A001350 -id:A001350 - cp's OEIS frontend

A060280 Number of orbits of length n under the map whose periodic points are counted by A001350.

1, 0, 1, 1, 2, 2, 4, 5, 8, 11, 18, 25, 40, 58, 90, 135, 210, 316, 492, 750, 1164, 1791, 2786, 4305, 6710, 10420, 16264, 25350, 39650, 61967, 97108, 152145, 238818, 374955, 589520, 927200, 1459960, 2299854, 3626200, 5720274, 9030450, 14263078, 22542396

Offset: 1

Thomas Ward, Mar 29 2001

Euler transform is A000045. 1/((1-x)*(1-x^3)*(1-x^4)*(1-x^5)^2*(1-x^6)^2*...) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 5*x^5 + 8*x^6 + ... - Michael Somos, Jan 28 2003

			a(7)=4 since the 7th term of A001350 is 29 and the 1st is 1, so there are (29-1)/7 = 4 orbits of length 7.
x + x^3 + x^4 + 2*x^5 + 2*x^6 + 4*x^7 + 5*x^8 + 8*x^9 + 11*x^10 + 18*x^11 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..4000
Michael Baake, Joachim Hermisson, and Peter Pleasants, The torus parametrization of quasiperiodic LI-classes J. Phys. A 30 (1997), no. 9, 3029-3056.
Michael Baake, John A.G. Roberts, and Alfred Weiss, Periodic orbits of linear endomorphisms on the 2-torus and its lattices, arXiv:0808.3489 [math.DS], Aug 26, 2008. [_Jonathan Vos Post_, Aug 27 2008]
Larry Ericksen, Primality Testing and Prime Constellations, Šiauliai Mathematical Seminar, Vol. 3 (11), 2008. Mentions this sequence.
N. Neumarker, Realizability of Integer Sequences as Differences of Fixed Point Count Sequences, JIS 12 (2009) 09.4.5.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
T. Ward, Exactly realizable sequences

Cf. A000032, A000045, A001350, A006206, A324485, A324489.

First column of A348422.

A060280:= func< n | n le 2 select 2-n else (&+[Lucas(d)*MoebiusMu(Floor(n/d)) : d in Divisors(n)])/n >;
[A060280(n): n in [1..50]]; // G. C. Greubel, Nov 06 2024

A060280 := proc(n)
    add( numtheory[mobius](d)*A001350(n/d), d=numtheory[divisors](n)) ;
    %/n;
end proc: # R. J. Mathar, Jul 15 2016

A001350[n_] := LucasL[n] - (-1)^n - 1;
a[n_] := (1/n)*DivisorSum[n, MoebiusMu[#]*A001350[n/#]& ];
Array[a, 50] (* Jean-François Alcover, Nov 23 2017 *)

{a(n) = if( n<3, n==1, sumdiv( n, d, moebius(n/d) * (fibonacci(d - 1) + fibonacci(d + 1))) / n)} /* Michael Somos, Jan 28 2003 */

A000032=BinaryRecurrenceSequence(1,1,2,1)
def A060280(n): return sum(A000032(k)*moebius(n/k) for k in (1..n) if (k).divides(n))//n - int(n==2)
[A060280(n) for n in range(1,41)] # G. C. Greubel, Nov 06 2024

a(n) = (1/n)* Sum_{d|n} mu(d)*A001350(n/d).
a(n) = A006206(n) except for n=2. - Michael Somos, Jan 28 2003
a(n) = A031367(n)/n. - R. J. Mathar, Jul 15 2016
G.f.: Sum_{k>=1} mu(k)*log(1 + x^k/(1 - x^k - x^(2*k)))/k. - Ilya Gutkovskiy, May 18 2019

A324487 a(n) = A001350(n)^3.

Original entry on oeis.org

0, 1, 1, 64, 125, 1331, 4096, 24389, 91125, 438976, 1771561, 7880599, 32768000, 141420761, 594823321, 2537716544, 10720765125, 45537538411, 192699928576, 817138135549, 3460080078125, 14662949322176, 62103840598801, 263115950765039, 1114512556032000, 4721424167332081, 19999831641819121

Offset: 0

N. J. A. Sloane, Mar 12 2019

Seiichi Manyama, Table of n, a(n) for n = 0..1000
M. Baake, J. Hermisson, P. Pleasants, The torus parametrization of quasiperiodic LI-classes, J. Phys. A 30 (1997), no. 9, 3029-3056. See Tables 5 and 6.
Index entries for linear recurrences with constant coefficients, signature (4,12,-44,-44,132,66,-132,-44,44,12,-4,-1).

Cf. A001350, A152152, A324488, A324489.

From Colin Barker, Mar 13 2019: (Start)
G.f.: x*(1 + x^2)*(1 - 3*x + 47*x^2 - 96*x^3 + 104*x^4 + 96*x^5 + 47*x^6 + 3*x^7 + x^8) / ((1 - x)*(1 + x)*(1 - 3*x + x^2)*(1 + x - x^2)*(1 - x - x^2)*(1 + 3*x + x^2)*(1 - 4*x - x^2)).
a(n) = 4*a(n-1) + 12*a(n-2) - 44*a(n-3) - 44*a(n-4) + 132*a(n-5) + 66*a(n-6) - 132*a(n-7) - 44*a(n-8) + 44*a(n-9) + 12*a(n-10) - 4*a(n-11) - a(n-12) for n>11. (End)

A329488 a(n) = A001350(n)^4.

Original entry on oeis.org

1, 1, 256, 625, 14641, 65536, 707281, 4100625, 33362176, 214358881, 1568239201, 10485760000, 73680216481, 500246412961, 3461445366016, 23639287100625, 162614549665681, 1113034787454976, 7639424429247601, 52333711181640625

Offset: 1

N. J. A. Sloane, Nov 15 2019

nonn

Felix Fröhlich, Table of n, a(n) for n = 1..1000
Michael Baake and Franz Gähler, Symmetry Structure of the Elser-Sloane Quasicrystal, arXiv:cond-mat/9809100 [cond-mat], 1998. See Table 1.

Cf. A001350.

a001350(n) = fibonacci(n+1) + fibonacci(n-1) - 1 - (-1)^n
a(n) = a001350(n)^4 \\ Felix Fröhlich, Nov 24 2019

A126069 Generates A001350, the associated Mersenne numbers; A001350(n)=Product[a(d)] for d|n.

Original entry on oeis.org

1, 1, 4, 5, 11, 4, 29, 9, 19, 11, 199, 4, 521, 29, 31, 49, 3571, 19, 9349, 25, 211, 199, 64079, 36, 15251

Offset: 1

John W. Layman, Feb 28 2007

nonn

A 2001 Iranian Mathematical Olympiad question shows that such a generating sequence {a(n)} exists for the sequence {S(n)} whenever gcd(S(m),S(n)) = S(gcd(m,n)).

			The divisors of 6 are 1,2,3,6 and a(1)*a(2)*a(3)*a(6)=1*1*4*4=16, which is, in fact, A001350(6).

Cf. A001350, A061446.

A092184 Sequence S_6 of the S_r family.

Original entry on oeis.org

0, 1, 6, 25, 96, 361, 1350, 5041, 18816, 70225, 262086, 978121, 3650400, 13623481, 50843526, 189750625, 708158976, 2642885281, 9863382150, 36810643321, 137379191136, 512706121225, 1913445293766, 7141075053841, 26650854921600, 99462344632561, 371198523608646

Offset: 0

Rainer Rosenthal, Apr 03 2004

The r-family of sequences is S_r(n) = 2*(T(n,(r-2)/2) - 1)/(r-4) provided r is not equal to 4 and S_4(n) = n^2 = A000290(n). Here T(n,x) are Chebyshev's polynomials of the first kind. See their coefficient triangle A053120. See also the R. Stephan link for the explicit formula for s_k(n) for k not equal to 4 (Stephan's s_k(n) is identical with S_r(n)).
An integer n is in this sequence iff mutually externally tangent circles with radii n, n+1, n+2 have Soddy circles (i.e., circles tangent to all three) of rational radius. - James R. Buddenhagen, Nov 16 2005
This sequence is a divisibility sequence, i.e., a(n) divides a(m) whenever n divides m. It is the case P1 = 6, P2 = 8, Q = 1 of the 3-parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Mar 25 2014
a(n) is the block size of the (n-1)-th design in a sequence of multi-set designs with 2 blocks, see A335649. - John P. McSorley, Jun 22 2020

			a(3)=25 because a(1)=1 and a(2)=6 and a(1)*a(3) = 1*25 = (6-1)^2 = (a(2)-1)^2.

Hugo Pfoertner, Table of n, a(n) for n = 0..250
Marco Abrate, Stefano Barbero, Umberto Cerruti, Nadir Murru, Polynomial sequences on quadratic curves, Integers, Vol. 15, 2015, #A38.
Peter Bala, Linear divisibility sequences and Chebyshev polynomials
Ioana-Claudia Lazăr, Lucas sequences in t-uniform simplicial complexes, arXiv:1904.06555 [math.GR], 2019.
R. Stephan, Boring proof of a nonlinearity
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (5,-5,1).

See A001110=S_36 for further references to S_r sequences.

Other members of this r-family are: A007877 (r=2), |A078070| (r=3), A004146 (r=5), A054493 (r=7). A098306, A100047. A001353, A001834. A001350, A052530.

[Floor(1/2*(-2+(2+Sqrt(3))^n+(2-Sqrt(3))^n)): n in [0..30]]; // Vincenzo Librandi, Oct 14 2015

A092184 := proc(n)
    option remember;
    if n <= 1 then
        n;
    else
        4*procname(n-1)-procname(n-2)+2 ;
    end if ;
end proc:
seq(A092184(n),n=0..10) ;# Zerinvary Lajos, Mar 09 2008

Table[Simplify[ -((2 + Sqrt[3])^n - 1)*((2 - Sqrt[3])^n - 1)]/2, {n, 0, 26}] (* Stefan Steinerberger, May 15 2007 *)
LinearRecurrence[{5,-5,1},{0,1,6},27] (* Ray Chandler, Jan 27 2014 *)
CoefficientList[Series[x (1 + x)/(1 - 5 x + 5 x^2 - x^3), {x, 0, 40}], x] (* Vincenzo Librandi, Oct 14 2015 *)

Vec(x*(1+x)/(1 - 5*x + 5*x^2 - x^3) + O(x^50)) \\ Michel Marcus, Oct 14 2015

S_r type sequences are defined by a(0)=0, a(1)=1, a(2)=r and a(n-1)*a(n+1) = (a(n)-1)^2. This sequence emanates from r=6.
a(n) = 1/2*(-2 + (2+sqrt(3))^n + (2-sqrt(3))^n). - Ralf Stephan, Apr 14 2004
G.f.: x*(1+x)/(1 - 5*x + 5*x^2 - x^3) = x*(1+x)/((1-x)*(1 - 4*x + x^2)). [from the Ralf Stephan link]
a(n) = T(n, 2)-1 = A001075(n)-1, with Chebyshev's polynomials T(n, 2) of the first kind.
a(n) = b(n) + b(n-1), n >= 1, with b(n):=A061278(n) the partial sums of S(n, 4) = U(n, 2) = A001353(n+1) Chebyshev's polynomials of the second kind.
An integer k is in this sequence iff k is nonnegative and (k^2 + 2*k)/3 is a square. - James R. Buddenhagen, Nov 16 2005
a(0)=0, a(1)=1, a(n+1) = 3 + floor(a(n)*(2+sqrt(3))). - Anton Vrba (antonvrba(AT)yahoo.com), Jan 16 2007
a(n) = 4*a(n-1) - a(n-2) + 2. - Zerinvary Lajos, Mar 09 2008
From Peter Bala, Mar 25 2014: (Start)
a(2*n) = 6*A001353(n)^2; a(2*n+1) = A001834(n)^2.
a(n) = u(n)^2, where {u(n)} is the Lucas sequence in the quadratic integer ring Z[sqrt(6)] defined by the recurrence u(0) = 0, u(1) = 1, u(n) = sqrt(6)*u(n-1) - u(n-2) for n >= 2.
Equivalently, a(n) = U(n-1,sqrt(6)/2)^2, where U(n,x) denotes the Chebyshev polynomial of the second kind.
a(n) = (1/2)*( ((sqrt(6) + sqrt(2))/2)^n - ((sqrt(6) - sqrt(2))/2)^n )^2.
a(n) = bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, -2; 1, 3] and T(n,x) denotes the Chebyshev polynomial of the first kind. Cf. A098306.
See the remarks in A100047 for the general connection between Chebyshev polynomials of the first kind and 4th-order linear divisibility sequences. (End)
exp( Sum_{n >= 1} 2*a(n)*x^n/n ) = 1 + Sum_{n >= 1} A052530(n)*x^n. Cf. A001350. - Peter Bala, Mar 19 2015
E.g.f.: exp(2*x)*cosh(sqrt(3)*x) - cosh(x) - sinh(x). - Stefano Spezia, Oct 13 2019

Extension and Chebyshev comments from Wolfdieter Lang, Sep 10 2004

A004146 Alternate Lucas numbers - 2.

Original entry on oeis.org

0, 1, 5, 16, 45, 121, 320, 841, 2205, 5776, 15125, 39601, 103680, 271441, 710645, 1860496, 4870845, 12752041, 33385280, 87403801, 228826125, 599074576, 1568397605, 4106118241, 10749957120, 28143753121, 73681302245, 192900153616, 505019158605, 1322157322201

Offset: 0

N. J. A. Sloane

This is the r=5 member in the r-family of sequences S_r(n) defined in A092184 where more information can be found.
Number of spanning trees of the wheel W_n on n+1 vertices. - Emeric Deutsch, Mar 27 2005
Also number of spanning trees of the n-helm graph. - Eric W. Weisstein, Jul 16 2011
a(n) is the smallest number requiring n terms when expressed as a sum of Lucas numbers (A000204). - David W. Wilson, Jan 10 2006
This sequence has a primitive prime divisor for all terms beyond the twelfth. - Anthony Flatters (Anthony.Flatters(AT)uea.ac.uk), Aug 17 2007
From Giorgio Balzarotti, Mar 11 2009: (Start)
Determinant of power series of gamma matrix with determinant 1:
a(n) = Determinant(A + A^2 + A^3 + A^4 + A^5 + ... + A^n)
where A is the submatrix A(1..2,1..2) of the matrix with factorial determinant
A = [[1,1,1,1,1,1,...],[1,2,1,2,1,2,...],[1,2,3,1,2,3,...],[1,2,3,4,1,2,...],
[1,2,3,4,5,1,...],[1,2,3,4,5,6,...],...]. Note: Determinant A(1..n,1..n)= (n-1)!.
See A158039, A158040, A158041, A158042, A158043, A158044, for sequences of matrix 2!,3!,... (End)
The previous comment could be rephrased as: a(n) = -det(A^n - I) where I is the 2 X 2 identity matrix and A = [1, 1; 1, 2]. - Peter Bala, Mar 20 2015
a(n) is also the number of points of Arnold's "cat map" that are on orbits of period n-1. This is a map of the two-torus T^2 into itself. If we regard T^2 as R^2 / Z^2, the action of this map on a two vector in R^2 is multiplication by the unit-determinant matrix A = [2, 1;1, 1], with the vector components taken modulo one. As such, an explicit formula for the n-th entry of this sequence is -det(I-A^n). - Bruce Boghosian, Apr 26 2009
7*a(n) gives the total number of vertices in a heptagonal hyperbolic lattice {7,3} with n total levels, in which an open heptagon is centered at the origin. - Robert M. Ziff, Apr 10 2011
The sequence is the case P1 = 5, P2 = 6, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Apr 03 2014
Determinants of the spiral knots S(3,k,(1,-1)). a(k) = det(S(3,k,(1,-1))). These knots are also the weaving knots W(k,3) and the Turk's Head Links THK(3,k). - Ryan Stees, Dec 14 2014
Even-indexed Fibonacci numbers (1, 3, 8, 21, ...) convolved with (1, 2, 2, 2, 2, ...). - Gary W. Adamson, Aug 09 2016
a(n) is the number of ways to tile a bracelet of length n with 1-color square, 2-color dominos, 3-color trominos, etc. - Yu Xiao, May 23 2020
a(n) is the number of face-labeled unfoldings of a pyramid whose base is a simple n-gon. Cf. A103536. - Rick Mabry, Apr 17 2023

			For k=3, b(3) = sqrt(5)*b(2) - b(1) = 5 - 1 = 4, so det(S(3,3,(1,-1))) = 4^2 = 16.
G.f. = x + 5*x^2 + 16*x^3 + 45*x^4 + 121*x^5 + 320*x^4 + 841*x^5 + ... - _Michael Somos_, Feb 10 2023

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (p. 193, Problem 3.3.40 (a)).
N. Hartsfield and G. Ringel, Pearls in Graph Theory, p. 102. Academic Press: 1990.
B. Hasselblatt and A. Katok, "Introduction to the Modern Theory of Dynamical Systems," Cambridge University Press, 1997.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 0..2375 (terms 0..200 from T. D. Noe)
Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, Polynomial sequences on quadratic curves, Integers, Vol. 15, 2015, #A38.
S. Barbero, U. Cerruti, and N. Murru, On polynomial solutions of the Diophantine equation (x + y - 1)^2 = wxy, Rendiconti Sem. Mat. Univ. Pol. Torino (2020) Vol. 78, No. 1, 5-12.
Joshua P. Bowman, Compositions with an Odd Number of Parts, and Other Congruences, J. Int. Seq (2024) Vol. 27, Art. 24.3.6. See pp. 25, 29.
N. Brothers, S. Evans, L. Taalman, L. Van Wyk, D. Witczak, and C. Yarnall, Spiral knots, Missouri J. of Math. Sci., 22 (2010).
M. DeLong, M. Russell, and J. Schrock, Colorability and determinants of T(m,n,r,s) twisted torus knots for n equiv. +/-1(mod m), Involve, Vol. 8 (2015), No. 3, 361-384.
N. Dowdall, T. Mattman, K. Meek, and P. Solis, On the Harary-Kauffman conjecture and turk's head knots, arxiv 0811.0044 [math.GT], 2008.
A. Flatters, Primitive divisors of some Lehmer-Pierce sequences, arXiv:0708.2190 [math.NT], 2007.
Hang Gu and Robert M. Ziff, Crossing on hyperbolic lattices, arXiv:1111.5626 [cond-mat.dis-nn], 2011 (see footnote 32).
Seong Ju Kim, R. Stees, and L. Taalman, Sequences of Spiral Knot Determinants, Journal of Integer Sequences, Vol. 19 (2016), #16.1.4.
Ioana-Claudia Lazăr, Lucas sequences in t-uniform simplicial complexes, arXiv:1904.06555 [math.GR], 2019.
B. R. Myers, Number of spanning trees in a wheel, IEEE Trans. Circuit Theory, 18 (1971), 280-282.
B. R. Myers, Number of spanning trees in a wheel, IEEE Trans. Circuit Theory, 18 (1971), 280-282. [Annotated scanned copy. See the last two pages of the scan, the first few pages are of a different article]
L. Oesper, p-Colorings of Weaving Knots, Undergraduate Thesis, Pomona College, 2005.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Franck Ramaharo, A one-variable bracket polynomial for some Turk's head knots, arXiv:1807.05256 [math.CO], 2018.
Mohammed A. Raouf, Fazirulhisyam Hashim, Jiun Terng Liew, and Kamal Ali Alezabi, Pseudorandom sequence contention algorithm for IEEE 802.11ah based internet of things network, PLoS ONE (2020) Vol. 15, No. 8, e0237386.
K. R. Rebman, The sequence: 1 5 16 45 121 320 ... in combinatorics, Fib. Quart., 13 (1975), 51-55.
Harry Richman, Farbod Shokrieh, and Chenxi Wu, Counting two-forests and random cut size via potential theory, arXiv:2308.03859 [math.CO], 2023. See p. 18.
Benoit Rittaud and Laurent Vivier, Circular words and applications, arXiv:1108.3618 [cs.FL], 2011.
Benoit Rittaud and Laurent Vivier, Circular words and three applications: factors of the Fibonacci word, F -adic numbers, and the sequence 1, 5, 16, 45, 121, 320, ... , HAL Id: hal-00566314.
Benoit Rittaud and Laurent Vivier, Circular words and three applications: factors of the Fibonacci word, F -adic numbers, and the sequence 1, 5, 16, 45, 121, 320, ... , Functiones et Approximatio Commentarii Mathematici, Volume 47, Number 2 (2012), 207-231.
Ryan Stees, Sequences of Spiral Knot Determinants, Senior Honors Projects. Paper 84. James Madison Univ., May 2016.
Eric Weisstein's World of Mathematics, Helm Graph
Eric Weisstein's World of Mathematics, Spanning Tree
Eric Weisstein's World of Mathematics, Wheel Graph
Eric Weisstein's World of Mathematics, Arnold's cat map
Wikipedia, Arnold's cat map
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences, Integers, Volume 12A (2012) The John Selfridge Memorial Volume
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (4,-4,1).

This is the r=5 member of the family S_r(n) defined in A092184.
Cf. A005248. Partial sums of A002878. Pairwise sums of A027941. Bisection of A074392.
Sequence A032170, the Möbius transform of this sequence, is then the number of prime periodic orbits of Arnold's cat map. - Bruce Boghosian, Apr 26 2009
Cf. A000032, A100047, A001350.
Cf. also A158039, A158040, A158041, A158042, A158043, A158044.
Cf. A103536 for the number of geometrically distinct edge-unfoldings of the regular pyramid. - Rick Mabry, Apr 17 2023

[Lucas(n)-2: n in [0..60 by 2]]; // Vincenzo Librandi, Mar 20 2015

Table[LucasL[2*n] - 2, {n, 0, 20}]
(* Second program: *)
LinearRecurrence[{4, -4, 1}, {0, 1, 5}, 30] (* Jean-François Alcover, Jan 08 2019 *)

a(n) = { we = quadgen(5);((1+we)^n) + ((2-we)^n) - 2;} /* Michel Marcus, Aug 18 2012 */

a(n) = A005248(n) - 2 = A000032(2*n) - 2.
a(n+1) = 3*a(n) - a(n-1) + 2.
G.f.: x*(1+x)/(1-4*x+4*x^2-x^3) = x*(1+x)/((1-x)*(1-3*x+x^2)).
a(n) = 2*(T(n, 3/2)-1) with Chebyshev's polynomials T(n, x) of the first kind. See their coefficient triangle A053120.
a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3), n>=3, a(0)=0, a(1)=1, a(2)=5.
a(n) = 2*T(n, 3/2) - 2, with twice the Chebyshev polynomials of the first kind, 2*T(n, x=3/2) = A005248(n).
a(n) = b(n) + b(n-1), n>=1, with b(n):=A027941(n-1), n>=1, b(-1):=0, the partial sums of S(n, 3) = U(n, 3/2) = A001906(n+1), with S(n, x) = U(n, x/2) Chebyshev's polynomials of the second kind.
a(2n) = A000204(2n)^2 - 4 = 5*A000045(2n)^2; a(2n+1) = A000204(2n+1)^2. - David W. Wilson, Jan 10 2006
a(n) = ((3+sqrt(5))/2)^n + ((3-sqrt(5))/2)^n - 2. - Felix Goldberg (felixg(AT)tx.technion.ac.il), Jun 09 2001
a(n) = b(n-1) + b(n-2), n>=1, with b(n):=A027941(n), b(-1):=0, partial sums of S(n, 3) = U(n, 3/2) = A001906(n+1), Chebyshev's polynomials of the second kind.
a(n) = n*Sum_{k=1..n} binomial(n+k-1,2*k-1)/k, n > 0. - Vladimir Kruchinin, Sep 03 2010
a(n) = floor(tau^(2*n)*(tau^(2*n) - floor(tau^(2*n)))), where tau = (1+sqrt(5))/2. - L. Edson Jeffery, Aug 26 2013
From Peter Bala, Apr 03 2014: (Start)
a(n) = U(n-1,sqrt(5)/2)^2, for n >= 1, where U(n,x) denotes the Chebyshev polynomial of the second kind.
a(n) = the bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, -3/2; 1, 5/2] and T(n,x) denotes the Chebyshev polynomial of the first kind.
See the remarks in A100047 for the general connection between Chebyshev polynomials of the first kind and 4th-order linear divisibility sequences. (End)
a(k) = det(S(3,k,(1,-1))) = b(k)^2, where b(1)=1, b(2)=sqrt(5), b(k)=sqrt(5)*b(k-1) - b(k-2) = b(2)*b(k-1) - b(k-2). - Ryan Stees, Dec 14 2014
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + Sum_{n >= 1} Fibonacci(2*n)*x^n. Cf. A001350. - Peter Bala, Mar 19 2015
E.g.f.: exp(phi^2*x) + exp(x/phi^2) - 2*exp(x), where phi = (1 + sqrt(5))/2. - G. C. Greubel, Aug 24 2015
a(n) = a(-n) for all n in Z. - Michael Somos, Aug 27 2015
From Peter Bala, Jun 03 2016: (Start)
a(n) = Lucas(2*n) - Lucas(0*n);
a(n)^2 = Lucas(4*n) - 3*Lucas(2*n) + 3*Lucas(0*n) - Lucas(-2*n);
a(n)^3 = Lucas(6*n) - 5*Lucas(4*n) + 10*Lucas(2*n) - 10*Lucas(0*n) + 5*Lucas(-2*n) - Lucas(-4*n) and so on (follows from Binet's formula for Lucas(2*n) and the algebraic identity (x + 1/x - 2)^m = f(x) + f(1/x) where f(x) = (x - 1)^(2*m - 1)/x^(m-1) ). (End)
Limit_{n->infinity} a(n+1)/a(n) = (3 + sqrt(5))/2 = A104457. - Ilya Gutkovskiy, Jun 03 2016
a(n) = (phi^n - phi^(-n))^2, where phi = A001622 = (1 + sqrt(5))/2. - Diego Rattaggi, Jun 10 2020
a(n) = 4*sinh(n*A002390)^2, where A002390 = arcsinh(1/2). - Gleb Koloskov, Sep 18 2021
a(n) = 5*F(n)^2 = L(n)^2 - 4 if n even and a(n) = L(n)^2 = 5*F(n)^2 - 4 if n odd. - Michael Somos, Feb 10 2023
a(n) = n^2 + Sum_{i=1..n-1} i*a(n-i). - Fern Gossow, Dec 03 2024

Correction to formula from Nephi Noble (nephi(AT)math.byu.edu), Apr 09 2002

Chebyshev comments from Wolfdieter Lang, Sep 10 2004

A098600 a(n) = Fibonacci(n-1) + Fibonacci(n+1) - (-1)^n.

Original entry on oeis.org

1, 2, 2, 5, 6, 12, 17, 30, 46, 77, 122, 200, 321, 522, 842, 1365, 2206, 3572, 5777, 9350, 15126, 24477, 39602, 64080, 103681, 167762, 271442, 439205, 710646, 1149852, 1860497, 3010350, 4870846, 7881197, 12752042, 20633240, 33385281, 54018522

Offset: 0

Paul Barry, Sep 17 2004

Row sums of A098599.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Aleksandar Petojević, Marjana Gorjanac Ranitović, Dragan Rastovac, and Milinko Mandić, The Golden Ratio, Factorials, and the Lambert W Function, Journal of Integer Sequences, Vol. 27 (2024), Article 24.5.7. See p. 4.
Index entries for linear recurrences with constant coefficients, signature (0,2,1).

Cf. A000032, A000035, A000045, A001350, A008346.
Cf. A020878, A098599, A099925, A181716, A182143.
First differences of A014217 and A062724.

[Fibonacci(n-1) + Fibonacci(n+1) - (-1)^n: n in [0..50]]; // Vincenzo Librandi, Aug 31 2014

Table[-(-1)^n + LucasL[n], {n, 0, 39}] (* Alonso del Arte, Aug 30 2014 *)
Table[Fibonacci[n - 1] + Fibonacci[n + 1] - (-1)^n, {n, 0, 40}] (* Vincenzo Librandi, Aug 31 2014 *)
CoefficientList[ Series[-(1 + 2x)/(-1 + 2x^2 + x^3), {x, 0, 40}], x] (* or *)
LinearRecurrence[{0, 2, 1}, {1, 2, 2}, 40] (* Robert G. Wilson v, Mar 09 2018 *)

a(n)=fibonacci(n-1) + fibonacci(n+1) - (-1)^n; \\ Joerg Arndt, Oct 18 2014

Vec((1+2*x)/((1+x)*(1-x-x^2)) + O(x^30)) \\ Colin Barker, Jun 03 2016

[lucas_number2(n,1,-1) -(-1)^n for n in range(51)] # G. C. Greubel, Mar 26 2024

G.f.: (1+2*x) / ((1+x)*(1-x-x^2)).
a(n) = Sum_{k = 0..n} binomial(k, n-k) + binomial(k-1, n-k-1).
a(n) = A020878(n) - 1 = A001350(n) + 1.
a(n) = Lucas(n) - (-1)^n. - Paul Barry, Dec 01 2004
a(n) = A181716(n+1). - Richard R. Forberg, Aug 30 2014
a(n) = [x^n] ( (1 + x + sqrt(1 + 6*x + 5*x^2))/2 )^n. exp( Sum_{n >= 1} a(n)*x^n/n ) = Sum_{n >= 0} Fibonacci(n+2)*x^n. Cf. A182143. - Peter Bala, Jun 29 2015
From Colin Barker, Jun 03 2016: (Start)
a(n) = (-(-1)^n + ((1/2)*(1-sqrt(5)))^n + ((1/2)*(1+sqrt(5)))^n).
a(n) = 2*a(n-2) + a(n-3) for n > 2. (End)
E.g.f.: (2*exp(3*x/2)*cosh(sqrt(5)*x/2) - 1)*exp(-x). - Ilya Gutkovskiy, Jun 03 2016
a(n) = A014217(n) + A000035(n). - Paul Curtz, Jul 27 2023

A050140 a(n) = 2floor(nphi)-n, where phi = (1+sqrt(5))/2.

Original entry on oeis.org

1, 4, 5, 8, 11, 12, 15, 16, 19, 22, 23, 26, 29, 30, 33, 34, 37, 40, 41, 44, 45, 48, 51, 52, 55, 58, 59, 62, 63, 66, 69, 70, 73, 76, 77, 80, 81, 84, 87, 88, 91, 92, 95, 98, 99, 102, 105, 106, 109, 110, 113, 116, 117, 120, 121, 124, 127, 128, 131, 134, 135, 138, 139

Offset: 1

Clark Kimberling

nonn

Old name was a(n) = last number in repeating block in continued fraction for n*phi.

Alfred Brousseau, Fibonacci and Related Number Theoretic Tables, The Fibonacci Association, 1972, 101-103.

Clark Kimberling, Table of n, a(n) for n = 1..1000
J.-P. Allouche and F. M. Dekking, Generalized Beatty sequences and complementary triples, arXiv:1809.03424 [math.NT], 2018.

Cf. A000201, A001622, A005206, A050141, A005614, A001350.

[-n + 2*Floor(n*(1+Sqrt(5))/2): n in [1..50]]; // G. C. Greubel, Oct 15 2017

Table[-n+2Floor[n*GoldenRatio],{n,1,100}]

for(n=1,50, print1(-n + 2*floor(n*(1+sqrt(5))/2), ", ")) \\ G. C. Greubel, Oct 15 2017

def A050140(n): return (n+isqrt(5*n**2)&-2)-n # Chai Wah Wu, Aug 25 2022

a(n) = -n + 2*floor(n*phi) = A283233(n)-n.
a(n) = floor(n*phi) + floor(n*sigma) where phi = (sqrt(5)+1)/2 and sigma = (sqrt(5)-1)/2.
a(n) = last number in repeating block in continued fraction for n*phi.

Formula and more terms from Vladeta Jovovic, Nov 23 2001

Name changed by Michel Dekking, Dec 27 2017

A031367 Inflation orbit counts.

Original entry on oeis.org

1, 0, 3, 4, 10, 12, 28, 40, 72, 110, 198, 300, 520, 812, 1350, 2160, 3570, 5688, 9348, 15000, 24444, 39402, 64078, 103320, 167750, 270920, 439128, 709800, 1149850, 1859010, 3010348, 4868640, 7880994, 12748470, 20633200, 33379200, 54018520, 87394452, 141421800

Offset: 1

N. J. A. Sloane

Also number of primitive Lucas strings of length n [Ashrafi et al.] - N. J. A. Sloane, Nov 19 2014

The preceding comment is true for all n except n=2, as there are 2 primitive Lucas strings of length 2. The sequence of the number of primitive Lucas strings is the Möbius transform of the Lucas numbers A000032. - Pontus von Brömssen, Jan 24 2019

Alois P. Heinz, Table of n, a(n) for n = 1..2000
A. R. Ashrafi, J. Azarija, K. Fathalikhani, S. Klavzar, et al., Orbits of Fibonacci and Lucas cubes, dihedral transformations, and asymmetric strings, 2014.
A. R. Ashrafi, J. Azarija, K. Fathalikhani, S. Klavzar and M. Petkovsek, Vertex and edge orbits of Fibonacci and Lucas cubes, 2014; See Table 3.
Michael Baake, Joachim Hermisson, Peter Pleasants, The torus parametrization of quasiperiodic LI-classes J. Phys. A 30 (1997), no. 9, 3029-3056.

Cf. A001350, A006206, A000032.

A031367 := proc(n)
    add( numtheory[mobius](d)*A001350(n/d), d=numtheory[divisors](n)) ;
end proc: # R. J. Mathar, Jul 15 2016
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(a(i)/i+j-1, j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= proc(n) a(n):= ((<<0|1>, <1|1>>^n)[1, 2]-b(n, n-1))*n end:
seq(a(n), n=1..40);  # Alois P. Heinz, Jun 22 2018

a[n_] := n*Sum[MoebiusMu[d]*Sum[Binomial[k-1, 2k-n/d]/(n-d*k), {k, 0, n/d-1} ], {d, Divisors[n]}];
Array[a, 40] (* Jean-François Alcover, Jul 09 2018 *)

If b(n) is the n-th term of A001350, then a(n) = Sum_{d|n} mu(d)b(n/d).
a(n) = n * A060280(n).
G.f.: Sum_{k>=1} mu(k) * x^k * (1 + x^(2*k)) / ((1 - x^(2*k)) * (1 - x^k - x^(2*k))). - Ilya Gutkovskiy, Feb 06 2020

More terms from James Sellers

cp's OEIS Frontend

A060280 Number of orbits of length n under the map whose periodic points are counted by A001350.

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A324487 a(n) = A001350(n)^3.

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A329488 a(n) = A001350(n)^4.

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A126069 Generates A001350, the associated Mersenne numbers; A001350(n)=Product[a(d)] for d|n.

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A092184 Sequence S_6 of the S_r family.

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A004146 Alternate Lucas numbers - 2.

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A098600 a(n) = Fibonacci(n-1) + Fibonacci(n+1) - (-1)^n.

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A050140 a(n) = 2floor(nphi)-n, where phi = (1+sqrt(5))/2.