A001076 -id:A001076 - cp's OEIS frontend

A117715 Triangle, read by rows, T(n, k) = Fibonacci(n, k), where Fibonacci(n, x) is the Fibonacci polynomial.

0, 1, 1, 0, 1, 2, 1, 2, 5, 10, 0, 3, 12, 33, 72, 1, 5, 29, 109, 305, 701, 0, 8, 70, 360, 1292, 3640, 8658, 1, 13, 169, 1189, 5473, 18901, 53353, 129949, 0, 21, 408, 3927, 23184, 98145, 328776, 927843, 2298912, 1, 34, 985, 12970, 98209, 509626, 2026009, 6624850, 18674305, 46866034

Offset: 0

Roger L. Bagula, Apr 13 2006

			Triangle begins as:
  0;
  1,  1;
  0,  1,   2;
  1,  2,   5,   10;
  0,  3,  12,   33,   72;
  1,  5,  29,  109,  305,   701;
  0,  8,  70,  360, 1292,  3640,  8658;
  1, 13, 169, 1189, 5473, 18901, 53353, 129949;

Steven Wolfram, The Mathematica Book, Cambridge University Press, 3rd ed. 1996, page 728

Alois P. Heinz, Rows n = 0..140, flattened
Eric Weisstein's World of Mathematics, Fibonacci Polynomial.
Wikipedia, Fibonacci Polynomial

Cf. A000045, A117716, A049310, A073133, A157103 (antidiagonals).
Main diagonal and first lower diagonal give: A084844, A084845.
Cf. A352361.

A117715:= func< n, k | k eq 0 select (n mod 2) else Evaluate(DicksonSecond(n-1, -1), k) >;
[A117715(n, k): k in [0..n], n in [0..13]]; // G. C. Greubel, Oct 01 2024

with(combinat):for n from 0 to 9 do seq(fibonacci(n,m), m = 0 .. n) od; # Zerinvary Lajos, Apr 09 2008

Table[Fibonacci[n, k], {n,0,12}, {k,0,n}]//Flatten

from sympy import fibonacci
def T(n, m): return 0 if n==0 else fibonacci(n, m)
for n in range(21): print([T(n, m) for m in range(n + 1)]) # Indranil Ghosh, Aug 12 2017

def A117715(n,k): return lucas_number1(n, k, -1)
flatten([[A117715(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Oct 01 2024

T(n, 1) = A000045(n).
T(n, 3) = A006190(n).
T(n, 4) = A001076(n).
T(n, 5) = A052918(n-1).
T(5, k) = A057721(k).
T(6, k) = A124152(k).
T(n, k) = (-1)^(n-1)*A352361(n-k, n). - G. C. Greubel, Oct 01 2024

Definition simplified by the Assoc. Editors of the OEIS, Nov 17 2009

A123747 Numerators of partial sums of a series for sqrt(5).

Original entry on oeis.org

1, 7, 41, 9, 239, 6227, 32059, 163727, 166301, 841229, 21215481, 106782837, 536618341, 538698461, 172897, 13538601629, 67813224223, 339532842359, 339895847771, 1700893049407, 42549895540939, 212857129279583, 1064706466190659, 1065035803419763, 5326468921246139

Offset: 0

Wolfdieter Lang, Nov 10 2006

Denominators are given by A123748.

The sum over central binomial coefficients scaled by powers of 5, r(n) = Sum_{k=0..n} binomial(2*k,k)/5^k, has the limit lim_{n -> infinity} r(n) = sqrt(5). From the expansion of 1/sqrt(1-x) for x=4/5.

			a(3) = 9 because r(3) = 1+2/5+6/25+4/25 = 9/5 = a(3)/A123748(3).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Wolfdieter Lang, Rationals and more.

Cf. A001077/A001076 continued fraction convergents for sqrt(5).

Cf. A123748, A123749.

List([0..25], n-> NumeratorRat(Sum([0..n], k-> Binomial(2*k,k)/5^k )) ); # G. C. Greubel, Aug 10 2019

[Numerator( (&+[Binomial(2*k,k)/5^k: k in [0..n]])): n in [0..25]]; // G. C. Greubel, Aug 10 2019

A123747:=n-> numer(sum(binomial(2*k,k)/5^k, k=0..n)); seq(A123747(n), n=0..25); # G. C. Greubel, Aug 10 2019

Table[Numerator[Sum[Binomial[2*k, k]/5^k, {k,0,n}]], {n, 0, 25}] (* G. C. Greubel, Aug 10 2019 *)

vector(25, n, n--; numerator(sum(k=0,n, binomial(2*k,k)/5^k))) \\ G. C. Greubel, Aug 10 2019

[numerator( sum(binomial(2*k,k)/5^k for k in (0..n)) ) for n in (0..25)] # G. C. Greubel, Aug 10 2019

a(n) = numerator(r(n)) with the rationals r(n) = Sum_{k=0..n} binomial(2*k,k)/5^k, in lowest terms.

r(n) = Sum_{k=0..n} ((2*k-1)!!/((2*k)!!))*(4/5)^k, n>=0, with the double factorials A001147 and A000165.

A270863 Self-composition of the Fibonacci sequence.

Original entry on oeis.org

0, 1, 2, 6, 17, 50, 147, 434, 1282, 3789, 11200, 33109, 97878, 289354, 855413, 2528850, 7476023, 22101326, 65338038, 193158521, 571033600, 1688143881, 4990651642, 14753839486, 43616704857, 128943855250, 381196100507, 1126928202714, 3331532438042, 9848993360069

Offset: 0

Oboifeng Dira, Mar 24 2016

This sequence has the same relation to the Fibonacci numbers A000045 as A030267 has to the natural numbers A000027.
From Oboifeng Dira, Jun 28 2020: (Start)
This sequence can be generated from a family of composition pairs of generating functions g(f(x)), where k is an integer and where
f(x) = x/(1-k*x-x^2) and g(x) = (x+(k-1)*x^2)/(1-(3-2*k)*x-(3*k-k^2-1)*x^2).
Some cases of k values are:
k=-5, f(x) g.f. 0,A052918(-1)^n and g(x) g.f. 0,A081571
k=-4, f(x) g.f. A001076(-1)^(n+1) and g(x) g.f. 0,A081570
k=-3, f(x) g.f. A006190(-1)^(n+1) and g(x) g.f. 0,A081569
k=-2, f(x) g.f. A215936(n+2) and g(x) g.f. 0,A081568
k=-1, f(x) g.f. A039834(n+2) and g(x) g.f. 0,A081567
k=0,  f(x) g.f. A000035 and g(x) g.f. 0,A001519(n+1)
k=1,  f(x) g.f. A000045 and g(x) g.f. A000045
k=2,  f(x) g.f. A000129 and g(x) g.f. 0,A039834(n+1)
k=3,  f(x) g.f. A006190 and g(x) g.f. 0,A001519(-1)^n
k=4,  f(x) g.f. A001076 and g(x) g.f. 0,A093129(-1)^n
k=5,  f(x) g.f. 0,A052918 and g(x) g.f. 0,A192240(-1)^n
k=6,  f(x) g.f. A005668 and g(x)=(x+5*x^2)/(1+9*x+19*x^2)
k=7,  f(x) g.f. 0,A054413 and g(x)=(x+6*x^2)/(1+11*x+29*x^2).
(End)

			a(5) = 3*a(4)+a(3)-3*a(2)-a(1) = 51+6-6-1 = 50.

Colin Barker, Table of n, a(n) for n = 0..1000
Oboifeng Dira, A Note on Composition and Recursion, Southeast Asian Bulletin of Mathematics (2017), Vol. 41, Issue 6, 849-853.
Oboifeng Dira, Family of composition pairs g(f(x)) generating A270683
Index entries for linear recurrences with constant coefficients, signature (3,1,-3,-1).

Cf. A000027, A000045, A001622, A030267, A000035, A001519, A000129, A039834.

I:=[0, 1, 2, 6]; [m le 4 select I[m] else 3*Self(m-1)+Self(m-2)-3*Self(m-3)-Self(m-4): m in [1..30]]; // Marius A. Burtea, Aug 03 2019

f:= x-> x/(1-x-x^2):
a:= n-> coeff(series(f(f(x)), x, n+1), x, n):
seq(a(n), n=0..30);

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,-3,1,3]^(n-1)*[1;2;6;17])[1,1] \\ Charles R Greathouse IV, Mar 24 2016

concat(0, Vec(x*(1-x-x^2)/(1-3*x-x^2+3*x^3+x^4) + O(x^40))) \\ Colin Barker, Mar 24 2016

a(n) = 3*a(n-1)+a(n-2)-3*a(n-3)-a(n-4) for n > 3, a(0)=0, a(1)=1, a(2)=2, a(3)=6.
G.f.: x*(1-x-x^2) / (1-3*x-x^2+3*x^3+x^4). - Colin Barker, Mar 24 2016
G.f.: B(B(x)) where B(x) is the g.f. of A000045. - Joerg Arndt, Mar 25 2016
a(n) = (phi*((phi^2 + 5^(1/4)*sqrt(3*phi))^n - (phi^2 - 5^(1/4)*sqrt(3*phi))^n) + (psi^2 + 5^(1/4)*sqrt(3*psi))^n - (psi^2 - 5^(1/4)*sqrt(3*psi))^n)/(2^n * 5^(3/4) * sqrt(3*phi)), where phi = (sqrt(5) + 1)/2 is the golden ratio, and psi = 1/phi = (sqrt(5) - 1)/2. - Vladimir Reshetnikov, Aug 01 2019
0 = a(n)*(a(n) +6*a(n+1) -a(n+2)) +a(n+1)*(8*a(n+1) -9*a(n+2) +a(n+3)) +a(n+2)*(-8*a(n+2) +6*a(n+3)) +a(n+3)*(-a(n+3)) if n>=0. - Michael Somos, Feb 05 2022

A056565 Fibonomial coefficients.

Original entry on oeis.org

1, 21, 714, 19635, 582505, 16776144, 488605194, 14169550626, 411591708660, 11948265189630, 346934172869802, 10072785423545712, 292460526776698763, 8491396839675395415, 246543315138161480670, 7158243695757340957617, 207835653079349665473587

Offset: 0

Wolfdieter Lang, Jul 10 2000

Index entries for linear recurrences with constant coefficients, signature (21,273,-1092,-1820,1092,273,-21,-1).

Cf. A010048, A000045, A001654-8, A001076, A049666 (signed), A049667.

[ &*[Fibonacci(n+k): k in [0..6]]/3120: n in [1..16] ]; // Bruno Berselli, Apr 11 2011

(Times@@@Partition[Fibonacci[Range[30]],7,1])/3120  (* Harvey P. Dale, Apr 10 2011 *)

b(n, k)=prod(j=1, k, fibonacci(n+j)/fibonacci(j));
vector(20, n, b(n-1, 7))  \\ Joerg Arndt, May 08 2016

a(n) = A010048(n+7, 7) =: Fibonomial(n+7, 7).
G.f.: 1/p(8, n) with p(8, n) = 1 - 21*x - 273*x^2 + 1092*x^3 + 1820*x^4 - 1092*x^5 - 273*x^6 + 21*x^7 + x^8 = (1 + x - x^2) * (1 - 4*x - x^2) * (1 + 11*x - x^2) * (1 - 29*x - x^2) (n=8 row polynomial of signed Fibonomial triangle A055870; see this entry for Knuth and Riordan references).
a(n) = 29*a(n-1) + a(n-2) + ((-1)^n) * A001657(n), n >= 2, a(0)=1, a(1)=21.
G.f.: exp( Sum_{k>=1} F(8*k)/F(k) * x^k/k ), where F(n) = A000045(n). - Seiichi Manyama, May 07 2025

Offset corrected by Seiichi Manyama, May 07 2025

A091870 A trinomial transform of Fibonacci(3n).

Original entry on oeis.org

0, 1, 8, 53, 336, 2105, 13144, 81997, 511392, 3189169, 19888040, 124023461, 773419248, 4823095913, 30077155576, 187563189565, 1169656805184, 7294059356257, 45486249993032, 283655347025429, 1768894026280080

Offset: 0

Paul Barry, Feb 06 2004

Binomial transform of A084326.

Second binomial transform of A001076(n) = Fibonacci(3n)/2.

Seiichi Manyama, Table of n, a(n) for n = 0..1258
S. Falcon, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014.
Index entries for linear recurrences with constant coefficients, signature (8,-11).

Cf. A084326.

a:=[0,1];; for n in [3..30] do a[n]:=8*a[n-1]-11*a[n-2]; od; a; # G. C. Greubel, May 21 2019

[n le 2 select n-1 else 8*Self(n-1) -11*Self(n-2): n in [1..30]]; // G. C. Greubel, May 21 2019

LinearRecurrence[{8, -11}, {0, 1}, 30] (* G. C. Greubel, May 21 2019 *)
CoefficientList[Series[x/(1 - 8 x + 11 x^2), {x, 0, 30}], x] (* Michael De Vlieger, Sep 22 2017 *)

my(x='x+O('x^30)); concat([0], Vec(x/(1 -8*x +11*x^2))) \\ G. C. Greubel, May 21 2019

[lucas_number1(n,8,11) for n in range(0, 30)] # Zerinvary Lajos, Apr 23 2009

G.f.: x/(1 - 8*x + 11*x^2).
a(n) = sqrt(5) * ((4+sqrt(5))^n - (4-sqrt(5))^n) / 10.
a(n) = Sum_{i=0..n} Sum_{j=0..n} (n!/(i!*j!*(n-i-j)!)) * Fibonacci(3*i) / 2.

A106707 a(n) = -A001353(n).

Original entry on oeis.org

0, -1, -4, -15, -56, -209, -780, -2911, -10864, -40545, -151316, -564719, -2107560, -7865521, -29354524, -109552575, -408855776, -1525870529, -5694626340, -21252634831, -79315912984, -296011017105, -1104728155436, -4122901604639, -15386878263120, -57424611447841

Offset: 0

Roger L. Bagula, May 30 2005

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 (4,-1).

Cf. A001076, A001353.

I:=[0,-1]; [n le 2 select I[n] else 4*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Feb 05 2018

a[0]:=0: a[1]:=-1: for n from 2 to 27 do a[n]:=4*a[n-1]-a[n-2] od: seq(a[n],n=0..27);

LinearRecurrence[{4,-1},{0,-1},30] (* Harvey P. Dale, Nov 01 2019 *)

x='x+O('x^30); Vec(-x/(1-4*x+x^2)) \\ G. C. Greubel, Feb 05 2018

G.f.: -x/(1-4*x+x^2).

a(n) = 4*a(n-1) - a(n-2); a(0)=0, a(1)=-1.

Edited by N. J. A. Sloane, Apr 30 2006

New name from Joerg Arndt, Sep 22 2023

A042937 Denominators of continued fraction convergents to sqrt(1000).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 53, 114, 281, 4329, 8939, 22207, 142181, 164388, 306569, 470957, 777526, 1248483, 78183472, 79431955, 157615427, 237047382, 394662809, 631710191, 4184923955, 9001558101, 22188040157, 341822160456, 705832361069, 1753486882594

Offset: 0

N. J. A. Sloane

			sqrt(1000) = 31.62... = 31 + 1/(1 + 1/(1 + ...)) with convergents 31/1, 32/1, 63/2, 95/3, 158/5, ... - _M. F. Hasler_, Nov 02 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 78960998, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1).

Cf. A042936 (numerators), A040968 (continued fraction), A010467 (decimals).

Analog for sqrt(m): A000129 (m=2), A002530 (m=3), A001076 (m=5), A041007 (m=6), A041009 (m=7), A041011 (m=8), A005663 (m=10), A041015 (m=11), A041017 (m=12), ..., A042933 (m=998), A042935 (m=999).

Denominator[Convergents[Sqrt[1000], 30]] (* Vincenzo Librandi, Feb 01 2014 *)

A42937=contfracpnqn(c=contfrac(sqrt(1000)),#c-1)[2,] \\ Possibly incorrect last term ignored. NB: a(n) = A42937[n+1]. For more terms use e.g. \p999, or compute any a(n) from this as in A042936. - M. F. Hasler, Nov 01 2019

More terms from Vincenzo Librandi, Feb 01 2014

A048878 Generalized Pellian with second term of 9.

Original entry on oeis.org

1, 9, 37, 157, 665, 2817, 11933, 50549, 214129, 907065, 3842389, 16276621, 68948873, 292072113, 1237237325, 5241021413, 22201322977, 94046313321, 398386576261, 1687592618365, 7148757049721, 30282620817249, 128279240318717, 543399582092117, 2301877568687185

Offset: 0

Barry E. Williams

			a(n) = 4a(n-1) + a(n-2); a(0)=1, a(1)=9.

Harry J. Smith, Table of n, a(n) for n = 0..1584
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (4,1).

Cf. A000045, A015448, A001077, A001076, A033887.

with(combinat): a:=n->5*fibonacci(n-1,4)+fibonacci(n,4): seq(a(n), n=1..16); # Zerinvary Lajos, Apr 04 2008

LinearRecurrence[{4,1},{1,9},31] (* or *) CoefficientList[ Series[ (1+5x)/(1-4x-x^2),{x,0,30}],x] (* Harvey P. Dale, Jul 12 2011 *)

{ default(realprecision, 2000); for (n=0, 2000, a=round(((7+sqrt(5))*(2+sqrt(5))^n - (7-sqrt(5))*(2-sqrt(5))^n )/10*sqrt(5)); if (a > 10^(10^3 - 6), break); write("b048878.txt", n, " ", a); ); } \\ Harry J. Smith, May 31 2009

a(n) = ( (7+sqrt(5))(2+sqrt(5))^n - (7-sqrt(5))(2-sqrt(5))^n )/2*sqrt(5).
G.f.: (1+5*x)/(1-4*x-x^2). - Philippe Deléham, Nov 03 2008
a(n) = F(3*n+3) + F(3*n-2); F = A000045. - Yomna Bakr and Greg Dresden, May 25 2024

A094292 Number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < 5 and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, s(0) = 2, s(n) = 4.

Original entry on oeis.org

0, 0, 1, 3, 9, 25, 68, 182, 483, 1275, 3355, 8811, 23112, 60580, 158717, 415715, 1088661, 2850645, 7463884, 19541994, 51163695, 133951675, 350695511, 918141623, 2403740304, 6293097000, 16475579353, 43133687427, 112925557953, 295643107825, 774003961940

Offset: 0

Herbert Kociemba, Jun 02 2004

In general, a(n,m,j,k) = (2/m)*Sum_{r=1..m-1} sin(j*r*Pi/m)*sin(k*r*Pi/m)*(1+2*cos(Pi*r/m))^n is the number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < m and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, s(0) = j, s(n) = k.
a(n+1) is an inverse Catalan transform of F(3n)/F(3). The g.f. may be obtained from that of A001076 under the mapping g(x)-> g(x(1-x)). - Paul Barry, Nov 17 2004
A transform of Fibonacci(2n): Fibonacci(2n) may be recovered as Sum_{k=0..2n} Sum_{j=0..k} binomial(0,2n-k)*binomial(k,j)(-1)^(k-j)*A094292(j). - Paul Barry, Jun 10 2005

É. Czabarka, R. Flórez, and L. Junes, A Discrete Convolution on the Generalized Hosoya Triangle, Journal of Integer Sequences, 18 (2015), #15.1.6.
Index entries for linear recurrences with constant coefficients, signature (4,-3,-2,1).

Cf. A000032, A000045, A049681.

Table[Sum[Fibonacci[n - 1 + i]/2, {i, 0, n - 1}], {n, 0, 27}]  (* Zerinvary Lajos, Jul 12 2009 *)
Table[Fibonacci[n] (LucasL[n] - 1)/2, {n, 0, 20}] (* Vladimir Reshetnikov, Sep 27 2016 *)

(numlib::fibonacci(2*n)-numlib::fibonacci(n))/2 $ n = 0..35; // Zerinvary Lajos, May 09 2008

a(n) = (fibonacci(2*n) - fibonacci(n))/2; \\ Altug Alkan, Dec 17 2017

a(n) = (2/5)*Sum_{k=1..4} sin(2*Pi*k/5)*sin(4*Pi*k/5)*(1+2*cos(Pi*k/5))^n.
From Herbert Kociemba, Jun 16 2004: (Start)
a(n) = 4*a(n-1) - 3*a(n-2) - 2*a(n-3) + a(n-4).
G.f.: (x^2-x^3)/(1 - 4x + 3x^2 + 2x^3 - x^4). (End)
a(n) = (Fibonacci(2*n) - Fibonacci(n))/2. - Vladeta Jovovic, Jul 17 2004
a(n+1) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-1)^k*F(3n-3k)/F(3). - Paul Barry, Nov 17 2004
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2k)*Fibonacci(2k). - Paul Barry, Jun 10 2005
a(n) = Sum_{k=0..n-1} Fibonacci(n+k-1)/2. - Gary Detlefs, Feb 22 2011
a(n) = Fibonacci(n)*(Lucas(n) - 1)/2. - Vladimir Reshetnikov, Sep 27 2016

a(0) = a(1) = 0 added and offset changed to 0 by Vladimir Reshetnikov, Oct 04 2016

A110526 a(n+3) = 3a(n+2) + 5a(n+1) + a(n), a(0) = 0, a(1) = 1, a(2) = 3.

Original entry on oeis.org

0, 1, 3, 14, 58, 247, 1045, 4428, 18756, 79453, 336567, 1425722, 6039454, 25583539, 108373609, 459077976, 1944685512, 8237820025, 34895965611, 147821682470, 626182695490, 2652552464431, 11236392553213, 47598122677284

Offset: 0

Creighton Dement, Jul 24 2005

A001076(n) = a(n) + a(n+1). Program "Superseeker" finds: A033887(n+1) = a(n+2) - a(n); Elements of even index in the sequence: A049661(n) = (F(6n+1)-1)/4; A015448(n+2) = a(n+2) + 2*a(n+1) + a(n)

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, A110527, A033887, A001076, A049661, A033887, A000045.

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

Table[(Fibonacci[3n+1]-(-1)^n)/4, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 28 2015 *)

concat(0, Vec(x/((1+x)*(1-x^2-4*x)) + O(x^100))) \\ Altug Alkan, Oct 28 2015

G.f.: -x/((1+x)*(x^2+4*x-1)).
a(n) = (-1)^n/2 * Sum_{k=0..n} (-1)^k*Fibonacci(3*k). - Gary Detlefs, Jan 03 2013
a(n) = (Fibonacci(3*n+1)-(-1)^n)/4, where Fibonacci(n) = A000045(n). - Vladimir Reshetnikov, Oct 28 2015

cp's OEIS Frontend

A117715 Triangle, read by rows, T(n, k) = Fibonacci(n, k), where Fibonacci(n, x) is the Fibonacci polynomial.

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A123747 Numerators of partial sums of a series for sqrt(5).

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A270863 Self-composition of the Fibonacci sequence.

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A056565 Fibonomial coefficients.

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A091870 A trinomial transform of Fibonacci(3n).

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A106707 a(n) = -A001353(n).

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A110526 a(n+3) = 3a(n+2) + 5a(n+1) + a(n), a(0) = 0, a(1) = 1, a(2) = 3.