A232221 - OEIS

%I #66 Dec 23 2024 14:53:43

%S 0,0,4,20,36,52,128,216,328,464,636,796,908,1092,1324,1520,1716,1948,

%T 2144,2436,2716,2972,3264,3580,3812,4032,4168,4268,4416,4720,5012,

%U 5328,5716,6128,6504,6700,6932,7248,7684,8180,8676,9268,9680,10140,10624,11024,11400

%N a(n) = Sum_{i=1..n} (A077068(i) - A077065(i)).

%C The sequence has wild fluctuations - see the successive plots in the links. This is typical behavior for a particle whose movement is governed by an arc-sine law (cf. Feller, Chap. III). - _N. J. A. Sloane_, Nov 23 2013

%C Negative stretches: terms 941-1031 and 13197-1431205. - _Hans Havermann_, Nov 23 2013

%C After reaching a local maximum of 21957005755012 at term 24118371, the sequence again descends with the first negative of a third such stretch at term 32437583. - _Hans Havermann_, Nov 28 2013

%C All terms are multiples of 4, cf. A008586. - _Reinhard Zumkeller_, Nov 22 2013

%C See A232361 and A232359 for peak values and where they occur: max{a(A232359(n)-1), a(A232359(n)+1)} < a(A232359(n)) = A232361(n). - _Reinhard Zumkeller_, Nov 24 2013

%D W. Feller, An Introduction to Probability Theory and its Applications, 3rd ed, Wiley, New York, 1968.

%H Reinhard Zumkeller, <a href="/A232221/b232221.txt">Table of n, a(n) for n = 1..10000</a>

%H OEIS "graph" command, <a href="/A232221/a232221.png">Pin plot of 200 terms, scatter plot of 10000 terms</a>

%H Hans Havermann, <a href="http://chesswanks.com/num/A232221/">Four graphs with an increasing number of terms</a>, personal web page.

%H Veikko Pohjola, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2013-November/011977.html">Re: Sister sequences</a>, SeqFan list, Nov 22 2013. (Early observation on how the sequence evolves.)

%o (Haskell)

%o a232221 n = a232221_list !! (n-1)

%o a232221_list = scanl1 (+) a232342_list

%o -- _Reinhard Zumkeller_, Dec 16 2013, Nov 22 2013

%Y Cf. A077065, A077068, A232359, A232361.

%Y Partial sums of A232342.

%K sign,look

%O 1,3

%A _N. J. A. Sloane_, Nov 22 2013, based on a posting by _V.J. Pohjola_ to the Sequence Fans Mailing List, Nov 22 2013