A001076 -id:A001076 - cp's OEIS frontend

A001657 Fibonomial coefficients: column 5 of A010048.

1, 8, 104, 1092, 12376, 136136, 1514513, 16776144, 186135312, 2063912136, 22890661872, 253854868176, 2815321003313, 31222272414424, 346260798314872, 3840089017377228, 42587248616222024, 472299787252290712, 5237885063192296801, 58089034826620525728

Offset: 0

N. J. A. Sloane

			G.f. = 1 + 8*x + 104*x^2 + 1092*x^3 + 12376*x^4 + 136136*x^5 + 1514513*x^6 + ...

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..200
A. Brousseau, A sequence of power formulas, Fib. Quart., 6 (1968), 81-83.
Alfred Brousseau, Fibonacci and Related Number Theoretic Tables, Fibonacci Association, San Jose, CA, 1972. See p. 17.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (8,40,-60,-40,8,1).

Cf. A010048, A001654-A001658, A065563.

with(combinat) : a:=n-> 1/30*fibonacci(n)*fibonacci(n+1)*fibonacci(n+2)*fibonacci(n+3)*fibonacci(n+4): seq(a(n), n=1..19); # Zerinvary Lajos, Oct 07 2007
A001657:=-1/(z**2+11*z-1)/(z**2-4*z-1)/(z**2+z-1); # Simon Plouffe in his 1992 dissertation

f[n_] := Times @@ Fibonacci[Range[n + 1, n + 5]]/30; t = Table[f[n], {n, 0, 20}] (* Vladimir Joseph Stephan Orlovsky, Feb 12 2010 *)
LinearRecurrence[{8,40,-60,-40,8,1},{1,8,104,1092,12376,136136},20] (* Harvey P. Dale, Nov 30 2019 *)

a(n)=(n->(n^5-n)/30)(fibonacci(n+3)) \\ Charles R Greathouse IV, Apr 24 2012

b(n, k)=prod(j=1, k, fibonacci(n+j)/fibonacci(j));
vector(20, n, b(n-1, 5))  \\ Joerg Arndt, May 08 2016

a(n) = A010048(5+n, 5) (or fibonomial(5+n, 5)).
G.f.: 1/(1-8*x-40*x^2+60*x^3+40*x^4-8*x^5-x^6) = 1/((1-x-x^2)*(1+4*x-x^2)*(1-11*x-x^2)) (see Comments to A055870).
a(n) = 11*a(n-1) + a(n-2) + ((-1)^n)*fibonomial(n+3, 3), n >= 2; a(0)=1, a(1)=8; fibonomial(n+3, 3)= A001655(n).
a(n) = Fibonacci(n+3)*(Fibonacci(n+3)^4-1)/30. - Gary Detlefs, Apr 24 2012
a(n) = (A049666(n+3) + 2*(-1)^n*A001076(n+3) - 3*A000045(n+3))/150, n >= 0, with A049666(n) = F(5*n)/5, A001076(n) = F(3*n)/2 and A000045(n) = F(n). From the partial fraction decomposition of the o.g.f. and recurrences. - Wolfdieter Lang, Aug 23 2012
a(n) = a(-6-n) * (-1)^n for all n in Z. - Michael Somos, Sep 19 2014
0 = a(n)*(-a(n+1) - 3*a(n+2)) + a(n+1)*(-8*a(n+1) + a(n+2)) for all n in Z. - Michael Somos, Sep 19 2014
G.f.: exp( Sum_{k>=1} F(6*k)/F(k) * x^k/k ), where F(n) = A000045(n). - Seiichi Manyama, May 07 2025

Corrected and extended by Wolfdieter Lang, Jun 27 2000

A081460 Consider the mapping f(r) = (1/2)*(r + N/r) from rationals to rationals where N = 5. Starting with a = 2 and applying the mapping to each new (reduced) rational number gives 2, 9/4, 161/72, 51841/23184, ..., tending to N^(1/2). Sequence gives values of the denominators.

Original entry on oeis.org

1, 4, 72, 23184, 2403763488, 25840354427429161536, 2986152136938872067784669198846010266752, 39878504028822311675150039382403961856254569551519724209276629577579916539865344

Offset: 1

Amarnath Murthy, Mar 22 2003

nonn

Related sequence pairs (numerator, denominator) can be obtained by choosing N = 2, 3, 6, etc.

The sequence satisfies the Pell equation A081459(n+1)^2 - 5*a(n+1)^2 = 1. - Vincenzo Librandi, Dec 20 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..11

Cf. A000032, A000045, A000129, A001333, A007283, A079613, A081459.

m:=8; f:=[ n eq 1 select 2 else (Self(n-1)+5/Self(n-1))/2: n in [1..m] ]; [ Denominator(f[n]): n in [1..m] ]; // Bruno Berselli, Dec 20 2011

Table[Fibonacci[2^(n - 1)*3], {n, 1, 8}]/2 (* Amiram Eldar, Apr 07 2023 *)

{r=2; N=5; for(n=1,8,a=numerator(r); b=denominator(r); print1(b,","); r=(1/2)*(r + N/r))}

a(n) = 2*a(n-1)*A081459(n-1). - Mario Catalani (mario.catalani(AT)unito.it), May 21 2003
a(n) = A000045(A007283(n-1))/2. - Ehren Metcalfe, Oct 07 2017
From Amiram Eldar, Apr 07 2023: (Start)
a(n) = A079613(n-1)/2.
a(n) = Product_{k=1..n-1} L(3*2^(n-1-k)), where L(k) is the k-th Lucas number (A000032). (End)
a(n) = A001076(2^(n-1)). - Robert FERREOL, Apr 18 2023

Edited and extended by Klaus Brockhaus and Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 06 2003

a(8) corrected by Vincenzo Librandi, Dec 20 2011

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

A147600 Expansion of 1/(1 - 3*x^2 + x^4).

Original entry on oeis.org

1, 0, 3, 0, 8, 0, 21, 0, 55, 0, 144, 0, 377, 0, 987, 0, 2584, 0, 6765, 0, 17711, 0, 46368, 0, 121393, 0, 317811, 0, 832040, 0, 2178309, 0, 5702887, 0, 14930352, 0, 39088169, 0, 102334155, 0, 267914296, 0, 701408733, 0, 1836311903, 0, 4807526976, 0

Offset: 0

Roger L. Bagula, Nov 08 2008

S(n,sqrt(5)), with the Chebyshev polynomials A049310, is an integer sequence in the real quadratic number field Q(sqrt(5)) with basis numbers <1,phi>, phi:=(1+sqrt(5))/2. S(n,sqrt(5)) = A(n) + 2*B(n)*phi, with A(n) = A005013(n+1)*(-1)^n and B(n) = a(n-1), n>=0, with a(-1)=0. - Wolfdieter Lang, Nov 24 2010
The sequence (s(n)) given by s(0) = 0 and s(n) = a(n-1) for n > 0 is the p-INVERT of (0, 1, 0, 1, 0, 1, ...) using p(S) = 1 - S^2; see A291219. - Clark Kimberling, Aug 30 2017
From Jean-François Alcover, Sep 24 2017: (Start)
Consider this array of successive differences:
    0,   0,    0,   1,    0,    3,    0,    8,    0,     21, ...
    0,   0,    1,  -1,    3,   -3,    8,   -8,    21,   -21, ...
    0,   1,   -2,   4,   -6,   11,  -16,   29,   -42,    76, ...
    1,  -3,    6, -10,   17,  -27,   45,  -71,   118,  -186, ...
   -4,   9,  -16,  27,  -44,   72, -116,  189,  -304,   495, ...
   13, -25,   43, -71,  116, -188,  305, -493,   799, -1291, ...
  -38,  68, -114, 187, -304,  493, -798, 1292, -2090,  3383, ...
...
First row = even-index Fibonacci numbers with interleaved zeros = this sequence right-shifted 3 positions.
Main diagonal = 0,0,-2,-10,-44,-188,-798,... = -A099919 right-shifted.
First upper subdiagonal = 0,1,4,17,72,305,1292,... = A001076 right-shifted.
Second upper subdiagonal = 0,-1,-6,-27,-116,-493,-2090,... = -A049651.
Third upper subdiagonal = 1,3,11,45,189,799,3383,... = A292278.
(End) (Comment based on an e-mail from Paul Curtz)

			G.f. = 1 + 3*x^2 + 8*x^4 + 21*x^6 + 55*x^8 + 144*x^10 + 377*x^12 + 987*x^14 + ...

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-1).

Cf. A000045, A001076, A001906, A005013, A049310.

Cf. A049651, A088305, A099919, A291219, A292278.

[(1+(-1)^n)*Fibonacci(n+2)/2: n in [0..60]]; // G. C. Greubel, Oct 25 2022

f[x_]= -1 -x +x^2; CoefficientList[Series[-1/(x^2*f[x]*f[1/x]), {x,0,60}], x]
(* or *)
M={{0,1,0,0}, {0,0,1,0}, {0,0,0,1}, {-1,0,3,0}}; v[0]= {1,0,3,0}; v[n_]:= v[n]= M.v[n-1]; Table[v[n][[1]], {n,0,60}]
LinearRecurrence[{0,3,0,-1}, {1,0,3,0}, 60] (* Jean-François Alcover, Sep 23 2017 *)

Vec(1/(1 - 3*x^2 + x^4)+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

[((n+1)%2)*fibonacci(n+2) for n in range(60)] # G. C. Greubel, Oct 25 2022

O.g.f.: 1/(1 - 3*x^2 + x^4).
a(2*k) = F(2*(k+1)), a(2*k+1) = 0, k>=0, with F(n)=A000045(n). - Richard Choulet, Nov 13 2008
a(n) + a(n-1) + a(n-2) = A005013(n + 1). - Michael Somos, Apr 13 2012
a(n) = (2^(-2-n)*((1 + (-1)^n)*((-3+sqrt(5))*(-1+sqrt(5))^n + (1+sqrt(5))^n*(3+sqrt(5)))))/sqrt(5). - Colin Barker, Mar 28 2016

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

A180250 a(n) = 5a(n-1) + 10a(n-2), with a(1)=0 and a(2)=1.

Original entry on oeis.org

0, 1, 5, 35, 225, 1475, 9625, 62875, 410625, 2681875, 17515625, 114396875, 747140625, 4879671875, 31869765625, 208145546875, 1359425390625, 8878582421875, 57987166015625, 378721654296875, 2473479931640625, 16154616201171875, 105507880322265625

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 16 2011

Nathaniel Johnston, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (5,10).

Cf. A001076, A006190, A007482, A015520, A015521, A015523, A015524, A015525, A015528, A015529, A015530, A015531, A015532, A015533, A015534, A015535, A015536, A015537, A015440, A015441, A015443, A015444, A015445, A015447, A030195, A053404, A057087, A057088, A083858, A085939, A090017, A091914, A099012, A180222, A180226.

[n le 2 select n-1 else 5*Self(n-1) +10*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 16 2018

Join[{a=0,b=1},Table[c=5*b+10*a;a=b;b=c,{n,100}]]
LinearRecurrence[{5,10}, {0,1}, 30] (* G. C. Greubel, Jan 16 2018 *)

a(n)=([0,1;10,5]^(n-1))[1,2] \\ Charles R Greathouse IV, Oct 03 2016

my(x='x+O('x^30)); concat([0], Vec(x^2/(1-5*x-10*x^2))) \\ G. C. Greubel, Jan 16 2018

A180250= BinaryRecurrenceSequence(5,10,0,1)
[A180250(n-1) for n in range(1,41)] # G. C. Greubel, Jul 21 2023

a(n) = ((5+sqrt(65))^(n-1) - (5-sqrt(65))^(n-1))/(2^(n-1)*sqrt(65)). - Rolf Pleisch, May 14 2011
G.f.: x^2/(1-5*x-10*x^2).
a(n) = (i*sqrt(10))^(n-1) * ChebyshevU(n-1, -i*sqrt(5/8)). - G. C. Greubel, Jul 21 2023

A368151 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 3x, p(n,x) = up(n-1,x) + vp(n-2,x) for n >=3, where u = p(2,x), v = 2 - x^2.

Original entry on oeis.org

1, 1, 3, 3, 6, 8, 5, 21, 25, 21, 11, 48, 101, 90, 55, 21, 123, 290, 414, 300, 144, 43, 282, 850, 1416, 1551, 954, 377, 85, 657, 2255, 4671, 6109, 5481, 2939, 987, 171, 1476, 5883, 13986, 22374, 24300, 18585, 8850, 2584, 341, 3303, 14736, 40320, 74295, 97713

Offset: 1

Clark Kimberling, Dec 31 2023

Because (p(n,x)) is a strong divisibility sequence, for each integer k, the sequence (p(n,k)) is a strong divisibility sequence of integers.

			First eight rows:
   1
   1     3
   3     6    8
   5    21    25    21
  11    48   101    90    55
  21   123   290   414   300  144
  43   282   850  1416  1551  954    377
  85   657  2255  4671  6109  5481  2939  987
Row 4 represents the polynomial p(4,x) = 5 + 21 x + 25 x^2 + 21 x^3, so (T(4,k)) = (5,21,25,21), k=0..3.

Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.

Cf. A001045 (column 1); A001906 (p(n,n-1)); A001076 (row sums), (p(n,1)); A077985 (alternating row sums), (p(n,-1)); A186446 (p(n,2)), A107839, (p(n,-2)); A190989, (p(n,3)); A023000, (p(n,-3)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300, A367301, A368150.

p[1, x_] := 1; p[2, x_] := 1 + 3 x; u[x_] := p[2, x]; v[x_] := 2 - x^2;
p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]

p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >=3, where p(1,x) = 1, p(2,x) = 1 + 3 x, u = p(2,x), and v = 2 - x^2.

p(n,x) = k*(b^n - c^n), where k = -1/sqrt(9 + 6 x + 5 x^2), b = (1/2) (3 x + 1 - 1/k), c = (1/2) (3 x + 1 + 1/k).

A015551 Expansion of x/(1 - 6x - 5x^2).

Original entry on oeis.org

0, 1, 6, 41, 276, 1861, 12546, 84581, 570216, 3844201, 25916286, 174718721, 1177893756, 7940956141, 53535205626, 360916014461, 2433172114896, 16403612761681, 110587537144566, 745543286675801, 5026197405777636

Offset: 0

Olivier Gérard

Let the generator matrix for the ternary Golay G_12 code be [I|B], where the elements of B are taken from the set {0,1,2}. Then a(n)=(B^n)1,2 for instance. - _Paul Barry, Feb 13 2004

Pisano period lengths: 1, 2, 4, 4, 1, 4, 42, 8, 12, 2, 10, 4, 12, 42, 4, 16, 96, 12, 360, 4, ... - R. J. Mathar, Aug 10 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Lucyna Trojnar-Spelina, Iwona Włoch, On Generalized Pell and Pell-Lucas Numbers, Iranian Journal of Science and Technology, Transactions A: Science (2019), 1-7.
Index entries for linear recurrences with constant coefficients, signature (6,5).

Cf. A001076, A006190, A007482, A015520, A015521, A015523, A015524, A015525, A015528, A015529, A015530, A015531, A015532, A015533, A015534, A015535, A015536, A015537, A015440, A015441, A015443, A015444, A015445, A015447, A015548, A030195, A053404, A057087, A057088, A057089, A083858, A085939, A090017, A091914, A099012, A135030, A135032, A180222, A180226, A180250.

I:=[0,1]; [n le 2 select I[n] else 6*Self(n-1)+5*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 14 2011

Join[{a=0,b=1},Table[c=6*b+5*a;a=b;b=c,{n,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 16 2011 *)
CoefficientList[Series[x/(1-6x-5x^2),{x,0,20}],x] (* or *) LinearRecurrence[ {6,5},{0,1},30] (* Harvey P. Dale, Oct 30 2017 *)

a(n)=([0,1; 5,6]^n*[0;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

[lucas_number1(n,6,-5) for n in range(0, 21)] # Zerinvary Lajos, Apr 24 2009

a(n) = 6*a(n-1) + 5*a(n-2).

a(n) = sqrt(14)*(3+sqrt(14))^n/28 - sqrt(14)*(3-sqrt(14))^n/28. - Paul Barry, Feb 13 2004

cp's OEIS Frontend

A001657 Fibonomial coefficients: column 5 of A010048.

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A081460 Consider the mapping f(r) = (1/2)*(r + N/r) from rationals to rationals where N = 5. Starting with a = 2 and applying the mapping to each new (reduced) rational number gives 2, 9/4, 161/72, 51841/23184, ..., tending to N^(1/2). Sequence gives values of the denominators.

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

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A147600 Expansion of 1/(1 - 3*x^2 + x^4).

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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).

A180250 a(n) = 5a(n-1) + 10a(n-2), with a(1)=0 and a(2)=1.

A368151 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 3x, p(n,x) = up(n-1,x) + vp(n-2,x) for n >=3, where u = p(2,x), v = 2 - x^2.

A015551 Expansion of x/(1 - 6x - 5x^2).