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, 365435296163, 956722026042

Offset

1,2

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

Kantaphon Kuhapatanakul, On the Sums of Reciprocal Generalized Fibonacci Numbers, J. Int. Seq., Vol. 16 (2013), Article 13.7.1. See Theorem 3, p.3.

Index entries for linear recurrences with constant coefficients, signature (4,-4,1).

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

Sum_{n>=1} 1/a(n) = phi + 1/2, where phi is the golden ratio (A001622). - Amiram Eldar, Dec 16 2025

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 *)

Prog

(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
(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

Crossrefs

Cf. A001622, A093468.

Essentially the same as A032908.

Cf. A001519, A055588.

Sequence in context: A238823 A002995 A384630 * A246640 A080408 A275774

Adjacent sequences: A093464 A093465 A093466 * A093468 A093469 A093470

Keyword

nonn,easy

Author

Amarnath Murthy, Apr 07 2004

Extensions

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

Status

approved