F(n+4) + F(n+1) + F(n-2) + F(n-4), n>3, where F = A000045.
a(n) = round((18*phi-9)/5 phi^n), for n > 4. - Thomas Baruchel, Sep 08 2004
G.f.: 9*x/(1-x-x^2). - Philippe Deléham, Nov 20 2008

A193917 Triangular array: the self-fusion of (p(n,x)), where p(n,x)=sum{F(k+1)*x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 2, 3, 6, 9, 3, 5, 9, 15, 24, 5, 8, 15, 24, 40, 64, 8, 13, 24, 39, 64, 104, 168, 13, 21, 39, 63, 104, 168, 273, 441, 21, 34, 63, 102, 168, 272, 441, 714, 1155, 34, 55, 102, 165, 272, 440, 714, 1155, 1870, 3025, 55, 89, 165, 267, 440, 712, 1155

Offset: 0

Clark Kimberling, Aug 09 2011

See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.  (Fusion is defined at A193822; fission, at A193742; see A202503 and A202453 for infinite-matrix representations of fusion and fission.)
First five rows of P (triangle of coefficients of polynomials p(n,x)):
1
1...1
1...1...2
1...1...2...3
1...1...2...3...5
First eight rows of A193917:
1
1...1
1...2...3
2...3...6...9
3...5...9...15...24
5...8...15..24...40...64
8...13..24..39...64...104..168
13..21..39..63...104..168..273..441
...
col 1:  A000045
col 2:  A000045
col 3:  A022086
col 4:  A022086
col 5:  A022091
col 6:  A022091
col 7:  A022355
col 8:  A022355
right edge, w(n,n):  A064831
w(n,n-1):  A001654
w(n,n-2):  A064831
w(n,n-3):  A059840
w(n,n-4):  A080097
w(n,n-5):  A080143
w(n,n-6):  A080144
Suppose n is an even positive integer and w(n+1,x) is the polynomial matched to row n+1 of A193917 as in the Mathematica program (and definition of fusion at A193722), where the first row is counted as row 0.

			First six rows:
1
1...1
1...2...3
2...3...6....9
3...5...9....15...24
5...8...15...24...40...64

Cf. A193722, A064831, A193918, A194000, A194001.

z = 12;
p[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
q[n_, x_] := p[n, x];
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193917 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193918 *)

cp's OEIS Frontend

A022089 Fibonacci sequence beginning 0, 6.

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A022090 Fibonacci sequence beginning 0, 7.

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A249783 Smallest index of a Fibonacci-like sequence containing n, see comments.

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A280154 a(n) = 5*Lucas(n).

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A050610 Sum_{i=0..y} (C(y,i) mod 2)*F(2i+x) = FL(y+x)*A050613(y), where A050613(y) = Product_{i=0..floor(log_2(y+1))} L(2^i)^bit(y,i).

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A097135 a(0) = 1; for n>0, a(n) = 3*Fibonacci(n).

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A126714 Dual Wythoff array read along antidiagonals downwards.

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A050610 Sum_{i=0..y} (C(y,i) mod 2)F(2i+x) = FL(y+x)A050613(y), where A050613(y) = Product_{i=0..floor(log_2(y+1))} L(2^i)^bit(y,i).