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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. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

LINKS

Muniru A Asiru, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2, -2, 2).

FORMULA

Or, 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).

EXAMPLE

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)

MAPLE

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

MATHEMATICA

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 *)

PROG

(GAP) 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

(PARI) 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

CROSSREFS

Cf. A093996.

Sequence in context: A070542 A098855 A143283 * A072944 A024722 A024965

Adjacent sequences:  A104764 A104765 A104766 * A104768 A104769 A104770

KEYWORD

nonn

AUTHOR

Don N. Page, Oct 13 2005

EXTENSIONS

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

STATUS

approved

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Last modified September 20 13:45 EDT 2020. Contains 337264 sequences. (Running on oeis4.)