OFFSET
0,2
COMMENTS
A sequence rich in perfect numbers, since according to the Euclid-Euler theorem all even perfect numbers are of the form 2^(k-1)*(2^k - 1) and in this sequence k = 2*n + 1.
Let G be a sequence satisfying G(i) = 2*G(i-1) + 8*G(i-2) for arbitrary integers i and without regard to the initial values of G. Then a(n) = (G(i+4*n+2) - G(i)*8^(2*n+1))/(2*G(i+2*n+1)) as long as G(i+2*n+1) != 0. - Klaus Purath, Oct 22 2020
In the binary system, the elements of the sequence consist of a total of 4*n+1 bits starting with 2*n+1 ones followed by 2*n zeros. - Martin Renner, Mar 22 2022
Michael Stifel and Petrus Bungus erroneously claimed that for n > 0 all these numbers were perfect. See links: Bungus, Hassler, and Stifel. - Stefano Spezia, Jul 28 2024
LINKS
Petrus Bungus, Numerorum Mysteria, Taupinart, (1618). (In Latin). See p. 468.
Uwe Hassler, Perfect Numbers, Euleriana: 3(2), pp.176-185, (2023). See pp. 177-178.
Michael Stifel, Arithmetica integra, Joh. Petreius, (1544). (In Latin). See p. 11.
Index entries for linear recurrences with constant coefficients, signature (20,-64).
FORMULA
a(n+1) = 16*a(n) + 12*2^(2n).
a(n) = Sum_{k=1..2^n} (2*k-1)^3. - Franz Vrabec, Jun 24 2006
a(n) = Sum_{k=2*n..4*n} 2^k. - Martin Renner, Mar 22 2022
G.f.: ( 1+8*x ) / ( (16*x-1)*(4*x-1) ). - R. J. Mathar, Nov 29 2011
E.g.f.: exp(4*x)*(2*exp(12*x) - 1). - Stefano Spezia, Jul 28 2024
MAPLE
seq(sum((2*k-1)^3, k=1..2^n), n=0..15);
seq(sum(2^k, k=2*n..4*n), n=0..15);
PROG
(PARI) a(n)=2^(4*n+1)-4^n \\ Charles R Greathouse IV, Nov 29 2011
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Marco Matosic, Jan 28 2003
STATUS
approved