A156216 -id:A156216 - cp's OEIS frontend

A203800 a(n) = (1/n) * Sum_{d|n} moebius(n/d) * Lucas(d)^(d-1), where Lucas(n) = A000032(n).

1, 1, 5, 85, 2928, 314925, 84974760, 63327890015, 123670531939440, 644385861467631972, 8853970669063185618000, 321538767413685546538468385, 30768712746239178236068160093280, 7755868453482819803691622493685140880, 5144106193113274410507722020733942141881664

Offset: 1

Paul D. Hanna, Jan 06 2012

nonn

			G.f.: F(x) = 1/((1-x-x^2) * (1-3*x^2+x^4) * (1-4*x^3-x^6)^5 * (1-7*x^4+x^8)^85 * (1-11*x^5-x^10)^2928 * (1-18*x^6+x^12)^314925 * (1-29*x^7-x^14)^84974760 * (1-47*x^8+x^16)^63327890015 * (1-76*x^9-x^18)^123670531939440 *...).
where F(x) = exp( Sum_{n>=1} Lucas(n)^n * x^n/n ) = g.f. of A156216:
F(x) = 1 + x + 5*x^2 + 26*x^3 + 634*x^4 + 32928*x^5 + 5704263*x^6 +...
so that the logarithm of F(x) begins:
log(F(x)) = x + 3^2*x^2/2 + 4^3*x^3/3 + 7^4*x^4/4 + 11^5*x^5/5 + 18^6*x^6/6 + 29^7*x^7/7 + 47^8*x^8/8 + 76^9*x^9/9 + 123^10*x^10/10 +...+ Lucas(n)^n*x^n +...

G. C. Greubel, Table of n, a(n) for n = 1..69

Cf. A156216, A000032 (Lucas), A203318, A203803, A203804, A203805, A203806, A203807, A203808.

a[n_] := 1/n DivisorSum[n, MoebiusMu[n/#] LucasL[#]^(#-1)&]; Array[a, 15] (* Jean-François Alcover, Dec 23 2015 *)

{a(n)=if(n<1, 0, sumdiv(n, d, moebius(n/d)*(fibonacci(d-1)+fibonacci(d+1))^(d-1))/n)}

{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=local(F=exp(sum(m=1, n, Lucas(m)^m*x^m/m)+x*O(x^n)));if(n==1,1,polcoeff(F*prod(k=1,n-1,(1 - Lucas(k)*x^k + (-1)^k*x^(2*k) +x*O(x^n))^a(k)),n)/Lucas(n))}

G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^a(n) = exp(Sum_{n>=1} Lucas(n)^n * x^n/n), which is the g.f. of A156216.

G.f.: Product_{n>=1} G_n(x^n)^a(n) = exp(Sum_{n>=1} Lucas(n)^n * x^n/n) where G_n(x^n) = Product_{k=0..n-1} G(u^k*x) where G(x) = 1/(1-x-x^2) and u is an n-th root of unity.

A067961 Number of binary arrangements without adjacent 1's on n X n torus connected n-s.

Original entry on oeis.org

1, 9, 64, 2401, 161051, 34012224, 17249876309, 23811286661761, 84590643846578176, 792594609605189126649, 19381341794579313317802199, 1242425797286480951825250390016, 208396491430277954192889648311785961, 91534759488004239323168528670973468727049

Offset: 1

R. H. Hardin, Feb 02 2002

			Neighbors for n=4:
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o o o o
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o o o o
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o o o o
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o o o o
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Vincenzo Librandi, Table of n, a(n) for n = 1..69
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 409.

Cf. circle A000204, line A000045, arrays: ne-sw nw-se A067965, e-w ne-sw nw-se A067963, n-s nw-se A067964, e-w n-s nw-se A066864, e-w ne-sw n-s nw-se A063443, n-s A067966, e-w n-s A006506, nw-se A067962, toruses: bare A002416, ne-sw nw-se A067960, ne-sw n-s nw-se A067959, e-w ne-sw n-s nw-se A067958, e-w n-s A027683, e-w ne-sw n-s A066866.
Cf. A156216. - Paul D. Hanna, Sep 13 2010
Cf. A215941.

[Lucas(n)^n: n in [1..15]]; // Vincenzo Librandi, Mar 15 2014

a:= n-> (<<0|1>, <1|1>>^n. <<2, 1>>)[1$2]^n:
seq(a(n), n=1..15);  # Alois P. Heinz, Aug 01 2021

Table[LucasL[n]^n,{n,15}] (* Harvey P. Dale, Mar 13 2014 *)

a(n) = L(n)^n, where L(n) = A000032(n) is the n-th Lucas number.
Logarithmic derivative of A156216. - Paul D. Hanna, Sep 13 2010
Sum_{n>=1} 1/a(n) = A215941. - Amiram Eldar, Nov 17 2020

Edited by Dean Hickerson, Feb 15 2002

A166168 G.f.: exp( Sum_{n>=1} Lucas(n^2)*x^n/n ) where Lucas(n) = A000204(n).

Original entry on oeis.org

1, 1, 4, 29, 585, 34212, 5600397, 2490542953, 2968152042068, 9416588994339205, 79216509536543420965, 1762508872870620792746360, 103525263562786817866762466405, 16031370626878431551103688398524485

Offset: 0

Paul D. Hanna, Oct 08 2009

nonn

Conjectured to consist entirely of integers.
The Lucas numbers (A000204) forms the logarithmic derivative of the Fibonacci numbers (A000045).
Note that Lucas(n^2) = [(1+sqrt(5))/2]^(n^2) + [(1-sqrt(5))/2]^(n^2).

			G.f.: A(x) = 1 + x + 4*x^2 + 29*x^3 + 585*x^4 + 34212*x^5 +...
log(A(x)) = x + 7*x^2/2 + 76*x^3/3 + 2207*x^4/4 + 167761*x^5/5 + 33385282*x^6/6 +...+ Lucas(n^2)*x^n/n +...

G. C. Greubel, Table of n, a(n) for n = 0..60
Sawian Jaidee, Patrick Moss, Tom Ward, Time-changes preserving zeta functions, arXiv:1809.09199 [math.DS], 2018.

Cf. A166169, A156216, A155200, A000204, A000045.

with(combinat): seq(coeff(series(exp(add((fibonacci(k^2-1)+fibonacci(k^2+1))*x^k/k,k=1..n)),x,n+1), x, n), n = 0 .. 15); # Muniru A Asiru, Dec 18 2018

CoefficientList[Series[Exp[Sum[LucasL[n^2]*x^n/n, {n, 1, 200}]], {x, 0, 50}], x](* G. C. Greubel, May 06 2016 *)

{a(n)=polcoeff(exp(sum(m=1,n,(fibonacci(m^2-1)+fibonacci(m^2+1))*x^m/m)+x*O(x^n)),n)}

a(n) = (1/n)*Sum_{k=1..n} Lucas(k^2)*a(n-k), a(0)=1.

Logarithmic derivative yields A166169.

A207834 G.f.: exp( Sum_{n>=1} 5L(n)x^n/n ), where L(n) = Fibonacci(n-1)^n + Fibonacci(n+1)^n.

Original entry on oeis.org

1, 5, 25, 130, 1295, 38861, 4227075, 1309117220, 1123176929475, 2564594183278115, 15604715134340991949, 251021373648740285348860, 10668788238489683954523431475, 1195322752666989652479885363067075, 352750492054485236937115646128341734205

Offset: 0

Paul D. Hanna, Feb 20 2012

nonn

Given g.f. A(x), note that A(x)^(1/5) is not an integer series.
Compare the definition to the g.f. of the Fibonacci numbers:
1/(1-x-x^2) = exp( Sum_{n>=1} Lucas(n)*x^n/n ), where Lucas(n) = Fibonacci(n-1) + Fibonacci(n+1).

			G.f.: A(x) = 1 + 5*x + 25*x^2 + 130*x^3 + 1295*x^4 + 38861*x^5 +...
such that, by definition,
log(A(x))/5 = x + 5*x^2/2 + 28*x^3/3 + 641*x^4/4 + 33011*x^5/5 +...+ (Fibonacci(n-1)^n + Fibonacci(n+1)^n)*x^n/n +...

Cf. A207835, A156216, A166168.

{L(n)=fibonacci(n-1)^n+fibonacci(n+1)^n}
{a(n)=polcoeff(exp(sum(m=1,n,5*L(m)*x^m/m)+x*O(x^n)),n)}
for(n=0,51,print1(a(n),", "))

A207835 G.f.: exp( Sum_{n>=1} 5L(n)x^n/n ), where L(n) = Fibonacci((n-1)^2) + Fibonacci((n+1)^2).

Original entry on oeis.org

1, 15, 200, 3525, 134355, 16781664, 6730280105, 7679335074975, 23795707614699850, 197148338964056588955, 4337960355881995023988299, 252594793852565664429620014530, 38838042059493582778244565420563025, 15744729667082405326504405819215652913325

Offset: 0

Paul D. Hanna, Feb 20 2012

nonn

Given g.f. A(x), note that A(x)^(1/5) is not an integer series.
Compare the definition to the g.f. of the Fibonacci numbers:
1/(1-x-x^2) = exp( Sum_{n>=1} Lucas(n)*x^n/n ), where Lucas(n) = Fibonacci(n-1) + Fibonacci(n+1).

			G.f.: A(x) = 1 + 15*x + 200*x^2 + 3525*x^3 + 134355*x^4 + 16781664*x^5 +...
such that, by definition,
log(A(x))/5 = 3*x + 35*x^2/2 + 990*x^3/3 + 75059*x^4/4 + 14931339*x^5/5 + 7778817074*x^6/6 +...+ (Fibonacci((n-1)^2) + Fibonacci((n+1)^2))*x^n/n +...

Cf. A207834, A156216, A166168.

{L(n)=fibonacci((n-1)^2)+fibonacci((n+1)^2)}
{a(n)=polcoeff(exp(sum(m=1,n,5*L(m)*x^m/m)+x*O(x^n)),n)}
for(n=0,21,print1(a(n),", "))

A171186 G.f.: exp( Sum_{n>=1} (x^n/n)*[Sum_{k=0..[n/2]} A034807(n,k)^n] ), where A034807 is a triangle of Lucas polynomials.

Original entry on oeis.org

1, 1, 3, 12, 82, 1350, 97888, 15395388, 3754569984, 3038160817708, 10054063262475469, 52672088781183258841, 474423679267205966998406, 20987531454245723696517676183, 2606758801245041424971290635855234

Offset: 0

Paul D. Hanna, Dec 13 2009

nonn

			G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 82*x^4 + 1350*x^5 +...
log(A(x)) = x + 5*x^2/2 + 28*x^3/3 + 273*x^4/4 + 6251*x^5/5 +...+ A171187(n)*x^n/n +...

Cf. A171187, A034807, A156216.

{a(n)=polcoeff(exp(sum(m=1,n,(x^m/m)*sum(k=0, m\2, (binomial(m-k, k)+binomial(m-k-1, k-1))^m))+x*O(x^n)),n)}

A163189 G.f.: A(x) = exp( Sum_{n>=1} (1 + A000204(n)x)^n x^n/n ).

Original entry on oeis.org

1, 1, 2, 5, 14, 40, 159, 812, 5133, 42942, 474619, 6708142, 121367878, 2819170132, 83571532538, 3148951107867, 151069353323782, 9219463980803329, 714951048370178409, 70448496563603216429, 8818161368662624534857

Offset: 0

Paul D. Hanna, Jul 22 2009

nonn

Compare to g.f. of Fibonacci sequence: exp( Sum_{n>=1} A000204(n)*x^n/n ), where A000204 is the Lucas numbers.

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 40*x^5 + 159*x^6 +...

Cf. A156216, A156100, A159308, A000204.

{a(n)=polcoeff(exp(sum(m=1, n+1, (1+(fibonacci(m-1)+fibonacci(m+1))*x+x*O(x^n))^m*x^m/m)), n)}

A353050 G.f. A(x) satisfies: 0 = Sum_{n>=1} (Lucas(n) - A(x))^n * x^n/n, where Lucas(n) = A000204(n).

Original entry on oeis.org

1, 2, 5, 298, 18949, 3962150, 1916564344, 2501114025582, 8336852053702202, 73027618049882652700, 1666798946804859125492899, 99738726494828465657124210156, 15634873312495144092899303952245929, 6430416165010536428917103922818349814504

Offset: 0

Paul D. Hanna, Apr 26 2022

nonn

It is conjectured that this is an integer sequence.

This sequence is a special case of a more general conjecture: 0 = Sum_{n>=1} (L(n) - F(x))^n * x^n/n is satisfied by an integer series F(x) if exp( Sum_{n>=1} L(n) * x^n/n ) yields an integer series.

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 298*x^3 + 18949*x^4 + 3962150*x^5 + 1916564344*x^6 + 2501114025582*x^7 + 8336852053702202*x^8 + ...
where
0 = (1 - A(x))*x + (3 - A(x))^2*x^2/2 + (4 - A(x))^3*x^3/3 + (7 - A(x))^4*x^4/4 + (11 - A(x))^5*x^5/5 + (18 - A(x))^6*x^6/6 + (29 - A(x))^7*x^7/7 + (47 - A(x))^8*x^8/8 + (76 - A(x))^9*x^9/9 + ... + (Lucas(n) - A(x))^n*x^n/n + ...
Related series.
exp( Sum_{n>=1} A(x)^n * x^n/n ) = 1/(1 - x*A(x)) = 1 + x + 3*x^2 + 10*x^3 + 319*x^4 + 19601*x^5 + 4002282*x^6 + 1924629400*x^7 + 2504975492897*x^8 + 8341867813691252*x^9 + ...
exp( Sum_{n>=1} Lucas(n)^n * x^n/n ) = 1 + x + 5*x^2 + 26*x^3 + 634*x^4 + 32928*x^5 + 5704263*x^6 + 2470113915*x^7 + 2978904483553*x^8 + 9401949327631932*x^9 + ... + A156216(n)*x^n + ...

Paul D. Hanna, Table of n, a(n) for n = 0..100

Cf. A156216, A316363, A000204.

{Lucas(n) = fibonacci(n-1) + fibonacci(n+1)}
{a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
A[#A] = polcoeff( sum(m=1,#A, (Lucas(m) - Ser(A))^m*x^m/m), #A));A[n+1]}
for(n=0,20,print1(a(n),", "))

A381422 Expansion of g.f. = exp( Sum_{n>=1} A066802(n)*x^n/n ).

Original entry on oeis.org

1, 20, 662, 26780, 1205961, 58050204, 2924165436, 152231599628, 8125577046740, 442293253888592, 24457749066666142, 1370114821790970340, 77591333270514869230, 4434803157977731784808, 255492958449660158603448, 14820943641891118200315756, 864962304943085638764540396

Offset: 0

Karol A. Penson, Apr 22 2025

nonn

Cf. A066802, A155200, A255881, A229451, A229452, A156216.

G.f. = 64/((1 + sqrt(1 - 4*x^(1/3)))^2*(1 + sqrt(1 + 4*(-1)^(1/3)*x^(1/3)))^2*(1 + sqrt(1 - 4*(-1)^(2/3)*x^(1/3)))^2).
The above g.f. denoted by h satisfies algebraic equation of order eight:
1 + (8*x - 1)*h + 4*x*(7*x + 3)*h^2 + 7*x^2*(8*x - 1)*h^3 + x^2*(70*x^2 - 40*x + 1)*h^4 + 7*x^4*(8*x - 1)*h^5 + 4*x^5*(7*x + 3)*h^6 + x^6*(8*x - 1)*h^7 + x^8*h^8 = 0.

cp's OEIS Frontend

A203800 a(n) = (1/n) * Sum_{d|n} moebius(n/d) * Lucas(d)^(d-1), where Lucas(n) = A000032(n).

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A067961 Number of binary arrangements without adjacent 1's on n X n torus connected n-s.

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A166168 G.f.: exp( Sum_{n>=1} Lucas(n^2)*x^n/n ) where Lucas(n) = A000204(n).

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A207834 G.f.: exp( Sum_{n>=1} 5*L(n)*x^n/n ), where L(n) = Fibonacci(n-1)^n + Fibonacci(n+1)^n.

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A207835 G.f.: exp( Sum_{n>=1} 5*L(n)*x^n/n ), where L(n) = Fibonacci((n-1)^2) + Fibonacci((n+1)^2).

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A171186 G.f.: exp( Sum_{n>=1} (x^n/n)*[Sum_{k=0..[n/2]} A034807(n,k)^n] ), where A034807 is a triangle of Lucas polynomials.

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A163189 G.f.: A(x) = exp( Sum_{n>=1} (1 + A000204(n)*x)^n * x^n/n ).

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A353050 G.f. A(x) satisfies: 0 = Sum_{n>=1} (Lucas(n) - A(x))^n * x^n/n, where Lucas(n) = A000204(n).

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A207834 G.f.: exp( Sum_{n>=1} 5L(n)x^n/n ), where L(n) = Fibonacci(n-1)^n + Fibonacci(n+1)^n.

A207835 G.f.: exp( Sum_{n>=1} 5L(n)x^n/n ), where L(n) = Fibonacci((n-1)^2) + Fibonacci((n+1)^2).

A163189 G.f.: A(x) = exp( Sum_{n>=1} (1 + A000204(n)x)^n x^n/n ).