A001077 -id:A001077 - cp's OEIS frontend

A374439 Triangle read by rows: the coefficients of the Lucas-Fibonacci polynomials. T(n, k) = T(n - 1, k) + T(n - 2, k - 2) with initial values T(n, k) = k + 1 for k < 2.

1, 1, 2, 1, 2, 1, 1, 2, 2, 2, 1, 2, 3, 4, 1, 1, 2, 4, 6, 3, 2, 1, 2, 5, 8, 6, 6, 1, 1, 2, 6, 10, 10, 12, 4, 2, 1, 2, 7, 12, 15, 20, 10, 8, 1, 1, 2, 8, 14, 21, 30, 20, 20, 5, 2, 1, 2, 9, 16, 28, 42, 35, 40, 15, 10, 1, 1, 2, 10, 18, 36, 56, 56, 70, 35, 30, 6, 2

Offset: 0

Peter Luschny, Jul 22 2024

There are several versions of Lucas and Fibonacci polynomials in this database. Our naming follows the convention of calling polynomials after the values of the polynomials at x = 1. This assumes a regular sequence of polynomials, that is, a sequence of polynomials where degree(p(n)) = n. This view makes the coefficients of the polynomials (the terms of a row) a refinement of the values at the unity.
A remarkable property of the polynomials under consideration is that they are dual in this respect. This means they give the Lucas numbers at x = 1 and the Fibonacci numbers at x = -1 (except for the sign). See the example section.
The Pell numbers and the dual Pell numbers are also values of the polynomials, at the points x = -1/2 and x = 1/2 (up to the normalization factor 2^n). This suggests a harmonized terminology: To call 2^n*P(n, -1/2) = 1, 0, 1, 2, 5, ... the Pell numbers (A000129) and 2^n*P(n, 1/2) = 1, 4, 9, 22, ... the dual Pell numbers (A048654).
Based on our naming convention one could call A162515 (without the prepended 0) the Fibonacci polynomials. In the definition above only the initial values would change to: T(n, k) = k + 1 for k < 1. To extend this line of thought we introduce A374438 as the third triangle of this family.
The triangle is closely related to the qStirling2 numbers at q = -1. For the definition of these numbers see A333143. This relates the triangle to A065941 and A103631.

			Triangle starts:
  [ 0] [1]
  [ 1] [1, 2]
  [ 2] [1, 2, 1]
  [ 3] [1, 2, 2,  2]
  [ 4] [1, 2, 3,  4,  1]
  [ 5] [1, 2, 4,  6,  3,  2]
  [ 6] [1, 2, 5,  8,  6,  6,  1]
  [ 7] [1, 2, 6, 10, 10, 12,  4,  2]
  [ 8] [1, 2, 7, 12, 15, 20, 10,  8,  1]
  [ 9] [1, 2, 8, 14, 21, 30, 20, 20,  5,  2]
  [10] [1, 2, 9, 16, 28, 42, 35, 40, 15, 10, 1]
.
Table of interpolated sequences:
  |  n | A039834 & A000045 | A000032 |   A000129   |   A048654  |
  |  n |     -P(n,-1)      | P(n,1)  |2^n*P(n,-1/2)|2^n*P(n,1/2)|
  |    |     Fibonacci     |  Lucas  |     Pell    |    Pell*   |
  |  0 |        -1         |     1   |       1     |       1    |
  |  1 |         1         |     3   |       0     |       4    |
  |  2 |         0         |     4   |       1     |       9    |
  |  3 |         1         |     7   |       2     |      22    |
  |  4 |         1         |    11   |       5     |      53    |
  |  5 |         2         |    18   |      12     |     128    |
  |  6 |         3         |    29   |      29     |     309    |
  |  7 |         5         |    47   |      70     |     746    |
  |  8 |         8         |    76   |     169     |    1801    |
  |  9 |        13         |   123   |     408     |    4348    |

Paolo Xausa, Rows n = 0..150 of the triangle, flattened
Peter Luschny, Illustrating the polynomials.

Triangles related to Lucas polynomials: A034807, A114525, A122075, A061896, A352362.
Triangles related to Fibonacci polynomials: A162515, A053119, A168561, A049310, A374441.
Sums include: A000204 (Lucas numbers, row), A000045 & A212804 (even sums, Fibonacci numbers), A006355 (odd sums), A039834 (alternating sign row).
Type m^n*P(n, 1/m): A000129 & A048654 (Pell, m=2), A108300 & A003688 (m=3), A001077 & A048875 (m=4).
Adding and subtracting the values in a row of the table (plus halving the values obtained in this way): A022087, A055389, A118658, A052542, A163271, A371596, A324969, A212804, A077985, A069306, A215928.
Columns include: A040000 (k=1), A000027 (k=2), A005843 (k=3), A000217 (k=4), A002378 (k=5).
Diagonals include: A000034 (k=n), A029578 (k=n-1), abs(A131259) (k=n-2).
Cf. A029578 (subdiagonal), A124038 (row reversed triangle, signed).
Cf. A039961, A065941 (qStirling2), A103631, A108299, A131259. A133607, A194005, A333143, A374438 (m=3).

function T(n,k) // T = A374439
  if k lt 0 or k gt n then return 0;
  elif k le 1 then return k+1;
  else return T(n-1,k) + T(n-2,k-2);
  end if;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 23 2025

A374439 := (n, k) -> ifelse(k::odd, 2, 1)*binomial(n - irem(k, 2) - iquo(k, 2), iquo(k, 2)):
# Alternative, using the function qStirling2 from A333143:
T := (n, k) -> 2^irem(k, 2)*qStirling2(n, k, -1):
seq(seq(T(n, k), k = 0..n), n = 0..10);

A374439[n_, k_] := (# + 1)*Binomial[n - (k + #)/2, (k - #)/2] & [Mod[k, 2]];
Table[A374439[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* Paolo Xausa, Jul 24 2024 *)

from functools import cache
@cache
def T(n: int, k: int) -> int:
    if k > n: return 0
    if k < 2: return k + 1
    return T(n - 1, k) + T(n - 2, k - 2)

from math import comb as binomial
def T(n: int, k: int) -> int:
    o = k & 1
    return binomial(n - o - (k - o) // 2, (k - o) // 2) << o

def P(n, x):
    if n < 0: return P(n, x)
    return sum(T(n, k)*x**k for k in range(n + 1))
def sgn(x: int) -> int: return (x > 0) - (x < 0)
# Table of interpolated sequences
print("|  n | A039834 & A000045 | A000032 |   A000129   |   A048654  |")
print("|  n |     -P(n,-1)      | P(n,1)  |2^n*P(n,-1/2)|2^n*P(n,1/2)|")
print("|    |     Fibonacci     |  Lucas  |     Pell    |    Pell*   |")
f = "| {0:2d} | {1:9d}         |  {2:4d}   |   {3:5d}     |    {4:4d}    |"
for n in range(10): print(f.format(n, -P(n, -1), P(n, 1), int(2**n*P(n, -1/2)), int(2**n*P(n, 1/2))))

from sage.combinat.q_analogues import q_stirling_number2
def A374439(n,k): return (-1)^((k+1)//2)*2^(k%2)*q_stirling_number2(n+1, k+1, -1)
print(flatten([[A374439(n, k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Jan 23 2025

T(n, k) = 2^k' * binomial(n - k' - (k - k') / 2, (k - k') / 2) where k' = 1 if k is odd and otherwise 0.
T(n, k) = (1 + (k mod 2))*qStirling2(n, k, -1), see A333143.
2^n*P(n, -1/2) = A000129(n - 1), Pell numbers, P(-1) = 1.
2^n*P(n, 1/2) = A048654(n), dual Pell numbers.
T(2*n, n) = (1/2)*(-1)^n*( (1+(-1)^n)*A005809(n/2) - 2*(1-(-1)^n)*A045721((n-1)/2) ). - G. C. Greubel, Jan 23 2025

A042936 Numerators of continued fraction convergents to sqrt(1000).

Original entry on oeis.org

31, 32, 63, 95, 158, 253, 1676, 3605, 8886, 136895, 282676, 702247, 4496158, 5198405, 9694563, 14892968, 24587531, 39480499, 2472378469, 2511858968, 4984237437, 7496096405, 12480333842, 19976430247, 132338915324, 284654260895, 701647437114, 10809365817605, 22320379072324

Offset: 0

N. J. A. Sloane

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. A042937 (denominators).

Analog for sqrt(m): A001333 (m=2), A002531 (m=3), A001077 (m=5), A041006 (m=6), A041008 (m=7), A041010 (m=8), A005667 (m=10), A041014 (m=11), ..., A042934 (m=999).

Numerator[Convergents[Sqrt[1000], 30]] (* Harvey P. Dale, Oct 29 2013 *)

A42936=contfracpnqn(c=contfrac(sqrt(1000)), #c)[1,][^-1] \\ Discards possibly incorrect last term. NB: a(n)=A42936[n+1]. Could be extended using: {A42936=concat(A42936, 78960998*A42936[-18..-1]-A42936[-36..-19])}
\\ But terms with arbitrarily large indices can be computed using:
A042936(n)={[A42936[n%18+i]|i<-[1, 19]]*([0, -1; 1, 78960998]^(n\18))[,1]} \\ Faster but longer with n=divrem(n,18). (End)

A115032 Expansion of (5-14x+x^2)/((1-x)(x^2-18*x+1)).

Original entry on oeis.org

5, 81, 1445, 25921, 465125, 8346321, 149768645, 2687489281, 48225038405, 865363202001, 15528312597605, 278644263554881, 5000068431390245, 89722587501469521, 1610006506595061125, 28890394531209630721, 518417095055178291845, 9302617316461999622481

Offset: 0

Creighton Dement, Feb 26 2006

Relates squares of numerators and denominators of continued fraction convergents to sqrt(5).
Sequence is generated by the floretion A*B*C with A = + 'i - 'k + i' - k' - 'jj' - 'ij' - 'ji' - 'jk' - 'kj' ; B = - 'i + 'j - i' + j' - 'kk' - 'ik' - 'jk' - 'ki' - 'kj' ; C = - 'j + 'k - j' + k' - 'ii' - 'ij' - 'ik' - 'ji' - 'ki' (apart from a factor (-1)^n)
Floretion Algebra Multiplication Program, FAMP Code: tesseq[A*B*C].
The sequence a(n-1), n >= 0, with a(-1) = 1, is also the curvature of circles inscribed in a special way in the larger segment of a circle of radius 5/4 (in some units) cut by a chord of length 2. For the smaller segment, the analogous curvature sequence is given in A240926. For more details see comments on A240926. See also the illustration in the present sequence, and the proof of the coincidence of the curvatures with a(n-1) in part I of the W. Lang link. - Kival Ngaokrajang, Aug 23 2014

			G.f. = 5 + 81*x + 1445*x^2 + 25921*x^3 + 465125*x^4 + 8346321*x^5 + ...

G. C. Greubel, Table of n, a(n) for n = 0..795
Creighton Dement, Floretions associated with A115032.
Wolfdieter Lang, A proof for the touching circle problem (part I).
Giovanni Lucca, Circle chains inside the arbelos and integer sequences, Int'l J. Geom. (2023) Vol. 12, No. 1, 71-82.
Kival Ngaokrajang, Illustration of initial terms.
Index entries for linear recurrences with constant coefficients, signature (19,-19,1).
Index entries for sequences related to Chebyshev polynomials.

Cf. A001076, A001077, A007805, A023039, A097924.

Cf. also A000045, A049660, A049310, A023039. - Wolfdieter Lang, Aug 22 2014

seq((9*combinat:-fibonacci(6*(n+1)) - combinat:-fibonacci(6*n) + 8)/16, n = 0 .. 20); # Robert Israel, Aug 25 2014

LinearRecurrence[{19,-19,1},{5,81,1445},30] (* Harvey P. Dale, Nov 14 2014 *)
CoefficientList[Series[(5 - 14*x + x^2)/((1 - x)*(x^2 - 18*x + 1)), {x, 0, 50}], x] (* G. C. Greubel, Dec 19 2017 *)

Vec((5-14*x+x^2)/((1-x)*(x^2-18*x+1)) + O(x^20)) \\ Michel Marcus, Aug 23 2014

sqrt(a(2*n)) = sqrt(5)*A007805(n) = sqrt(5)*Fibonacci(6*n+3)/2 = sqrt(5)*A001076(2*n+1); sqrt(a(2*n+1)) = A023039(2*n+1) = A001077(2*n).
From Wolfdieter Lang, Aug 22 2014: (Start)
O.g.f.: (5-14*x+x^2)/((1-x)*(x^2-18*x+1)) (see the name).
a(n) = (9*F(6*(n+1)) - F(6*n) + 8)/16, n >= 0 with F(n) = A000045(n) (Fibonacci). From the partial fraction decomposition of the o.g.f.: (1/2)*((9 - x)/(1 - 18*x + x^2) + 1/(1 - x)). For F(6*n)/8 see A049660(n). a(-1) = 1 with F(-6) = -F(6) = -8.
a(n) = (9*S(n, 18) - S(n-1, 18) + 1)/2, with the Chebyshev S-polynomials (see A049310). From A049660.
a(n) = (A023039(n+1) + 1)/2.
(End)
a(n) = 19*a(n-1) - 19*a(n-2) + a(n-3). - Colin Barker, Aug 23 2014
From Wolfdieter Lang, Aug 24 2014: (Start)
a(n) = 18*a(n-1) - a(n-2) - 8, n >= 1, a(-1) = 1, a(0) = 5. See the Chebyshev S-polynomial formula above.
The o.g.f. for the sequence a(n-1) with a(-1) = 1, n >= 0, [1, 5,  81, 1445, ..] is (1-14*x+5*x^2)/((1-x)*(1-18*x+x^2)).
(See the Colin Barker formula from Aug 04 2014 in the history of A240926.) (End)

More terms from Michel Marcus, Aug 23 2014
Edited (comment by Kival Ngaokrajang rewritten, Chebyshev index link added) by Wolfdieter Lang, Aug 26 2014
Partially edited by Jon E. Schoenfield and N. J. A. Sloane, Mar 15 2024

A041010 Numerators of continued fraction convergents to sqrt(8).

Original entry on oeis.org

2, 3, 14, 17, 82, 99, 478, 577, 2786, 3363, 16238, 19601, 94642, 114243, 551614, 665857, 3215042, 3880899, 18738638, 22619537, 109216786, 131836323, 636562078, 768398401, 3710155682, 4478554083, 21624372014, 26102926097, 126036076402, 152139002499

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..199 [1 removed by _Georg Fischer_, Jul 01 2019]
Index entries for linear recurrences with constant coefficients, signature (0,6,0,-1).

Cf. A040005 (continued fraction), A041011 (denominators), A010466 (decimals).

Analog for other sqrt(m): A001333 (m=2), A002531 (m=3), A001077 (m=5), A041006 (m=6), A041008 (m=7), A005667 (m=10), A041014 (m=11), A041016 (m=12), ..., A042934 (m=999), A042936 (m=1000).

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[8],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 16 2011*)
CoefficientList[Series[(2 + 3*x + 2*x^2 - x^3)/(1 - 6*x^2 + x^4), {x, 0, 30}], x]  (* Vincenzo Librandi, Oct 28 2013 *)
a0[n_] := -((3-2*Sqrt[2])^n*(1+Sqrt[2]))+(-1+Sqrt[2])*(3+2*Sqrt[2])^n // Simplify
a1[n_] := ((3-2*Sqrt[2])^n+(3+2*Sqrt[2])^n)/2 // Simplify
Flatten[MapIndexed[{a0[#], a1[#]} &,Range[20]]] (* Gerry Martens, Jul 11 2015 *)

A041010=contfracpnqn(c=contfrac(sqrt(8)),#c)[1,][^-1] \\ Discard possibly incorrect last element. NB: a(n)=A041010[n+1]! For more terms use:
A041010(n)={n<#A041010|| A041010=extend(A041010, [-1,0,6,0]~, n\.8); A041010[n+1]}
extend(A,c,N)={for(n=#A+1,#A=Vec(A,N), A[n]=A[n-#c..n-1]*c);A} \\ (End)

a(n) = 6*a(n-2) - a(n-4).
a(2n) = a(2n-1) + a(2n-2), a(2n+1) = 4*a(2n) + a(2n-1).
a(2n) = A001333(2n), a(2n+1) = 2*A001333(2n+1).
G.f.: (2+3*x+2*x^2-x^3)/(1-6*x^2+x^4).
a(n) = A001333(n+1)*A000034(n+1). - R. J. Mathar, Jul 08 2009
From Gerry Martens, Jul 11 2015: (Start)
Interspersion of 2 sequences [a0(n),a1(n)] for n>0:
a0(n) = -((3-2*sqrt(2))^n*(1+sqrt(2))) + (-1+sqrt(2))*(3+2*sqrt(2))^n.
a1(n) = ((3-2*sqrt(2))^n + (3+2*sqrt(2))^n)/2. (End)

Entry improved by Michael Somos
Initial term 1 removed and b-file, program and formulas adapted by Georg Fischer, Jul 01 2019
Cross-references added by M. F. Hasler, Nov 02 2019

A059973 Expansion of x(1 + x - 2x^2) / ( 1 - 4*x^2 - x^4).

Original entry on oeis.org

0, 1, 1, 2, 4, 9, 17, 38, 72, 161, 305, 682, 1292, 2889, 5473, 12238, 23184, 51841, 98209, 219602, 416020, 930249, 1762289, 3940598, 7465176, 16692641, 31622993, 70711162, 133957148, 299537289, 567451585, 1268860318, 2403763488, 5374978561

Offset: 0

H. Peter Aleff (hpaleff(AT)earthlink.net), Mar 05 2001

Based on fact that cube root of (2 +- 1 sqrt(5)) = sixth root of (9 +- 4 sqrt(5)) = ninth root of (38 +- 17 sqrt(5)) = ... = phi or 1/phi, where phi is the golden ratio.
Osler gives the first three of the above equalities with phi on page 27, stating they are simplified expressions from Ramanujan, but without hinting that the series continues.
Bisections: A001076 and A001077.

			G.f. = x + x^2 + 2*x^3 + 4*x^4 + 9*x^5 + 17*x^6 + 38*x^7 + 72*x^8 + 161*x^9 + ... - _Michael Somos_, Aug 11 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
T. J. Osler, Cardan polynomials and the reduction of radicals, Math. Mag., 74 (No. 1, 2001), 26-32.
Index entries for linear recurrences with constant coefficients, signature (0,4,0,1).

Cf. A000045, A001076, A001077.

Cf. A179319, A183555, A183556.

I:=[0,1,1,2]; [n le 4 select I[n] else 4*Self(n-2)+Self(n-4): n in [1..40]]; // Vincenzo Librandi, Oct 10 2015

CoefficientList[ Series[(x +x^2 -2x^3)/(1 -4x^2 -x^4), {x, 0, 33}], x]
LinearRecurrence[{0,4,0,1}, {0,1,1,2}, 50] (* Vincenzo Librandi, Oct 10 2015 *)

{a(n) = if( n<0, n = -n; polcoeff( (-2*x + x^2 + x^3) / (1 + 4*x^2 - x^4) + x*O(x^n), n), polcoeff( (x + x^2 - 2*x^3) / ( 1 - 4*x^2 - x^4) + x*O(x^n), n))} /* Michael Somos, Aug 11 2009 */

a(n) = if (n < 4, fibonacci(n), 4*a(n-2) + a(n-4));
vector(50, n, a(n-1)) \\ Altug Alkan, Oct 04 2015

def a(n): return fibonacci(n) if (n<4) else 4*a(n-2) + a(n-4)
[a(n) for n in [0..40]] # G. C. Greubel, Jul 12 2021

From Michael Somos, Aug 11 2009: (Start)
a(2*n) = A001076(n).
a(2*n+1) = A001077(n). (End)
Recurrence: a(n) = 4*a(n-2) + a(n-4) for n >= 4; a(0)=0, a(1)=a(2)=1, a(3)=2. - Werner Schulte, Oct 03 2015
From Altug Alkan, Oct 06 2015: (Start)
a(2n) = Sum_{k=0..2n-1} a(k).
a(2n+1) = A001076(n-1) + Sum_{k=0..2n} a(k), n>0. (End)

Edited by Randall L Rathbun, Jan 11 2002
More terms from Sascha Kurz, Jan 31 2003
I made the old definition into a comment and gave the g.f. as an explicit definition. - N. J. A. Sloane, Jan 05 2011
Moved g.f. from Michael Somos, into name to match terms. - Paul D. Hanna, Jan 12 2011

A097924 a(n) = 4*a(n-1) + a(n-2), n>=2, a(0) = 2, a(1) = 7.

Original entry on oeis.org

2, 7, 30, 127, 538, 2279, 9654, 40895, 173234, 733831, 3108558, 13168063, 55780810, 236291303, 1000946022, 4240075391, 17961247586, 76085065735, 322301510526, 1365291107839, 5783465941882, 24499154875367, 103780085443350, 439619496648767, 1862258072038418

Offset: 0

Creighton Dement, Sep 04 2004; corrected Sep 16 2004

Previous name was: Sequence relates numerators and denominators in the continued fraction convergents to sqrt(5).

Floretion Algebra Multiplication Program, FAMP Code: 2lesforcycseq[ ( - 'i + 'j - i' + j' - 'kk' - 'ik' - 'jk' - 'ki' - 'kj' )*( .5'i + .5i' ) ], 2vesforcycseq = A000004.

			G.f. = 2 + 7*x + 30*x^2 + 127*x^3 + 538*x^4 + 2279*x^5 + 9654*x^6 + 40895*x^7 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Mark W. Coffey, James L. Hindmarsh, Matthew C. Lettington, and John Pryce, On Higher Dimensional Interlacing Fibonacci Sequences, Continued Fractions and Chebyshev Polynomials, arXiv:1502.03085 [math.NT], 2015 (see p. 31).
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (4,1).

Cf. A001076, A001077, A097924.

I:=[2,7]; [n le 2 select I[n] else 4*Self(n-1) + Self(n-2): n in [1..30]]; // G. C. Greubel, Dec 20 2017

a[n_] := Expand[((2Sqrt[5] + 3)*(2 + Sqrt[5])^n + (2Sqrt[5] - 3)*(2 - Sqrt[5])^n)/(2Sqrt[5])]; Table[ a[n], {n, 0, 20}] (* Robert G. Wilson v, Sep 17 2004 *)
a[ n_] := (3 I ChebyshevT[ n + 1, -2 I] + 4 ChebyshevT[ n, -2 I]) I^n / 5; (* Michael Somos, Feb 23 2014 *)
a[ n_] := If[ n < 0, SeriesCoefficient[ (2 + 7 x) / (1 + 4 x - x^2), {x, 0, -n}], SeriesCoefficient[ (2 - x) / (1 - 4 x - x^2), {x, 0, n}]]; (* Michael Somos, Feb 23 2014 *)
LinearRecurrence[{4,1}, {2,7}, 50] (* G. C. Greubel, Dec 20 2017 *)

{a(n) = ( 3*I*polchebyshev( n+1, 1, -2*I) + 4*polchebyshev( n, 1, -2*I)) * I^n / 5}; \\ Michael Somos, Feb 23 2014

{a(n) = if( n<0, polcoeff( (2 + 7*x) / (1 + 4*x - x^2) + x * O(x^-n), -n), polcoeff( (2 - x) / (1 - 4*x - x^2) + x * O(x^n), n))}; \\ Michael Somos, Feb 23 2014

a(n) = A001077(n+1) - 2*A001076(n).
A048875(n) + A001077(n+1)/2 = a(n)/2 + A048876(n).
a(n) = ((2*sqrt(5)+3)*(2+sqrt(5))^n + (2*sqrt(5)-3)*(2-sqrt(5))^n)/(2*sqrt(5)).
a(n+1) = A001077(n+1) + A015448(n+2) - Creighton Dement, Mar 08 2005
From Philippe Deléham, Nov 20 2008: (Start)
a(n) = 4*a(n-1) + a(n-2) for n>=2, a(0)=2, a(1)=7.
G.f.: (2-x)/(1-4*x-x^2). (End)
G.f.: G(0)*(2-x)/2, where G(k) = 1 + 1/(1 - x*(8*k + 4 +x)/(x*(8*k + 8 +x) + 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Feb 15 2014
a(-1 - n) = -(-1)^n * A048875(n). - Michael Somos, Feb 23 2014
E.g.f.: exp(2*x)*(10*cosh(sqrt(5)*x) + 3*sqrt(5)*sinh(sqrt(5)*x))/5. - Stefano Spezia, Aug 21 2025

Edited, corrected and extended by Robert G. Wilson v, Sep 17 2004

Better name (using formula from Philippe Deléham) from Joerg Arndt, Feb 16 2014

A138367 Count of post-period decimal digits up to which the rounded n-th convergent to sqrt(5) agrees with the exact value.

Original entry on oeis.org

0, 2, 4, 5, 6, 7, 8, 10, 8, 12, 14, 14, 16, 18, 19, 20, 21, 23, 24, 24, 26, 28, 29, 30, 31, 33, 33, 34, 35, 37, 39, 40, 41, 42, 44, 44, 46, 47, 48, 49, 51, 53, 53, 55, 56, 57, 59, 60, 60, 61, 64, 65, 66, 68, 69, 70, 72, 73, 74, 75, 76, 77, 79, 80, 81, 83, 83, 85, 85, 88, 89, 90, 91, 92

Offset: 1

Artur Jasinski, Mar 17 2008

This is a measure of the quality of the n-th convergent to A002163 if the convergent and the exact value are compared rounded to an increasing number of digits.
The sequence of rounded values of sqrt(5) is 2, 2.2, 2.24, 2.236, 2.2361, 2.23607, 2.236068, 2.2360680 etc, and the n-th convergent (provided by A001077 and A001076) is to be represented by its equivalent sequence.
a(n) represents the maximum number of post-period digits of the two sequences if compared at the same level of rounding. Counting only post-period digits (which is one less than the full number of decimal digits) is just a convention taken from A084407.

			For n=3, the 3rd convergent is 161/72 = 2.236111111..., with a sequence of rounded representations 2, 2.2, 2.24, 2.236, 2.2361, 2.23611, 2.236111, 2.2361111 etc.
Rounded to 1, 2, 3, or 4 post-period decimal digits, this is the same as the rounded version of the exact sqrt(5), but disagrees if both are rounded to 5 decimal digits, where 2.23607 <> 2.23611.
So a(3) = 4 (digits), the maximum rounding level of agreement.

Cf. A138335, A138336, A138337, A138339, A138343, A138366, A138369, A138370.

Definition and values replaced as defined via continued fractions by R. J. Mathar, Oct 01 2009

A075155 Cubes of Lucas numbers.

Original entry on oeis.org

8, 1, 27, 64, 343, 1331, 5832, 24389, 103823, 438976, 1860867, 7880599, 33386248, 141420761, 599077107, 2537716544, 10749963743, 45537538411, 192900170952, 817138135549, 3461452853383, 14662949322176, 62113250509227, 263115950765039, 1114577054530568

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Sep 06 2002

G. C. Greubel, Table of n, a(n) for n = 0..1000
Mohammad K. Azarian, Identities Involving Lucas or Fibonacci and Lucas Numbers as Binomial Sums, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 45, 2012, pp. 2221-2227.
Index entries for linear recurrences with constant coefficients, signature (3,6,-3,-1).

Cf. A000032, A061084, A075150, A075151.

Third row of array A103324.

[ Lucas(n)^3 : n in [0..120]]; // Vincenzo Librandi, Apr 14 2011

CoefficientList[Series[(8 - 23*x - 24*x^2 + x^3)/((x^2 + 4*x - 1)*(x^2 - x - 1)), {x,0,50}], x] (* or *) Table[LucasL[n]^3, {n,0,30}] (* or *) LinearRecurrence[{3,6,-3,-1}, {8, 1, 27, 64}, 30] (* G. C. Greubel, Dec 21 2017 *)

a(n)=(fibonacci(n-1)+fibonacci(n+1))^3 \\ Charles R Greathouse IV, Feb 09 2016

from sympy import lucas
def a(n):  return lucas(n)**3
print([a(n) for n in range(25)]) # Michael S. Branicky, Aug 01 2021

a(n) = 3*(-1)^n*L(n) + L(3*n).
a(n) = (-1)^n*A075151(n).
a(n) = A000032(n)^3 = A000032(n) * A001254(n).
a(n) = L(n)*C(n)^2, L(n) = Lucas numbers (A000032), C(n) = reflected Lucas numbers (comment to A061084).
a(n) = 3*a(n-1) + 6*a(n-2) - 3*a(n-3) - a(n-4), n>=4.
G.f.: ( 8-23*x-24*x^2+x^3 )/( (x^2+4*x-1)*(x^2-x-1) ).
a(n) = 2*A001077(n) + 3*A061084(n+1). - R. J. Mathar, Nov 17 2011
a(n) = L(3*n) + (F(n+4) - F(n-4))*(-1)^n, n>3 and F(n)=A000045(n). - J. M. Bergot, Feb 09 2016
a(n) + Sum_{i=0..n+1} a(i) = 19/2 + (5/2)*L(3*n+2). - Greg Dresden, Feb 24 2025

Simpler definition from Ralf Stephan, Nov 01 2004

A108404 Expansion of (1-4x)/(1-8x+11x^2).

Original entry on oeis.org

1, 4, 21, 124, 761, 4724, 29421, 183404, 1143601, 7131364, 44471301, 277325404, 1729418921, 10784771924, 67254567261, 419404046924, 2615432135521, 16310012568004, 101710347053301, 634272638178364, 3955367287840601

Offset: 0

Philippe Deléham, Jul 04 2005

Binomial transform of A098648. Second binomial transform of A001077. Third binomial transform of A084057. 4th binomial transform of (1, 0, 5, 0, 25, 0, 125, 0, 625, 0, 3125, ...).

Seiichi Manyama, Table of n, a(n) for n = 0..1258
Index entries for linear recurrences with constant coefficients, signature (8,-11).

Cf. A001077, A084057, A098648.

CoefficientList[Series[(1-4x)/(1-8x+11x^2),{x,0,30}],x] (* or *) LinearRecurrence[{8,-11},{1,4},30] (* Harvey P. Dale, Jan 03 2012 *)

E.g.f.: exp(4x)cosh(sqrt(5)x).
a(n) = 8a(n-1) - 11a(n-2), a(0) = 1, a(1) = 4.
a(n) = ((4+sqrt(5))^n + (4-sqrt(5))^n)/2.
a(n+1)/a(n) converges to 4 + sqrt(5) = 6.2360679774997896964... = 4+A002163.
a(n) = A091870(n+1)-4*A091870(n). - R. J. Mathar, Nov 10 2013

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.

cp's OEIS Frontend

A374439 Triangle read by rows: the coefficients of the Lucas-Fibonacci polynomials. T(n, k) = T(n - 1, k) + T(n - 2, k - 2) with initial values T(n, k) = k + 1 for k < 2.

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A042936 Numerators of continued fraction convergents to sqrt(1000).

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A115032 Expansion of (5-14*x+x^2)/((1-x)*(x^2-18*x+1)).

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A041010 Numerators of continued fraction convergents to sqrt(8).

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A059973 Expansion of x*(1 + x - 2*x^2) / ( 1 - 4*x^2 - x^4).

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A097924 a(n) = 4*a(n-1) + a(n-2), n>=2, a(0) = 2, a(1) = 7.

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A115032 Expansion of (5-14x+x^2)/((1-x)(x^2-18*x+1)).

A059973 Expansion of x(1 + x - 2x^2) / ( 1 - 4*x^2 - x^4).