A014445 -id:A014445 - cp's OEIS frontend

A099930 a(n) = Pell(n) * Pell(n-1) * Pell(n-2) / 10.

1, 12, 174, 2436, 34307, 482664, 6791772, 95567064, 1344731653, 18921807828, 266250046986, 3746422451772, 52716164405255, 741772724044560, 10437534301224120, 146867252940711408, 2066579075472320521, 29078974309550454492, 409172219409185308518, 5757490046038128779316

Offset: 3

Ralf Stephan, Nov 03 2004

Michael A. Allen and Kenneth Edwards, Fence tiling derived identities involving the metallonacci numbers squared or cubed, Fib. Q. 60:5 (2022) 5-17.
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006.
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
Ronald Orozco López, Generating Functions of Generalized Simplicial Polytopic Numbers and (s,t)-Derivatives of Partial Theta Function, arXiv:2408.08943 [math.CO], 2024. See p. 13.
Ronald Orozco López, Simplicial d-Polytopic Numbers Defined on Generalized Fibonacci Polynomials, arXiv:2501.11490 [math.CO], 2025. See p. 10.
Index entries for linear recurrences with constant coefficients, signature (12,30,-12,-1).

Cf. A000129. Third column of triangle A099927. Cf. A001656, A084175.

Drop[CoefficientList[Series[x^3/((1+2x-x^2)(1-14x-x^2)),{x,0,20}],x],3] (* or *) LinearRecurrence[{12,30,-12,-1},{1,12,174,2436},20] (* Harvey P. Dale, Feb 26 2012 *)

G.f.: x^3 / ((1+2*x-x^2)*(1-14*x-x^2)).
a(n) = 12*a(n-1) + 30*a(n-2) - 12*a(n-3) - a(n-4); a(3)=1, a(4)=12, a(5)=174, a(6)=2436. - Harvey P. Dale, Feb 26 2012
From Peter Bala, Mar 30 2015: (Start)
The following remarks assume an offset of 0.
The o.g.f. A(x) = 1/( (1 + 2*x - x^2)*(1 - 14*x - x^2) ). Hence A(x) (mod 4) = 1/(1 + 2*x - x^2)^2 (mod 4). It follows by Theorem 1 of Heninger et al. that sqrt(A(x)) = 1 + 6*x + 69*x^2 + 804*x^3 + ... has integral coefficients.
Sum_{n >= 0} a(n+3)*x^n = exp( Sum_{n >= 1} Pell(4*n)/Pell(n)*x^n/n ). Cf. A001656, A084175. (End)
a(n+1) = (1/2)*Sum_{k=1..n-1} ( A014445(k)*A110272(n-k) ) for n > 1. - Michael A. Allen, Jan 25 2023

A074723 Largest power of 2 dividing F(3n) where F(k) is the k-th Fibonacci number.

Original entry on oeis.org

2, 8, 2, 16, 2, 8, 2, 32, 2, 8, 2, 16, 2, 8, 2, 64, 2, 8, 2, 16, 2, 8, 2, 32, 2, 8, 2, 16, 2, 8, 2, 128, 2, 8, 2, 16, 2, 8, 2, 32, 2, 8, 2, 16, 2, 8, 2, 64, 2, 8, 2, 16, 2, 8, 2, 32, 2, 8, 2, 16, 2, 8, 2, 256, 2, 8, 2, 16, 2, 8, 2, 32, 2, 8, 2, 16, 2, 8, 2, 64, 2, 8, 2, 16, 2, 8, 2, 32, 2, 8, 2

Offset: 1

Benoit Cloitre, Sep 04 2002

nonn

If m == 1 or 2 (mod 3) then F(m) is odd.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000045, A001620, A006519, A014445, A297402.

seq(`if`(n::odd,2,2^(2+padic:-ordp(n,2))),n=1..100); # Robert Israel, Oct 10 2016

Table[2^(Length@ NestWhileList[#/2 &, Fibonacci[3 n], IntegerQ] - 2), {n, 120}] (* Michael De Vlieger, Oct 10 2016 *)
a[n_] := If[EvenQ[n], 2^(FactorInteger[n][[1]][[2]] + 2), 2]; Array[a, 90] (* Frank M Jackson, Jul 28 2018 *)

a(n) = 2^valuation(fibonacci(3*n), 2); \\ Michel Marcus, Oct 10 2016

It appears that 4 never appears and : a(2k+1)=2 a(2^m*(2k+1))=2^(m+2) for k>=0 and m >=1.
From Robert Israel, Oct 10 2016: (Start)
a(2k+1)=2 follows from F(n+6) = 5 F(n) + 8 F(n+1) == F(n) mod 4.
a(2*(2k+1))=8 follows from F(n+12) = 89 F(n) + 144 F(n+1) == 9 F(n) mod 16.
a(2^m*(2k+1)) = 2^(m+2) for m > 2 follows from F(2n) = F(n) (2 F(n-1) + F(n)).
G.f. 2*x/(1-x^2) + Sum_{m>=1} 2^(m+2)*x^(2^m)/(1 - x^(2^(m+1))). (End)
a(n) = A006519(A014445(n)). - Michel Marcus, Oct 10 2016
As proved above, for m > 0, a(2m-1) = 2 and a(2m) = 2^(k+2) where k is the exponent of the even prime in the prime factorization of 2m. Also a(n) = 2*A297402(n). - Frank M Jackson, Jul 28 2018
Sum_{k=1..n} a(k) ~ (2*n/log(2)) * (log(n) + gamma + log(2) - 1), where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 27 2023

A087567 a(n) = (1/5)Sum_{k=0..n} binomial(n,k)Fibonacci(k)*5^k.

Original entry on oeis.org

1, 7, 68, 609, 5555, 50456, 458737, 4169823, 37904764, 344559985, 3132110411, 28471412592, 258809985953, 2352626740919, 21385776919540, 194400346514241, 1767132187070947, 16063531893267208, 146020234807218449, 1327348749622606095, 12065825708695393196

Offset: 0

Benoit Cloitre, Oct 25 2003

Index entries for linear recurrences with constant coefficients, signature (7,19).

Cf. A014445, A057088, A015553.

LinearRecurrence[{7,19},{1,7},25] (* Paolo Xausa, Jan 07 2024 *)

[lucas_number1(n,7,-19) for n in range(1, 20)] # Zerinvary Lajos, Apr 29 2009

a(n) = 7*a(n-1) + 19*a(n-2).

G.f.: 1 / (-19*x^2-7*x+1). - Colin Barker, Aug 08 2013

More terms from Colin Barker, Aug 08 2013

A107227 Numbers having no odd terms in their Zeckendorf representation.

Original entry on oeis.org

2, 8, 10, 34, 36, 42, 44, 144, 146, 152, 154, 178, 180, 186, 188, 610, 612, 618, 620, 644, 646, 652, 654, 754, 756, 762, 764, 788, 790, 796, 798, 2584, 2586, 2592, 2594, 2618, 2620, 2626, 2628, 2728, 2730, 2736, 2738, 2762, 2764, 2770, 2772, 3194, 3196, 3202

Offset: 1

Reinhard Zumkeller, May 15 2005

nonn

A107016(a(n))=0, A107015(a(n))>0; subsequence of A107225.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Zeckendorf Representation

Cf. A107228.

Cf. A035516, A014437, A014445, A011655.

a107227 n = a107227_list !! (n-1)
a107227_list = filter ((all even) . a035516_row) [1..]
-- Reinhard Zumkeller, Mar 10 2013

A107228 Numbers having no even terms in their Zeckendorf representation.

Original entry on oeis.org

1, 3, 4, 5, 6, 13, 14, 16, 17, 18, 19, 21, 22, 24, 25, 26, 27, 55, 56, 58, 59, 60, 61, 68, 69, 71, 72, 73, 74, 76, 77, 79, 80, 81, 82, 89, 90, 92, 93, 94, 95, 102, 103, 105, 106, 107, 108, 110, 111, 113, 114, 115, 116, 233, 234, 236, 237, 238, 239, 246, 247, 249, 250

Offset: 1

Reinhard Zumkeller, May 15 2005

nonn

A107015(a(n))=0, A107016(a(n))>0; subsequence of A107224.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Zeckendorf Representation

Cf. A107227.

Cf. A035516, A014437, A014445, A011655.

a107228 n = a107228_list !! (n-1)
a107228_list = filter ((all odd) . a035516_row) [1..]
-- Reinhard Zumkeller, Mar 10 2013

A195614 Denominators a(n) of Pythagorean approximations b(n)/a(n) to 2.

Original entry on oeis.org

8, 136, 2448, 43920, 788120, 14142232, 253772064, 4553754912, 81713816360, 1466294939560, 26311595095728, 472142416783536, 8472251907007928, 152028391909359160, 2728038802461456960, 48952670052396866112, 878420022140682133064

Offset: 1

Clark Kimberling, Sep 22 2011

See A195500 for a discussion and references.

Colin Barker, Table of n, a(n) for n = 1..797
Index entries for linear recurrences with constant coefficients, signature (17,17,-1).

Cf. A007805, A014445, A195500, A195615.

r = 2; z = 32;
p[{f_, n_}] := (#1[[2]]/#1[[
      1]] &)[({2 #1[[1]] #1[[2]], #1[[1]]^2 - #1[[
         2]]^2} &)[({Numerator[#1], Denominator[#1]} &)[
     Array[FromContinuedFraction[
        ContinuedFraction[(#1 + Sqrt[1 + #1^2] &)[f], #1]] &, {n}]]]];
{a, b} = ({Denominator[#1], Numerator[#1]} &)[
  p[{r, z}]]  (* A195614, A195615 *)
Sqrt[a^2 + b^2] (* A007805 *)
(* Peter J. C. Moses, Sep 02 2011 *)

Vec(8*x/((x+1)*(x^2-18*x+1)) + O(x^50)) \\ Colin Barker, Jun 04 2015

From Colin Barker, Jun 04 2015: (Start)
G.f.: 8*x / ((x+1)*(x^2-18*x+1)).
a(n) = 17*a(n-1) + 17*a(n-2) - a(n-3). (End)
a(n) = (-4*(-1)^n - (-2+sqrt(5))*(9+4*sqrt(5))^(-n) + (2+sqrt(5))*(9+4*sqrt(5))^n)/10. - Colin Barker, Mar 04 2016
a(n) = A014445(n) * A014445(n+1) / 2. - Diego Rattaggi, Jun 01 2020
a(n) is the numerator of continued fraction [4, ..., 4, 8, 4, ..., 4] with (n-1) 4's before and after the middle 8. - Greg Dresden and Hexuan Wang, Aug 30 2021

A199334 Triangle T(n,k) = Fibonacci(n+k), related to A000045 (Fibonacci numbers).

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 2, 3, 5, 8, 3, 5, 8, 13, 21, 5, 8, 13, 21, 34, 55, 8, 13, 21, 34, 55, 89, 144, 13, 21, 34, 55, 89, 144, 233, 377, 21, 34, 55, 89, 144, 233, 377, 610, 987, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584

Offset: 0

Philippe Deléham, Nov 07 2011

			Triangle begins :
   0;
   1,  1;
   1,  2,  3;
   2,  3,  5,  8;
   3,  5,  8, 13, 21;
   5,  8, 13, 21, 34,  55;
   8, 13, 21, 34, 55,  89, 144;
  13, 21, 34, 55, 89, 144, 233, 377;
  ...

Cf. A000045, A001906, A014445.

Row sums give A362067.

T(n,0) = Fibonacci(n) = A000045(n),
T(n,n) = Fibonacci(2n) = A001906(n),
T(2n,n) = Fibonacci(3n) = A014445(n).
T(n,k) = T(n,k-1) + T(n-1,k-1) = T(n-1,k-1) + T(n-1,k).

A014729 Squares of even Fibonacci numbers.

Original entry on oeis.org

0, 4, 64, 1156, 20736, 372100, 6677056, 119814916, 2149991424, 38580030724, 692290561600, 12422650078084, 222915410843904, 4000054745112196, 71778070001175616, 1288005205276048900, 23112315624967704576, 414733676044142633476, 7442093853169599697984

Offset: 0

Mohammad K. Azarian

Colin Barker, Table of n, a(n) for n = 0..700
Index entries for linear recurrences with constant coefficients, signature (17,17,-1)

Cf. A014445.

[Fibonacci(3*n)^2: n in [0..20]]; // Vincenzo Librandi, Nov 19 2018

(Table[Fibonacci@ n, {n, 0, 55}] /. n_ /; OddQ@ n -> Nothing)^2 (* or *)
CoefficientList[Series[4 (-x^2 + x)/((1 + x) (1 - 18 x + x^2)), {x, 0, 18}], x] (* Michael De Vlieger, Mar 04 2016 *)
LinearRecurrence[{17,17,-1},{0,4,64},20] (* Harvey P. Dale, Aug 02 2024 *)

numlib::fibonacci(3*n)^2 $ n = 0..25; // Zerinvary Lajos, May 09 2008

concat(0, Vec(4*x*(1-x)/((1+x)*(1-18*x+x^2)) + O(x^40))) \\ Colin Barker, Mar 04 2016

[fibonacci(3*n)^2 for n in range(0, 17)] # Zerinvary Lajos, May 15 2009

a(n) = (1/5)*(Fibonacci(6*n+3) - 2*Fibonacci(6*n) - 2*(-1)^n). - Ralf Stephan, May 14 2004
G.f.: 4*(-x^2+x)/((1+x)*(1-18*x+x^2)). - Ralf Stephan, May 14 2004
a(n) = Fibonacci(3*n)^2. - Gary Detlefs, Nov 28 2010
a(n) = (-1)^(n+1)*(Fibonacci(n)*Fibonacci(7*n)-Fibonacci(4*n)^2). - Gary Detlefs, Nov 28 2010
a(n) = (-2*(-1)^n+(9+4*sqrt(5))^(-n)+(9+4*sqrt(5))^n)/5. - Colin Barker, Mar 04 2016
a(n) = A014445(n)^2. - Sean A. Irvine, Nov 18 2018

More terms from James Sellers

A087603 a(n) = (1/8)Sum_{k=0..n} binomial(n,k)Fibonacci(k)*8^k.

Original entry on oeis.org

1, 10, 155, 2100, 29525, 410750, 5731375, 79905000, 1114275625, 15537531250, 216660471875, 3021168937500, 42128015328125, 587444444843750, 8191485291484375, 114224297381250000, 1592774664844140625, 22210083004410156250, 309703436610529296875

Offset: 0

Benoit Cloitre, Oct 25 2003

More generally a(n)=(1/x)*sum(k=0,n,binomial(n,k)*Fibonacci(k)*x^k) satisfies the recurrence formula a(n)=(x+2)*a(n-1)+(x^2-x-1)*a(n-2).

Harvey P. Dale, Table of n, a(n) for n = 0..873
Index entries for linear recurrences with constant coefficients, signature (10, 55).

Cf. A014445, A057088, A015553.

LinearRecurrence[{10,55},{1,10},30] (* Harvey P. Dale, Nov 26 2014 *)

Vec(1/(1-10*x-55*x^2) + O(x^50)) \\ Colin Barker, Mar 30 2016

a(n) = 10*a(n-1)+55*a(n-2).
G.f.: -1/(-1+10*x+55*x^2). - R. J. Mathar, Dec 05 2007
a(n) = ((-(5-4*sqrt(5))^(1+n)+(5+4*sqrt(5))^(1+n)))/(8*sqrt(5)). - Colin Barker, Mar 30 2016

A099134 Expansion of x/(1-2x-19x^2).

Original entry on oeis.org

0, 1, 2, 23, 84, 605, 2806, 17107, 87528, 500089, 2663210, 14828111, 80257212, 442248533, 2409384094, 13221490315, 72221278416, 395650872817, 2163506035538, 11844378654599, 64795371984420, 354633938406221

Offset: 0

Paul Barry, Sep 29 2004

Binomial transform is A099133. Binomial transform of x/(1-20x^2), or (0,1,0,20,0,400,0,8000,....). The inverse binomial transform of k^(n-1)Fib(n) has g.f. x/(1-(k-2)x-(k^2+k-1)x^2).

4*a(n) = (-1)^(n+1)*b(n;4) = 3^n*b(n;4/3), where b(n;d), n=0,1,..., d \in C, denote one of the delta-Fibonacci numbers defined in comments to A014445 (see also Witula-Slota's paper). Our first identity is equivalent to the second formula given below. We note that the sequence (4/3)^n*F(n) is the binomial transform of the sequence 3^(-n)*b(n;4). - Roman Witula, Jul 24 2012

R. Witula, D. Slota, \delta-Fibonacci Numbers, Appl. Anal. Discrete Math., 3 (2009), 310-329.

Index entries for linear recurrences with constant coefficients, signature (2,19).

Cf. A015447.

Join[{a=0,b=1},Table[c=2*b+19*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2011 *)
CoefficientList[Series[x/(1-2x-19x^2),{x,0,30}],x] (* or *) LinearRecurrence[ {2,19},{0,1},30] (* Harvey P. Dale, Dec 25 2019 *)

a(n) = 2a(n-1) + 19a(n-2).
a(n) = sum{k=0..n, (-1)^(n-k)binomial(n, k)4^(k-1)*Fib(k)}.
a(n) = sum{k=0..n, binomial(n, 2k+1)20^k}.

cp's OEIS Frontend

A099930 a(n) = Pell(n) * Pell(n-1) * Pell(n-2) / 10.

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A074723 Largest power of 2 dividing F(3n) where F(k) is the k-th Fibonacci number.

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A087567 a(n) = (1/5)*Sum_{k=0..n} binomial(n,k)*Fibonacci(k)*5^k.

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A107227 Numbers having no odd terms in their Zeckendorf representation.

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A107228 Numbers having no even terms in their Zeckendorf representation.

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A195614 Denominators a(n) of Pythagorean approximations b(n)/a(n) to 2.

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A199334 Triangle T(n,k) = Fibonacci(n+k), related to A000045 (Fibonacci numbers).

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A014729 Squares of even Fibonacci numbers.

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A087567 a(n) = (1/5)Sum_{k=0..n} binomial(n,k)Fibonacci(k)*5^k.

A087603 a(n) = (1/8)Sum_{k=0..n} binomial(n,k)Fibonacci(k)*8^k.