[seqfan] Re: [2] close sequences
Christian G.Bower
bowerc at usa.net
Mon Aug 3 08:29:54 CEST 1998
Wouter:
> at that point, how certain can you be that there is no
> fault in the algorithm to calculate them?
In general I can't but in the cases I listed the match-up
can be easily verified by the formula.
I:
> Here are some examples:
>
> A001399 A008761 18 terms
A001399 has generating function 1/((1-x)*(1-x^2)*(1-x^3))
A008761 had generating function (1+x^18)/((1-x)*(1-x^2)*(1-x^3))
So A008761 is just a convolution of 1 [17 0's] 1 [all 0's] with
A001399 and the correspondence is trivial.
A008632/A008638
A008633/A008639
A008634/A008640
are similar.
A002620/A025699
%I A002620 M0998 N0374
%S A002620 0,0,1,2,4,6,9,12,16,20,25,30,36,42,49,56,64,72,81,90,100,110,
%T A002620 121,132,144,156,169,182,196,210,225,240,256,272,289,306,
%U A002620 324,342,361,380,400,420,441,462,484,506,529,552,576,600
%N A002620 Quarter-squares: [n/2]*ceiling(n/2) .
%R A002620 AMS 26 304 1955. GKP 99.
%O A002620 0,4
%K A002620 nonn,easy,nice
%p A002620 series(x^2/((1-x)^2*(1-x^2)),x,60);
%A A002620 njas
%P A002620 My ref Latt 73 p 106.
%I A025699
%S A025699 1,2,4,6,9,12,16,20,25,30,36,42,49,56,64,72,81,90,100,111,122,134,146,
%T A025699 159,172,186,200,215,230,246,262,279,296,314,332,351,371,391,412,433,455,
%U A025699 477,500,523,547,571,596,621,647,673,700,727,755,784,813,843,873,904,935
%N A025699 Index of 3^n within sequence of numbers of form 3^i*8^j.
%O A025699 1,2
%K A025699 nonn
%A A025699 wilson at ctron.com
A002620 is partial sums of 0 0 1 1 2 2 3 3 4 4...
Can calculate A025699 as follows:
Write the powers of 3 in base 8.
Add up all the digits:
3^n (base 8) sum of digits
1 1
3 2
11 4
33 6
121 9
363 12
1331 16
...
754502703 90
2705710511 100
10521531733 111
...
Also A025699 is sum{k=0..n}([k*log(3)/log(8)]+1)
Since log(3)/log(8)=0.52832... is close to 1/2 the sequence follows the quarter-squares for a while before diverging.
Incidentally, Neil (and Dave), the starting
index of A025699 should be 0, not 1.
I have other close sequences I did not post because I haven't verified them yet.
Christian
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