A093467 a(1) = 1, a(2) = 2; for n >= 2, a(n+1) = a(n) + Sum_{i = 1..n} (a(i) - a(1)).

1, 2, 3, 6, 14, 35, 90, 234, 611, 1598, 4182, 10947, 28658, 75026, 196419, 514230, 1346270, 3524579, 9227466, 24157818, 63245987, 165580142, 433494438, 1134903171, 2971215074, 7778742050, 20365011075, 53316291174, 139583862446

Offset: 1

Author

Amarnath Murthy, Apr 07 2004

Keywords

• nonn
• easy

Comments

If the "man or boy" program A(k, x1, x2, x3) from the program section is run with k > 0 and arbitrary x1, x2, and x3, the result is A055588(k-1)*x1 + A001519(k-1)*x2. - Eric M. Schmidt, Jun 24 2021

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• K. Kuhapatanakul, On the Sums of Reciprocal Generalized Fibonacci Numbers, J. Int. Seq. 16 (2013) #13.7.1. See Theorem 3 p.3.
• Index entries for linear recurrences with constant coefficients, signature (4,-4,1).

Crossrefs

Cf. A093468.

Essentially the same as A032908.

Cf. A001519, A055588.

Programs

Magma

I:=[2,3,6]; [1] cat [n le 3 select I[n] else  4*Self(n-1)-4*Self(n-2)+Self(n-3): n in [1..30]]; // Vincenzo Librandi, Feb 08 2017

Mathematica

a[1] = 1; a[2] = 2; a[n_] := a[n] = a[n - 1] + Sum[a[i] - a[1], {i, n - 1}]; Table[ a[n], {n, 30}]
Join[{1}, LinearRecurrence[{4, -4, 1}, {2, 3, 6}, 30]] (* Vincenzo Librandi, Feb 08 2017 *)

PARI

a(n)=if(n==1,1,if(n==2,2,a(n-1)+sum(i=1,n-1,a(i)-a(1)))) \\ Edward Jiang, Sep 06 2014
(ALGOL 60) begin integer procedure A(k, x1, x2, x3);
    value k; integer k;
    integer x1, x2, x3;
    begin integer procedure b;
        begin
            k:= k - 1;
            B:= A := A (k, B, x1, x2);
        end;
        A := if k <= 0 then x2 + x3 else B;
    end;
    integer i;
    for i:= 0 step 1 until 20 do
        print (A (i, 1, 1, 0));
end
comment The above is a simplified Man or Boy Test program (cf. A132343), omitting the negative parameters from the original. - Leonid Broukhis, Feb 07 2017

Formula

a(n) = 3*a(n-1) - a(n-2) - 1, n > 3. - Robert G. Wilson v, Apr 08 2004

G.f.: x - x^2*(2*x-1)*(x-2) / ( (x-1)*(x^2-3*x+1) ). - R. J. Mathar, Sep 06 2014

a(n) = A055588(n-2) + A001519(n-2), n > 1. - Eric M. Schmidt, Jun 24 2021

Extensions

More terms from Robert G. Wilson v, Apr 08 2004