A130205 a(n) = n^2 - a(n-1) - a(n-2), with a(1) = 1 and a(2) = 2.

1, 2, 6, 8, 11, 17, 21, 26, 34, 40, 47, 57, 65, 74, 86, 96, 107, 121, 133, 146, 162, 176, 191, 209, 225, 242, 262, 280, 299, 321, 341, 362, 386, 408, 431, 457, 481, 506, 534, 560, 587, 617, 645, 674, 706, 736, 767, 801, 833, 866, 902, 936, 971, 1009, 1045, 1082

Offset: 1

Author

Zak Seidov, May 16 2007

Keywords

• nonn
• easy

Comments

Any three consecutive terms sum up to a perfect square. First 9 terms coincide with A076991.

Changing a(1) leaves a(5+3m) constant for m >= 0. Changing a(2) leaves a(4+3m) constant for m >= 0. - Richard R. Forberg, Jun 05 2013

Examples

			1+2+6=3^2, 2+6+8=4^2, 6+8+11=5^2.
G.f. = x + 2*x^2 + 6*x^3 + 8*x^4 + 11*x^5 + 17*x^6 + 21*x^7 + 26*x^8 + ...
		

Links

• Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Crossrefs

Cf. A076991, A001840.

Programs

Maple

A130205 := proc(n)
    option remember;
    if n <= 2 then
        n;
    else
        n^2-procname(n-1)-procname(n-2) ;
    end if;
end proc:
seq(A130205(n),n=1..50) ; # R. J. Mathar, Aug 06 2016

Mathematica

a[1]=1;a[2]=2;a[n_]:=a[n]=n^2-a[n-1]-a[n-2]; Table[a[n],{n,100}]
a[ n_] := Quotient[ (n + 1)^2, 3] + 1 - Mod[n, 3]; (* Michael Somos, Aug 04 2016 *)

PARI

a(n)=(n^2+2*n+4)\3 - n%3 \\ Charles R Greathouse IV, Aug 03 2016

Formula

a(1)=1, a(2)=2; n>2: a(n)=n^2-a(n-1)-a(n-2).

G.f.: x*(1+3*x^2-3*x^3+x^4)/(1+x+x^2)/(1-x)^3. - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009; checked and corrected by R. J. Mathar, Sep 16 2009

a(n) = floor((n^2+2*n+1)/3) + 1 - (n mod 3). - Ivan Neretin, May 25 2015

For n>6, a(n)=2*a(n-3)-a(n-6)+6. - Zak Seidov, Aug 05 2016

a(n) = (3*n^2+6*n+1 +8*A049347(n)+7*A049347(n-1))/9.. - R. J. Mathar, Aug 06 2016