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
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).
FORMULA
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 *)
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 *)
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
KEYWORD
nonn
AUTHOR
Don N. Page, Oct 13 2005
EXTENSIONS
More terms from Robert G. Wilson v, Oct 14 2005
STATUS
approved