duplicate hunting, pt. 15

[Joshua Zucker]

On 5/12/07, Joshua Zucker <joshua.zucker at gmail.com> wrote:
> On 5/12/07, Jon Schoenfield <jonscho at hiwaay.net> wrote:
> > > A025067 and A024371
> >
> > Of A025067 and A024371, is it the case that one of the definitions really
> > yields the sequence 1, 2, 3, 5, 11, 18, etc., and the other doesn't?  Or are
> > they both wrong?
>
> My amateurish calculations seem to show that they are two different
> definitions of the same sequence.

To clarify a bit:
I think these two sequences should be merged, since the terms are the same.
And the definition should be
  SUM from k=1 to floor(n/2) of A023532(k)*Fib(n-k)
and in the comments it can indicate
Also equals the SUM from k = 1 to floor((n+1)/2) of A023532(k)*Fib(n-k+1)

And while we're at it, we might as well add a few more terms since
it's not quite 3 lines full yet.

0 1 2 3 5 11 18 34 55 89 144 246 398 665 1076 1775 2872 4647 7519
12255 19829 32228 52146 84607 136897 221881 359011 580892 939903
1521782 2462295 3985674 6448956 10437214 16887767 27329162 44219513
71555440 115779134 187334574 303113708 490465993 793590647 1284085297
2077693655 3361825320 5439547632 8801447977 14241041977

And finally, you might note I inserted a 0 at the beginning -- should
we have offset 0, and include that leading term?  Or stick with offset
1 and beginning with 1?

Thanks,
--Joshua

Neil:
Here it is in OEIS comment format (also submitted via the web form, so
you can ignore this copy if you like).
I think you should also kill A025067 (maybe replacing it with a
pointer to here).
A quick check doesn't find any cross-refs (inside OEIS or, by Google,
outside of OEIS) to A025067 so I think there's no harm in killing it.

%I A024371
%S A024371 0 1 2 3 5 11 18 34 55 89 144 246 398 665 1076 1775 2872
4647 7519 12255 19829 32228 52146 84607 136897 221881 359011 580892
939903 1521782 2462295 3985674 6448956 10437214 16887767 27329162
44219513 71555440 115779134 187334574 303113708 490465993 793590647
1284085297 2077693655 3361825320 5439547632 8801447977 14241041977
%N A024371 Sum from k=1 to floor(n/2) of A023532(k)*Fib(n-k)
%F A024371 Sum from k=1 to floor(n/2) of A023532(k)*Fib(n-k).
Also equals the sum from k = 1 to floor((n+1)/2) of A023532(k)*Fib(n-k+1)
%e A024371 a(5) = 1*Fib(5) + 1*Fib(3) = 8 + 3 = 11.
%O A024371 0
%K A024371 ,nonn,
%A A024371 Joshua Zucker (joshua.zucker at stanfordalumni.org), May 14 2007




Dear Seqfans,  A correspondent,
"fabio mercurio" <mercurio.fabio at gmail.com>
writes to say that this number

%I A122989
%S A122989 1,0,2,3,4,7,6,3,2
%N A122989 Decimal expansion of Sum_{n >= 1} 1/A007504(n), where A007504(n) is the sum of the f\
irst n primes.
%e A122989 1/2+1/5+1/10+1/17+1/28+1/41+1/58+1/77+1/100+... = 1.02347632...
%Y A122989 Cf. A007504.
%Y A122989 Adjacent sequences: A122986 A122987 A122988 this_sequence A122990 A122991 A122992
%Y A122989 Sequence in context: A026237 A125150 A072275 this_sequence A077223 A055265 A117922
%K A122989 cons,nonn,more
%O A122989 1,3
%A A122989 Pierre CAMI (pierrecami(AT)tele2.fr), Oct 28 2006

is really equal to 1/2 + Pi/6.  But the agreement is
not very close.  Can someone compute more decimal places?

Thanks

Neil