A324489 -id:A324489 - 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)

A324485 a(n) = A324484(n)/n.

Original entry on oeis.org

1, 0, 5, 6, 24, 40, 120, 250, 640, 1452, 3600, 8510, 20880, 50460, 124024, 303750, 750120, 1853120, 4600200, 11437548, 28527320, 71281800, 178526880, 447893250, 1125750120, 2833844040, 7144449920, 18036271740, 45591631800, 115381449692, 292329067800, 741410192250

Offset: 1

N. J. A. Sloane, Mar 12 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..2000
M. Baake, J. Hermisson, and P. Pleasants, The torus parametrization of quasiperiodic LI-classes, J. Phys. A 30 (1997), no. 9, 3029-3056. See Table 4.

Cf. A001350, A060280, A152152, A324483, A324484, A324489.

a001350(n) = fibonacci(n+1)+fibonacci(n-1)-1-(-1)^n;
a(n) = sumdiv(n, d, moebius(n/d)*a001350(d)^2)/n; \\ Seiichi Manyama, Apr 29 2021

f(x) = ((1-x-x^2)*(1+x-x^2))^2/((1-3*x+x^2)*(1-x)^2*(1+x)^4);
my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, moebius(k)*log(f(x^k))/k)) \\ Seiichi Manyama, Apr 29 2021

From Seiichi Manyama, Apr 29 2021: (Start)
a(n) = (1/n) * Sum_{d|n} mu(n/d) * A001350(d)^2 = (1/n) * Sum_{d|n} mu(n/d) * A152152(d).
G.f.: Sum_{k>=1} mu(k) * log(f(x^k))/k , where f(x) = ((1-x-x^2) * (1+x-x^2))^2/((1-3*x+x^2) * (1-x)^2 * (1+x)^4). (End)

More terms from Seiichi Manyama, Apr 29 2021

A324488 Inflation orbit counts b^{(3)}_n for Danzer's F-type tiling and other 3D cut and project patterns with tau-inflation.

Original entry on oeis.org

1, 0, 63, 124, 1330, 4032, 24388, 91000, 438912, 1770230, 7880598, 32763780, 141420760, 594798932, 2537715150, 10720674000, 45537538410, 192699485568, 817138135548, 3460078306440, 14662949297724, 62103832718202, 263115950765038, 1114512523173000, 4721424167330750

Offset: 1

N. J. A. Sloane, Mar 12 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000
M. Baake, J. Hermisson, and P. Pleasants, The torus parametrization of quasiperiodic LI-classes, J. Phys. A 30 (1997), no. 9, 3029-3056. See Tables 5 and 6.

Cf. A001350, A031367, A324487, A324489.

a001350(n) = fibonacci(n+1)+fibonacci(n-1)-1-(-1)^n;
a(n) = sumdiv(n, d, moebius(n/d)*a001350(d)^3); \\ Seiichi Manyama, Apr 29 2021

a(n) = Sum_{d|n} mu(n/d) * A001350(d)^3 = Sum_{d|n} mu(n/d) * A324487(d). - Seiichi Manyama, Apr 29 2021

More terms from Seiichi Manyama, Apr 29 2021

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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A324485 a(n) = A324484(n)/n.

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A324488 Inflation orbit counts b^{(3)}_n for Danzer's F-type tiling and other 3D cut and project patterns with tau-inflation.

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