A001076 -id:A001076 - cp's OEIS frontend

A163070 a(n) = ((4+sqrt(5))(2+sqrt(5))^n + (4-sqrt(5))(2-sqrt(5))^n)/2.

4, 13, 56, 237, 1004, 4253, 18016, 76317, 323284, 1369453, 5801096, 24573837, 104096444, 440959613, 1867934896, 7912699197, 33518731684, 141987625933, 601469235416, 2547864567597, 10792927505804, 45719574590813

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jul 20 2009

nonn

Binomial transform of A163069. Second binomial transform of A163141. Inverse binomial transform of A163071.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,1).

Cf. A000032, A000045, A001076, A014448, A163069, A163071, A163141.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-5); S:=[ ((4+r)*(2+r)^n+(4-r)*(2-r)^n)/2: n in [0..21] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 21 2009

LinearRecurrence[{4,1},{4,13},30] (* Harvey P. Dale, Sep 19 2011 *)

x='x+O('x^30); Vec((4-3*x)/(1-4*x-x^2)) \\ G. C. Greubel, Jan 08 2018

a(n) = 4*a(n-1) + a(n-2) for n > 1; a(0) = 4, a(1) = 13.
G.f.: (4-3*x)/(1-4*x-x^2).
a(n) = 2*A000032(3*n) + 5*A000045(3*n)/2 = 2*A014448(n) + 5*A001076(n). - Diego Rattaggi, Aug 09 2020

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 21 2009

A189800 a(n) = 6a(n-1) + 8a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 6, 44, 312, 2224, 15840, 112832, 803712, 5724928, 40779264, 290475008, 2069084160, 14738305024, 104982503424, 747801460736, 5326668791808, 37942424436736, 270267896954880, 1925146777223168, 13713023838978048, 97679317251653632, 695780094221746176

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,8).

Sequences of the form a(n) = c*a(n-1) + d*a(n-2), with a(0)=0, a(1)=1:
c/d...1.......2.......3.......4.......5.......6.......7.......8.......9......10
.A000045,A001045,A006130,A006131,A015440,A015441,A015442,A015443,A015445,A015446
.A000129,A002605,A015518,A063727,A002532,A083099,A015519,A003683,A002534,A083102
.A006190,A007482,A030195,A015521,A015523,A083858,A015524,A015525,A099012,A015528
.A001076,A090017,A015530,A057087,A015531,A085939,A015532,A180222,A015533,A180226
.A052918,A015535,A015536,A015537,A057088,A015540,A015541,A015544,A015545,A180250
.A005668,A135030,A090018,A135032,A015551,A057089,A015552,A189800,A189801,A190005
.A054413,A015555,A015559,A015561,A015562,A015564,A057090,A015565,A015566,A015568
.A041025,A190331,A015574,A190510,A015575,A190560,A015576,A057091,A015577,A190953
.A099371,A015579,A181353,A015580,A015581,A153191,A015583,A015584,A057092,A015585
A041041,A191014,A015588,A190954,A190955,A190956,A015589,A190957,A015591,A057093

I:=[0,1]; [n le 2 select I[n] else 6*Self(n-1)+8*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 14 2011

LinearRecurrence[{6, 8}, {0, 1}, 50]
CoefficientList[Series[-(x/(-1+6 x+8 x^2)),{x,0,50}],x] (* Harvey P. Dale, Jul 26 2011 *)

a(n)=([0,1; 8,6]^n*[0;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

G.f.: x/(1 - 2*x*(3+4*x)). - Harvey P. Dale, Jul 26 2011

A203319 a(n) = nFibonacci(n) Sum_{d|n} 1/(d*Fibonacci(d)).

Original entry on oeis.org

1, 3, 7, 19, 26, 81, 92, 267, 358, 848, 980, 3061, 3030, 7976, 11042, 25099, 27150, 78642, 79440, 219884, 270704, 584862, 659112, 1977909, 1950651, 4735370, 6204499, 14189096, 14912642, 43168586, 41734340, 110786987, 135815060, 290854380, 339428752, 953889058, 893839230

Offset: 1

Paul D. Hanna, Jan 01 2012

nonn

			L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 19*x^4/4 + 26*x^5/5 + 81*x^6/6 +...
where
L(x) = x*(1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 8*x^5 +...+ F(n+1)*x^n +...) +
x^2/2*(1 + 3*x^2 + 8*x^4 + 21*x^6 + 55*x^8 +...+ F(2*n+2)*x^(2*n) +...) +
x^3/3*(1 + 4*x^3 + 17*x^6 + 72*x^9 +...+ F(3*n+3)/2*x^(3*n) +...) +
x^4/4*(1 + 7*x^4 + 48*x^8 + 329*x^12 +...+ F(4*n+4)/3*x^(4*n) +...) +
x^5/5*(1 + 11*x^5 + 122*x^10 + 1353*x^15 +...+ F(5*n+5)/5*x^(5*n) +...) +
x^6/6*(1 + 18*x^6 + 323*x^12 + 5796*x^18 +...+ F(6*n+6)/8*x^(6*n) +...) +...
here F(n) = Fibonacci(n) = A000045(n).
Equivalently,
L(x) = x/(1-x-x^2) + (x^2/2)/(1-3*x^2+x^4) + (x^3/3)/(1-4*x^3-x^6) + (x^4/4)/(1-7*x^4+x^8) +...+ (x^n/n)/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) +...
here Lucas(n) = A000032(n).
Exponentiation of the l.g.f. equals the g.f. of A203318:
exp(L(x)) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 16*x^5 + 36*x^6 + 64*x^7 +...+ A203318(n)*x^n +...

Paul D. Hanna, Table of n, a(n) for n = 1..200

Cf. A203318, A203321; A203414 (Pell variant).

Cf. A000032 (Lucas), A000045 (Fibonacci), A001906, A001076, A004187, A049666, A049660, A049667.

a[n_] := n Fibonacci[n] DivisorSum[n, 1/(# Fibonacci[#]) &]; Array[a, 40] (* Jean-François Alcover, Dec 23 2015 *)

{a(n)=if(n<1,0, n*fibonacci(n)*sumdiv(n,d,1/(d*fibonacci(d))) )}

{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=n*polcoeff(sum(m=1,n+1,(x^m/m)/(1-Lucas(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))),n)}

{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=local(L=x); L=sum(m=1, n, x^m/m*exp(sum(k=1, floor((n+1)/m), Lucas(m*k)*x^(m*k)/k)+x*O(x^n))); n*polcoeff(L,n)}

{a(n)=local(A=1+x+x*O(x^n),F=1/(1-x-x^2+x*O(x^n))); A=exp(sum(m=1, n+1, x^m/m*round(prod(k=0, m-1, subst(F, x, exp(2*Pi*I*k/m)*x+x*O(x^n)))))); n*polcoeff(log(A), n)}

Equals the logarithmic derivative of A203318.
L.g.f.: L(x) = Sum_{n>=1} a(n)*x^n/n satisfies:
(1) L(x) = Sum_{n>=1} x^n/n * Sum_{k>=0} F(n*k+n)/F(n) * x^(n*k) where F(n) = Fibonacci(n).
(2) L(x) = Sum_{n>=1} x^n/n * exp( Sum_{k>=1} Lucas(n*k)*x^(n*k)/k ) where Lucas(n) = A000032(n).
(3) L(x) = Sum_{n>=1} x^n/n * 1/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) where Lucas(n) = A000032(n).
(4) L(x) = Sum_{n>=1} x^n/n * G_n(x^n) where G_n(x^n) = Product_{k=0..n-1} G(u^k*x) where G(x) = 1/(1-x-x^2) and u is an n-th root of unity.

A232970 Expansion of (1-3x)/(1-5x+3*x^2+x^3).

Original entry on oeis.org

1, 2, 7, 28, 117, 494, 2091, 8856, 37513, 158906, 673135, 2851444, 12078909, 51167078, 216747219, 918155952, 3889371025, 16475640050, 69791931223, 295643364940, 1252365390981, 5305104928862, 22472785106427, 95196245354568, 403257766524697, 1708227311453354, 7236167012338111, 30652895360805796

Offset: 0

N. J. A. Sloane, Dec 05 2013

For n > 2, a(n) is the number of tilings of (a 2 X (n+1) rectangle missing the top right and top left 1 X 1 cells) using 1 X 1 squares, dominoes and right trominoes. Compare with similar tiling sequences A001076 and A110679. - Greg Dresden and Yilin Zhu, Jul 10 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. Dziemianczuk, Counting Lattice Paths With Four Types of Steps, Graphs and Combinatorics, September 2013, DOI 10.1007/s00373-013-1357-1.
Hermann Stamm-Wilbrandt, 6 interlaced bisections
Index entries for linear recurrences with constant coefficients, signature (5,-3,-1).

Cf. A001076, A110679.

I:=[1,2,7]; [n le 3 select I[n] else 5*Self(n-1)- 3*Self(n-2)-Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 24 2017

LinearRecurrence[{5, -3, -1}, {1, 2, 7}, 30] (* Vincenzo Librandi, Jun 24 2017 *)
CoefficientList[Series[(1-3x)/(1-5x+3x^2+x^3),{x,0,30}],x] (* Harvey P. Dale, Oct 19 2024 *)

Vec((1-3*x)/(1-5*x+3*x^2+x^3) + O(x^30)) \\ Felix Fröhlich, Apr 15 2019

[(fibonacci(3*n+1) +1)/2 for n in (0..30)] # G. C. Greubel, Apr 19 2019

a(n) = 5*a(n-1) - 3*a(n-2) - a(n-3). - N. J. A. Sloane, Jun 23 2017
a(n) = (Fibonacci(3*n+1) + 1)/2 = Sum_{k=0..n} Fibonacci(3*k-1). - Ehren Metcalfe, Apr 15 2019
a(2*n) = A294262(2*n); a(2*n+1) = A254627(2*n+2). See "6 interlaced bisections" link. - Hermann Stamm-Wilbrandt, Apr 18 2019

A296239 a(n) = distance from n to nearest Fibonacci number.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 2, 2, 1, 0, 1, 2, 3, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 16, 15, 14, 13, 12, 11, 10, 9

Offset: 0

Rémy Sigrist, Dec 09 2017

The Fibonacci numbers correspond to sequence A000045.
This sequence is analogous to:
- A051699 (distance to nearest prime),
- A053188 (distance to nearest square),
- A053646 (distance to nearest power of 2),
- A053615 (distance to nearest oblong number),
- A053616 (distance to nearest triangular number),
- A061670 (distance to nearest power),
- A074989 (distance to nearest cube),
- A081134 (distance to nearest power of 3),
The local maxima of the sequence correspond to positive terms of A004695.
a(n) = 0 iff n = A000045(k) for some k >= 0.
a(n) = 1 iff n = A061489(k) for some k > 4.
For any n >= 0, abs(a(n+1) - a(n)) <= 1.
For any n > 0, a(n) < n, and a^k(n) = 0 for some k > 0 (where a^k denotes the k-th iterate of a); k equals A105446(n) for n = 1..80 (and possibly more values).
a(n) > max(a(n-1), a(n+1)) iff n = A001076(k) for some k > 1.

			For n = 42:
- A000045(9) = 34 <= 42 <= 55 = A000045(10),
- a(42) = min(42 - 34, 55 - 42) = min(8, 13) = 8.

Cf. A000045, A001076, A004695, A051699, A053188, A053615, A053616, A053646, A061489, A061670, A074989, A081134, A105446.

fibPi[n_] := 1 + Floor[ Log[ GoldenRatio, 1 + n*Sqrt@5]]; f[n_] := Block[{m = fibPi@ n}, Min[n - Fibonacci[m -1], Fibonacci[m] - n]]; Array[f, 81, 0] (* Robert G. Wilson v, Dec 11 2017 *)
With[{nn=80,fibs=Fibonacci[Range[0,20]]},Table[Abs[n-Nearest[fibs,n]][[1]],{n,0,nn}]] (* Harvey P. Dale, Jul 02 2022 *)

a(n) = for (i=1, oo, if (n<=fibonacci(i), return (min(n-fibonacci(i-1), fibonacci(i)-n))))

a(n) = abs(n - Fibonacci(floor(log(sqrt(20)*n)/log((1 + sqrt(5))/2)-1))). - Jon E. Schoenfield, Dec 14 2017

A015541 Expansion of x/(1 - 5x - 7x^2).

Original entry on oeis.org

0, 1, 5, 32, 195, 1199, 7360, 45193, 277485, 1703776, 10461275, 64232807, 394392960, 2421594449, 14868722965, 91294775968, 560554940595, 3441838134751, 21133075257920, 129758243232857, 796722742969725, 4891921417478624, 30036666288181195

Offset: 0

Olivier Gérard

Pisano period lengths: 1, 3, 8, 6, 8, 24, 6, 6, 24, 24, 5, 24, 12, 6, 8, 12, 16, 24, 120, 24, ... - R. J. Mathar, Aug 10 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,7).

Cf. A001076, A006190, A007482, A015520, A015521, A015523, A015524, A015525, A015528, A015529, A015530, A015531, A015532, A015533, A015534, A015535, A015536, A015537, A015443, A015447, A030195, A053404, A057087, A057088, A083858, A085939, A090017, A091914, A099012, A180222, A180226.

[n le 2 select n-1 else 5*Self(n-1) + 7*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 13 2012

Join[{a=0,b=1},Table[c=5*b+7*a;a=b;b=c,{n,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 16 2011 *)
LinearRecurrence[{5, 7}, {0, 1}, 30] (* Vincenzo Librandi, Nov 13 2012 *)

x='x+O('x^30); concat([0], Vec(x/(1-5*x-7*x^2))) \\ G. C. Greubel, Jan 24 2018

[lucas_number1(n,5,-7) for n in range(0, 21)] # Zerinvary Lajos, Apr 24 2009

a(n) = 5*a(n-1) + 7*a(n-2).

A015544 Lucas sequence U(5,-8): a(n+1) = 5a(n) + 8a(n-1), a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 5, 33, 205, 1289, 8085, 50737, 318365, 1997721, 12535525, 78659393, 493581165, 3097180969, 19434554165, 121950218577, 765227526205, 4801739379641, 30130517107845, 189066500576353, 1186376639744525, 7444415203333449, 46713089134623445

Offset: 0

Olivier Gérard

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,8).

Cf. A001076, A006190, A007482, A015520, A015521, A015523, A015524, A015525, A015528, A015529, A015530, A015531, A015532, A015533, A015534, A015535, A015536, A015537, A015441, A015443, A015447, A030195, A053404, A057087, A057088, A083858, A085939, A090017, A091914, A099012, A180222, A180226, A015555 (binomial transform).

[n le 2 select n-1 else 5*Self(n-1) + 8*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 13 2012

a[n_]:=(MatrixPower[{{1,2},{1,-6}},n].{{1},{1}})[[2,1]]; Table[Abs[a[n]],{n,-1,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
LinearRecurrence[{5, 8}, {0, 1}, 30] (* Vincenzo Librandi, Nov 13 2012 *)

A015544(n)=imag((2+quadgen(57))^n) \\ M. F. Hasler, Mar 06 2009

x='x+O('x^30); concat([0], Vec(x/(1 - 5*x - 8*x^2))) \\ G. C. Greubel, Jan 01 2018

[lucas_number1(n,5,-8) for n in range(0, 21)] # Zerinvary Lajos, Apr 24 2009

a(n) = 5*a(n-1) + 8*a(n-2).

G.f.: x/(1 - 5*x - 8*x^2). - M. F. Hasler, Mar 06 2009

More precise definition by M. F. Hasler, Mar 06 2009

A024551 a(n) = floor(a(n-1)/(sqrt(5) - 2)) for n > 0 and a(0) = 1.

Original entry on oeis.org

1, 4, 16, 67, 283, 1198, 5074, 21493, 91045, 385672, 1633732, 6920599, 29316127, 124185106, 526056550, 2228411305, 9439701769, 39987218380, 169388575288, 717541519531, 3039554653411, 12875760133174, 54542595186106, 231046140877597

Offset: 0

Clark Kimberling

Clark Kimberling, Table of n, a(n) for n = 0..250
Index entries for linear recurrences with constant coefficients, signature (5,-3,-1).

a[0] = 1;
a[n_] := Floor[a[n - 1]/FractionalPart[Sqrt[5]]]
Table[a[n], {n, 0, 60}]
(* Clark Kimberling, Aug 16 2012 *)
a[0]=1;
a[1]=4;
a[2]=16;
a[n_]:=Floor[a[n-1]^2/a[n-2]]+3
Table[a[n],{n,0,60}]
With[{c=Sqrt[5]-2},NestList[Floor[#/c]&,1,30]] (* Harvey P. Dale, Jul 18 2018 *)

a(n)=([0,1,0; 0,0,1; -1,-3,5]^n*[1;4;16])[1,1] \\ Charles R Greathouse IV, Jan 20 2017

step(n)=2*n + sqrtint(5*n^2)
a(n)=if(n, step(a(n-1)), 1) \\ Charles R Greathouse IV, Jan 20 2017

a(n) = 5*a(n-1) - 3*a(n-2) - a(n-3). - Clark Kimberling, Aug 16 2012
G.f.: (-x^2-x+1)/[(1-x)(1-4x-x^2)].
a(n) = (3*Fibonacci(3*n+2) + 1)/4 = 1 + 3*Sum_{k=0..n} A001076(k). - Ehren Metcalfe, Apr 15 2019

A099014 a(n) = Fibonacci(n)*(Fibonacci(n-1)^2 + Fibonacci(n+1)^2).

Original entry on oeis.org

0, 1, 5, 20, 87, 365, 1552, 6565, 27825, 117844, 499235, 2114729, 8958240, 37947545, 160748653, 680941780, 2884516383, 12219006325, 51760543280, 219261176861, 928805254905, 3934482189716, 16666734024715, 70601418270865

Offset: 0

Paul Barry, Sep 22 2004

Form the matrix A=[1,1,1,1;3,2,1,0;3,1,0,0;1,0,0,0]=(binomial(3-j,i)). Then a(n)=(2,3)-element of A^n.

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..173 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (3,6,-3,-1).

Cf. A000045, A056570, A066258, A066259.

[Fibonacci(n)*(Fibonacci(n-1)^2+Fibonacci(n+1)^2): n in [0..30]]; // Vincenzo Librandi, Jun 05 2011

CoefficientList[Series[x*(1 + 2*x - x^2)/(1 - 3*x - 6*x^2 + 3*x^3 + x^4), {x, 0, 50}], x] (* G. C. Greubel, Dec 31 2017 *)
Join[{0},#[[2]](#[[1]]^2+#[[3]]^2)&/@Partition[Fibonacci[ Range[ 0,30]],3,1]] (* or *) LinearRecurrence[{3,6,-3,-1},{0,1,5,20},30] (* Harvey P. Dale, Oct 17 2021 *)

a(n)=fibonacci(n)*(fibonacci(n-1)^2+fibonacci(n+1)^2) \\ Charles R Greathouse IV, Jun 05 2011

G.f.: x*(1+2*x-x^2)/(1-3*x-6*x^2+3*x^3+x^4) = x*(1+2*x-x^2)/((1+x-x^2)*(1-4*x-x^2)).
a(n) = Sum_{k=0..n} (-1)^(k+1)*Fib(k)*(0^(n-k) + 6*A001076(n-k)).
a(n) = ((-1)^n*Fib(n) + 3*Fib(3*n))/5. - Ehren Metcalfe, May 21 2016

A110527 a(n+3) = 3a(n+2) + 5a(n+1) + a(n), a(0) = 0, a(1) = 1, a(2) = 8.

Original entry on oeis.org

0, 1, 8, 29, 128, 537, 2280, 9653, 40896, 173233, 733832, 3108557, 13168064, 55780809, 236291304, 1000946021, 4240075392, 17961247585, 76085065736, 322301510525, 1365291107840, 5783465941881, 24499154875368

Offset: 0

Creighton Dement, Jul 24 2005

A048878(n) = a(n) + a(n+1). Compare with A110526.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,5,1).

Cf. A110526, A110528, A033887, A001076, A049661, A033887.

seriestolist(series(-x*(1+5*x)/((1+x)*(x^2+4*x-1)), x=0,25)); -or- Floretion Algebra Multiplication Program, FAMP Code: 1lesseq[(- 'i + 'j - i' + j' - 'kk' - 'ik' - 'jk' - 'ki' - 'kj')(+ .5'i + .5i' + .5'jj' + .5'kk')], apart from initial term.

LinearRecurrence[{3,5,1},{0,1,8},30] (* Harvey P. Dale, Feb 12 2015 *)

x='x+O('x^50); concat(0, Vec(-x*(1+5*x)/((1+x)*(x^2+4*x-1)))) \\ G. C. Greubel, Aug 30 2017

G.f.: -x*(1+5*x)/((1+x)*(x^2+4*x-1)).
a(n) = (-1)^n + 3*A001076(n) - A015448(n). - Ehren Metcalfe, Nov 18 2017
a(n) = (-1)^n + 2*A110526(n) + A110679(n-2) for n >= 2. - Yomna Bakr and Greg Dresden, May 25 2024

cp's OEIS Frontend

A163070 a(n) = ((4+sqrt(5))*(2+sqrt(5))^n + (4-sqrt(5))*(2-sqrt(5))^n)/2.

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A189800 a(n) = 6*a(n-1) + 8*a(n-2), with a(0)=0, a(1)=1.

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A203319 a(n) = n*Fibonacci(n) * Sum_{d|n} 1/(d*Fibonacci(d)).

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A232970 Expansion of (1-3*x)/(1-5*x+3*x^2+x^3).

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A296239 a(n) = distance from n to nearest Fibonacci number.

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A015541 Expansion of x/(1 - 5*x - 7*x^2).

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A015544 Lucas sequence U(5,-8): a(n+1) = 5*a(n) + 8*a(n-1), a(0)=0, a(1)=1.

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A163070 a(n) = ((4+sqrt(5))(2+sqrt(5))^n + (4-sqrt(5))(2-sqrt(5))^n)/2.

A189800 a(n) = 6a(n-1) + 8a(n-2), with a(0)=0, a(1)=1.

A203319 a(n) = nFibonacci(n) Sum_{d|n} 1/(d*Fibonacci(d)).

A232970 Expansion of (1-3x)/(1-5x+3*x^2+x^3).

A015541 Expansion of x/(1 - 5x - 7x^2).

A015544 Lucas sequence U(5,-8): a(n+1) = 5a(n) + 8a(n-1), a(0)=0, a(1)=1.

A110527 a(n+3) = 3a(n+2) + 5a(n+1) + a(n), a(0) = 0, a(1) = 1, a(2) = 8.