A258160 -id:A258160 - cp's OEIS frontend

A022087 Fibonacci sequence beginning 0, 4.

0, 4, 4, 8, 12, 20, 32, 52, 84, 136, 220, 356, 576, 932, 1508, 2440, 3948, 6388, 10336, 16724, 27060, 43784, 70844, 114628, 185472, 300100, 485572, 785672, 1271244, 2056916, 3328160, 5385076, 8713236, 14098312, 22811548, 36909860, 59721408, 96631268

Offset: 0

N. J. A. Sloane

For n > 1, this sequence gives the number of binary strings of length n that do not contain 0000, 0101, 1010, or 1111 as contiguous substrings (see A230127). - Nathaniel Johnston, Oct 11 2013

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 18.

Vincenzo Librandi, 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. A000045, A119457.
Cf. similar sequences listed in A258160.
Cf. sequences of the form m*Fibonacci listed in A022086.

[4*Fibonacci(n): n in [0..40]]; // Vincenzo Librandi, Oct 12 2013

a:= n-> (Matrix([[4,0]]). Matrix([[1,1],[1,0]])^n)[1,2]: seq(a(n), n=0..40); # Alois P. Heinz, Aug 17 2008

4*Fibonacci[Range[0,50]] (* Vladimir Joseph Stephan Orlovsky, Jul 22 2008 *)
Table[4 Fibonacci(n), {n, 0, 40}] (* Bruno Berselli, May 22 2015 *)
LinearRecurrence[{1,1},{0,4},40] (* Harvey P. Dale, Jul 31 2025 *)

a(n)=4*fibonacci(n) \\ Charles R Greathouse IV, Jun 05 2011

def A022087(n): return 4*fibonacci(n)
print([A022087(n) for n in range(41)]) # G. C. Greubel, Apr 12 2025

a(n) = 4*F(n) = F(n-2) + F(n) + F(n+2), where F = A000045.
a(n) = round( phi^n*(8*phi-4)/5 ) for n>2. - Thomas Baruchel, Sep 08 2004
a(n) = A119457(n+2,n-1) for n>1. - Reinhard Zumkeller, May 20 2006
G.f.: 4*x/(1-x-x^2). - Philippe Deléham, Nov 19 2008
a(n) = F(n+9) - 17*F(n+3), where F=A000045. - Manuel Valdivia, Dec 15 2009
G.f.: Q(0) -1, where Q(k) = 1 + x^2 + (4*k+5)*x - x*(4*k+1 + x)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 07 2013
a(n) = Fibonacci(n+3) - Fibonacci(n-3), where Fibonacci(-3..-1) = 2,-1,1. - Bruno Berselli, May 22 2015

A022091 Fibonacci sequence beginning 0, 8.

Original entry on oeis.org

0, 8, 8, 16, 24, 40, 64, 104, 168, 272, 440, 712, 1152, 1864, 3016, 4880, 7896, 12776, 20672, 33448, 54120, 87568, 141688, 229256, 370944, 600200, 971144, 1571344, 2542488, 4113832, 6656320, 10770152, 17426472, 28196624, 45623096, 73819720, 119442816

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, A258160.

Sequences of the form m*Fibonacci are listed in A022086.

A022091:= func< n | 8*Fibonacci(n) >;
[A022091(n): n in [0..40]]; // G. C. Greubel, Apr 13 2025

8*Fibonacci[Range[0,41]] (* Vladimir Joseph Stephan Orlovsky, Sep 17 2008; modified by G. C. Greubel, Apr 13 2025 *)
LinearRecurrence[{1,1},{0,8},40] (* Harvey P. Dale, Jan 19 2018 *)

def A022091(n): return 8*fibonacci(n)
print([A022091(n) for n in range(41)]) # G. C. Greubel, Apr 13 2025

a(n) = round( (16phi-8)/5 phi^n) (works for n>4). - Thomas Baruchel, Sep 08 2004
a(n) = 8*F(n) = F(n+4) + F(n) + F(n-4) for n>3, where F=A000045.
G.f.: 8*x/(1-x-x^2). - Philippe Deléham, Nov 20 2008

A022379 Fibonacci sequence beginning 3, 9.

Original entry on oeis.org

3, 9, 12, 21, 33, 54, 87, 141, 228, 369, 597, 966, 1563, 2529, 4092, 6621, 10713, 17334, 28047, 45381, 73428, 118809, 192237, 311046, 503283, 814329, 1317612, 2131941, 3449553, 5581494, 9031047, 14612541, 23643588, 38256129, 61899717, 100155846, 162055563, 262211409

Offset: 0

N. J. A. Sloane

Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (1, 1).

Cf. A000032, A000045.

Cf. similar sequences listed in A258160.

[3*Lucas(n+1): n in [0..40]]; // Bruno Berselli, May 22 2015

LinearRecurrence[{1, 1}, {3, 9}, 30] (* Alonso del Arte, Oct 09 2013 *)
Table[3 LucasL[n + 1], {n, 0, 40}] (* Bruno Berselli, May 22 2015 *)
Table[LucasL[n + 4] + LucasL[n - 4] - 4 LucasL[n], {n, 1, 40}] (* Bruno Berselli, Dec 30 2016 *)

Vec((3+6*x)/(1-x-x^2)+O(x^99)) \\ Charles R Greathouse IV, Oct 21 2012

G.f.: (3 + 6*x)/(1 - x - x^2). - Philippe Deléham, Nov 19 2008
a(n+2) = 3*L(n+3) = L(n) + 4*L(n+1) + 2*L(n+2), where L=A000032. - J. M. Bergot, Oct 21 2012
a(n) = Fibonacci(n+4) - Fibonacci(n-4), where n>0 and Fibonacci(-3..-1) = 2,-1,1. - Bruno Berselli, May 22 2015
a(n) = L(n+4) + L(n-4) - 4*L(n) for n>0. - Bruno Berselli, Dec 29 2016

More terms from Bruno Berselli, May 22 2015

A022345 Fibonacci sequence beginning 0, 11.

Original entry on oeis.org

0, 11, 11, 22, 33, 55, 88, 143, 231, 374, 605, 979, 1584, 2563, 4147, 6710, 10857, 17567, 28424, 45991, 74415, 120406, 194821, 315227, 510048, 825275, 1335323, 2160598, 3495921, 5656519, 9152440, 14808959, 23961399, 38770358, 62731757, 101502115, 164233872

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. A000045.

Cf. similar sequences listed in A258160.

[11*Fibonacci(n): n in [0..40]]; // Bruno Berselli, May 22 2015

Table[11 Fibonacci(n), {n, 0, 40}] (* Bruno Berselli, May 22 2015 *)

x='x+O('x^50); concat([0], Vec(11*x/(1-x-x^2))) \\ G. C. Greubel, Aug 25 2017

a(n) = 11*F(n) = F(n+4) + F(n+2) + F(n) + F(n-2) + F(n-4) with n > 3 and F = A000045.
G.f.: 11*x/(1-x-x^2). - Philippe Deléham, Nov 20 2008
a(n) = Fibonacci(n+5) - Fibonacci(n-5), where Fibonacci(-5..-1) = 5, -3, 2, -1, 1. - Bruno Berselli, May 22 2015

More terms from Bruno Berselli, May 22 2015

A022352 Fibonacci sequence beginning 0, 18.

Original entry on oeis.org

0, 18, 18, 36, 54, 90, 144, 234, 378, 612, 990, 1602, 2592, 4194, 6786, 10980, 17766, 28746, 46512, 75258, 121770, 197028, 318798, 515826, 834624, 1350450, 2185074, 3535524, 5720598, 9256122, 14976720

Offset: 0

N. J. A. Sloane

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. A258160.

Cf. sequences with formula Fibonacci(n+k)+Fibonacci(n-k) listed in A280154.

a={};b=0;c=18;AppendTo[a, b];AppendTo[a, c];Do[b=b+c;AppendTo[a, b];c=b+c;AppendTo[a, c], {n, 4!}];a (* Vladimir Joseph Stephan Orlovsky, Sep 17 2008 *)
LinearRecurrence[{1,1},{0,18},40] (* or *) Table[9(LucasL[n]- Fibonacci[n]),{n,40}] (* Harvey P. Dale, Oct 09 2013 *)

for(n=0,50, print1(18*fibonacci(n), ", ")) \\ G. C. Greubel, Aug 26 2017

G.f.: 18*x/(1-x-x^2). - Philippe Deléham, Nov 20 2008

a(n) = 9*(Lucas(n) - Fibonacci(n)). - Harvey P. Dale, Oct 09 2013

A022363 Fibonacci sequence beginning 0, 29.

Original entry on oeis.org

0, 29, 29, 58, 87, 145, 232, 377, 609, 986, 1595, 2581, 4176, 6757, 10933, 17690, 28623, 46313, 74936, 121249, 196185, 317434, 513619, 831053, 1344672, 2175725, 3520397, 5696122, 9216519, 14912641, 24129160, 39041801, 63170961, 102212762, 165383723

Offset: 0

N. J. A. Sloane

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. A000045.

Cf. similar sequences listed in A258160.

[29*Fibonacci(n): n in [0..40]]; // Bruno Berselli, May 22 2015

Table[29 Fibonacci(n), {n, 0, 40}] (* Bruno Berselli, May 22 2015 *)
LinearRecurrence[{1,1},{0,29},40] (* Harvey P. Dale, Feb 13 2023 *)

for(n=0,50, print1(29*fibonacci(n), ", ")) \\ G. C. Greubel, Aug 26 2017

G.f.: 29*x/(1-x-x^2). - Philippe Deléham, Nov 20 2008
a(n) = Fibonacci(n+7) - Fibonacci(n-7), where Fibonacci(-7..-1) = 13, -8, 5, -3, 2, -1, 1. - Bruno Berselli, May 22 2015
a(n) = a(n-1) + a(n-2) for n>=2, with a(0)=0, a(1)=29. - Wesley Ivan Hurt, May 12 2023

More terms from Bruno Berselli, May 22 2015

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A022087 Fibonacci sequence beginning 0, 4.

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A022091 Fibonacci sequence beginning 0, 8.

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A022379 Fibonacci sequence beginning 3, 9.

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A022345 Fibonacci sequence beginning 0, 11.

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A022352 Fibonacci sequence beginning 0, 18.

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A022363 Fibonacci sequence beginning 0, 29.

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