A093996 -id:A093996 - cp's OEIS frontend

A298949 Expansion of Product_{k>=2} 1/(1 + x^{F_k}) where F_k are the Fibonacci numbers.

1, -1, 0, -1, 2, -2, 2, -2, 2, -3, 4, -3, 3, -5, 5, -5, 7, -7, 7, -9, 10, -11, 12, -12, 13, -16, 18, -17, 18, -21, 23, -25, 26, -27, 29, -32, 35, -36, 37, -40, 43, -46, 50, -51, 52, -58, 63, -64, 67, -71, 73, -79, 85, -85, 88, -96, 100, -104, 111, -113, 117

Offset: 0

Seiichi Manyama, Jan 30 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A000045, A000119, A003107, A093996.

Convolution inverse of A000119.

A104767 a(n)=n for n <= 3, a(n) = 2a(n-1) - 2a(n-2) + 2a(n-3) for n >= 4.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 10, 16, 24, 36, 56, 88, 136, 208, 320, 496, 768, 1184, 1824, 2816, 4352, 6720, 10368, 16000, 24704, 38144, 58880, 90880, 140288, 216576, 334336, 516096, 796672, 1229824, 1898496, 2930688, 4524032, 6983680, 10780672, 16642048, 25690112, 39657472

Offset: 0

Don N. Page, Oct 13 2005

nonn

Also a(n) for n > 0 is the number of terms in the expansion of (x - y) * (x - y) * (x^2 - y^2) * (x^3 - y^3) * ... * (x^F_n-1 - y^F_n-1), where F_n is the n-th Fibonacci number. In this definition one can take y=1. In other words the sequence gives the number of nonzero terms in the polynomial Product {k=1..n-1}, (1 - x^F_k). - Robert G. Wilson v, May 12 2013

Also a(n) for n > 0 is the number of terms in the expansion of Product_{k=2..n+1} (x^F_k - y^F_k) with coefficient +1 (same with -1). We can take y=1 and the Product_{k=2..n+1} (x^F_k - 1) has a(n) terms with coefficient +1 and same with -1. Note that no coefficient is greater than 1 in absolute value. - Michael Somos, May 17 2018

			From _Michael Somos_, May 17 2018: (Start)
For n=3, (x - y) * (x - y) = x^2 - 2*x*y + y^2 has a(3) = 3 terms.
For n=4, (x - y) * (x - y) * (x^2 - y^2) = x^4 - 2*x^3*y + 2*x*y^3 - y^4 has a(4) = 4 terms.
for n=2, (x - y) * (x^2 - y^2) = x^3 - x^2*y - x*y^2 + y^3 has a(2) = 2 terms with + sign and also with - sign.
For n=3, (x - y) * (x^2 - y^2) * (x^3 - y^3) = x^6 - x^5*y - x^4*y^2 + x^2*y^4 + x*y^5 - y^6 has a(3) = 3 terms with + sign and also with - sign. (End)

Muniru A Asiru, Table of n, a(n) for n = 0..1000
Richard P. Stanley, Theorems and Conjectures on Some Rational Generating Functions, arXiv:2101.02131 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (2,-2,2).

Cf. A093996.

a:=[0,1,2,3,4];; for n in [5..50] do a[n]:=2*a[n-1]-2*a[n-2]+2*a[n-3]; od; a; # Muniru A Asiru, May 17 2018

f:=proc(n) option remember; if n <= 4 then RETURN(n); fi; 2*f(n-4)+f(n-1); end;

a[n_] := a[n] = If[n < 4, n, 2a[n - 1] - 2a[n - 2] + 2a[n - 3]]; Table[ a[n], {n, 0, 39}] (* Robert G. Wilson v *)
Join[{0}, LinearRecurrence[{2, -2, 2}, {1, 2, 3}, 41]] (* Robert G. Wilson v, May 12 2013 *)
Join[{0}, LinearRecurrence[{1, 0, 0, 2}, {1, 2, 3, 4}, 41]] (* Robert G. Wilson v, May 12 2013 *)
a[n_] := Length@ ExpandAll@ Product[1 - x^Fibonacci[k], {k, n-1}]; a[1] = 1; (* Robert G. Wilson v, May 12 2013 *)
nxt[{a_,b_,c_}]:={b,c,2c-2b+2a}; Join[{0},NestList[nxt,{1,2,3},40][[All,1]]] (* Harvey P. Dale, Nov 30 2021 *)

a=vector(100); a[1]=1;a[2]=2;a[3]=3; for(n=4, #a, a[n] = 2*a[n-1]-2*a[n-2]+2*a[n-3]); concat(0,a) \\ Altug Alkan, May 18 2018

a(n) = n for n <= 4; for n >= 5, a(n) = 2a(n-4) + a(n-1).

G.f.: (x + x^3)/(-2*x^3 + 2*x^2 - 2*x + 1). a(n) = A077943(n-3) + A077943(n-1).

More terms from Robert G. Wilson v, Oct 14 2005

A151661 Exponents in g.f. Product_{k>=2} (1 - x^{F_k}) where F_k are the Fibonacci numbers.

Original entry on oeis.org

0, 1, 2, 4, 7, 8, 11, 12, 13, 14, 18, 19, 20, 22, 23, 24, 29, 30, 31, 33, 36, 38, 39, 40, 47, 48, 49, 51, 54, 55, 58, 59, 62, 64, 65, 66, 76, 77, 78, 80, 83, 84, 87, 88, 89, 90, 94, 95, 96, 97, 100, 101, 104, 106, 107, 108, 123, 124, 125, 127, 130, 131, 134, 135, 136, 137, 141, 142

Offset: 1

N. J. A. Sloane, May 30 2009

nonn

			1 - x - x^2 + x^4 + x^7 - x^8 + x^11 - x^12 - x^13 + x^14 + x^18 - x^19 - x^20 + x^22 + x^23 - x^24 + x^29 - x^30 - x^31 + x^33 + x^36 - x^38 - x^39 + x^40 + x^47 - ...

F. Ardila, The Coefficients of a Fibonacci power series, arXiv:math/0409418 [math.CO], 2004.
N. Robbins, Fibonacci Partitions, The Fibonacci Quarterly, 34.4 (1996), pp. 306-313.
Yufei Zhao, The coefficients of a truncated Fibonacci power series, Fib. Q., 46/47 (2008/2009), 53-55.

Cf. A000045, A093996.

kmax = 150; Exponent[#, x]& /@ List @@ (Product[1 - x^Fibonacci[k], {k, 2, Ceiling[FindRoot[Fibonacci[x] == kmax, {x, 5}][[1, 2]]]}] + O[x]^kmax // Normal) (* Jean-François Alcover, Oct 08 2018 *)

A357380 Expansion of Product_{k>=1} (1 - x^Fibonacci(k)).

Original entry on oeis.org

1, -2, 0, 1, 1, -1, 0, 1, -2, 1, 0, 1, -2, 0, 2, -1, 0, 0, 1, -2, 0, 1, 1, 0, -2, 1, 0, 0, 0, 1, -2, 0, 1, 1, -1, 0, 1, -1, -1, 0, 2, -1, 0, 0, 0, 0, 0, 1, -2, 0, 1, 1, -1, 0, 1, -2, 1, 0, 1, -2, 1, 0, -1, 1, 1, 0, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 0, 1, 1, -1, 0, 1, -2, 1, 0, 1, -2, 0, 2, -1, 0, 0, 1, -2, 0, 2, -1, 0, -1

Offset: 0

Ilya Gutkovskiy, Sep 26 2022

sign

Convolution inverse of A007000.

First differences of A093996.

Cf. A000045, A007000, A093996, A357381.

nmax = 100; CoefficientList[Series[Product[(1 - x^Fibonacci[k]), {k, 1, 21}], {x, 0, nmax}], x]

A329003 Numbers k such that the coefficient of x^k in the expansion of Product_{j>=2} (1 - x^Fibonacci(j)) is zero.

Original entry on oeis.org

3, 5, 6, 9, 10, 15, 16, 17, 21, 25, 26, 27, 28, 32, 34, 35, 37, 41, 42, 43, 44, 45, 46, 50, 52, 53, 56, 57, 60, 61, 63, 67, 68, 69, 70, 71, 72, 73, 74, 75, 79, 81, 82, 85, 86, 91, 92, 93, 98, 99, 102, 103, 105, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120

Offset: 1

Ilya Gutkovskiy, Nov 01 2019

nonn

Numbers k such that number of partitions of k into an even number of distinct Fibonacci parts equals number of partitions of k into an odd number of distinct Fibonacci parts (1 counted as single Fibonacci number).
Positions of 0's in A093996.
Complement of A151661.

Cf. A000045, A000119, A003107, A090864, A093996, A151661.

Flatten[Position[Rest[CoefficientList[Series[Product[(1 - x^Fibonacci[j]), {j, 2, 21}], {x, 0, 130}], x]], 0]]

A357520 Expansion of Product_{k>=0} (1 - x^Lucas(k)).

Original entry on oeis.org

1, -1, -1, 0, 0, 2, 0, -1, 0, 0, 1, -1, -1, 1, 0, 1, -1, -1, 0, 1, 1, -1, 0, 0, 0, 1, -1, -1, 0, 0, 2, 0, 0, -1, -1, 1, 0, 0, 0, 0, 1, -1, -1, 0, 0, 2, 0, -1, 0, 0, 1, 0, -2, 0, 0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, 1, -1, -1, 0, 0, 2, 0, -1, 0, 0, 1, -1, -1, 1, 0, 1, -1, -1, 1, 0, 0, -1, 0, 2, 0, 0, -1, -1, 1

Offset: 0

Ilya Gutkovskiy, Oct 02 2022

sign

Convolution inverse of A067593.

Cf. A000032, A067593, A093996, A357380, A357382.

nmax = 100; CoefficientList[Series[Product[(1 - x^LucasL[k]), {k, 0, 20}], {x, 0, nmax}], x]

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A298949 Expansion of Product_{k>=2} 1/(1 + x^{F_k}) where F_k are the Fibonacci numbers.

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A104767 a(n)=n for n <= 3, a(n) = 2a(n-1) - 2a(n-2) + 2a(n-3) for n >= 4.

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A151661 Exponents in g.f. Product_{k>=2} (1 - x^{F_k}) where F_k are the Fibonacci numbers.

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A357380 Expansion of Product_{k>=1} (1 - x^Fibonacci(k)).

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A329003 Numbers k such that the coefficient of x^k in the expansion of Product_{j>=2} (1 - x^Fibonacci(j)) is zero.

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A357520 Expansion of Product_{k>=0} (1 - x^Lucas(k)).

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