A001650

[franktaw at netscape.net]

It looks good to me, but you have to look at references to A001650, too.

A070015 has repeated references to A001650, but it's all a mistake: the 
desired reference is A001065 instead.

A000122 is OK; "partial sums" doesn't really specify the offset.

A001670 has problems, though not specifically with with A001650.  The 
formula, "a(n) = 2*floor( 1/2 + sqrt(n)); n > 0" is correct for offset 
1 instead of the current 2.  The first g.f., 
"(1-2x^2+x^3)/(1-2x+x^3-x^4)", is for A049868 instead of this sequence. 
  The asymptotic following also belongs with that sequence.  The final 
g.f. - "Sum(i=0, inf, 2*x^(i^2+i)/(1-x))" - is for offset 0.

A069896 has a reference to A001650.  It's hard to be sure, because the 
connection is not direct, but I think this one is also supposed to be 
A001065.  Ditto for A070016.  A070010 clearly is intending to reference 
A001065, which strongly implies that A069896 and A070016 are, too.

A095875 has references to A001650 that do depend on the offset.  It 
would be simplified if the proposed change is made.

----
If the offset for A001670 is changed to 1, and the first g.f. and 
asymptotic are moved to A049868, the only other change needed to 
A001670 is to change the remaining g.f. to "Sum(i=0, inf, 
2*x^(i^2+i+1)/(1-x))" - or maybe "Sum_{i=0}^{infinity} 
2*x^(i^2+i+1)/(1-x)".

There is only one reference to A001670, and it does not need to change.

Note, by the way,that A001650 has two crossrefs to A001670; I think one 
would suffice.

Franklin T. Adams-Watters


-----Original Message-----
From: davidwwilson at comcast.net

It seems to me that A001650 "wants" to start at index 0 instead of 
index 1. If we do this, the formulae simplify and special cases 
disappear, so that the current
 
%F A001650 a(n) = 1 + 2*floor(sqrt(n-1)), n > 0. - Antonio Esposito 
(antonio.b.esposito(AT)italtel.it), Jan 21 2002
%F A001650 G.f.: theta_3(x)*x/(1-x). a(n+1)=a(n)+A000122(n). - Michael 
Somos, Apr 29 2003.
%o A001650 (PARI) a(n)=if(n<1,0,1+2*sqrtint(n-1))
 
becomes
 
%F A001650 a(n) = 1 + 2*floor(sqrt(n)). - Antonio Esposito 
(antonio.b.esposito(AT)italtel.it), Jan 21 2002
%F A001650 G.f.: theta_3(x)/(1-x). a(n)=a(n-1)+A000122(n). - Michael 
Somos, Apr 29 2003.
%o A001650 (PARI) a(n)=1+2*sqrtint(n)
 
My interest in making this change is that it also makes 
A001650 consistent with a batch of sequences I plan to submit in the 
near future.
 


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