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

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

Author

Don N. Page, Oct 13 2005

Keywords

• nonn

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

Examples

			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)
		

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).

Crossrefs

Cf. A093996.

Programs

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

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 *)

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

Formula

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).

Extensions

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