[SeqFan] Sequence A002061 vs. A034111.

[Wouter Meeussen]

Patrick,

from one non-mathematician to an other in the style of Magritte
(ceci n'est pas de la mathematique) (;-))



f[n_]:=Sqrt[n^2 - n + 1]

Limit[f[k]-k,k->Infinity] equals  -1/2

or an other way:
Series[f[n]-n ,{n,\[Infinity],4}]
        equals
-(1/2) - 15/(256*n^4) + 3/(128*n^3) + 3/(16*n^2) + 3/(8*n) +...

so, from the moment that f[n]-n+1/2 < 1/10 , the decimal expansion will
start with 5.
That happens at n>=  Solve[f[n]-n+1/2==1/10,n]
or n>= 21/5 or 4.2
So from the fifth term onwards.


wouter.
========================================================


At 15:37 4-10-98 +0100, Patrick De Geest wrote:
>Hi to all,
>
>Last month I submitted the following sequence to Sloane's table :
>
>%I A034111
>%S A034111
21,31,43,57,73,91,111,133,157,183,211,241,273,307,343,381,421,463,507,
>%T A034111 553,601,651,703,757,813,871,931,993,1057,1123,1191,1261,1333,1407,
>%U A034111
1483,1561,1641,1723,1807,1893,1981,2071,2163,2257,2353,2451,2551,2653
>%N A034111 Decimal part of square root of a(n) starts with 5 : first term
of runs.
>%C A034111 Is basically sequence A002061 (n^2-n+1 = central polygonal
numbers) starting from the sixth term.
>%O A034111 0,1
>%K A034111 nonn,base
>%Y A034111 Cf. A034101.
>%A A034111 Patrick De Geest (Patrick.DeGeest at ping.be), September 1998.
>
>When I checked for possible presence I came across the next sequence :
>
>%I A002061 M2638 N1049
>%S A002061
1,1,3,7,13,21,31,43,57,73,91,111,133,157,183,211,241,273,307,343,381,
>%T A002061 421,463,507,553,601,651,703,757,813,871,931,993,1057,1123,1191,1261
>%N A002061 Central polygonal numbers: n^2 - n + 1.
>%R A002061 HO50 22. HO70 87.
>%O A002061 0,3
>%A A002061 njas
>%K A002061 nonn,easy
>
>I never anticipated that such different approaches can yield the same terms.
>But then again, I am not a genuine mathematician. I would appreciate it
very much
>if someone could explain this phenomenon or give me more insight into this
matter.
>Thanks in advance.
>
>-- 
>Patrick De Geest
>[mailto:Patrick.DeGeest at ping.be]
>---------------------------------------------------
>URL : http://www.ping.be/~ping6758/index.shtml
>---------------------------------------------------
>
>
>
Dr. Wouter L. J. MEEUSSEN
w.meeussen.vdmcc at vandemoortele.be
eu000949 at pophost.eunet.be