A232359 -id:A232359 - cp's OEIS frontend

A232221 a(n) = Sum_{i=1..n} (A077068(i) - A077065(i)).

0, 0, 4, 20, 36, 52, 128, 216, 328, 464, 636, 796, 908, 1092, 1324, 1520, 1716, 1948, 2144, 2436, 2716, 2972, 3264, 3580, 3812, 4032, 4168, 4268, 4416, 4720, 5012, 5328, 5716, 6128, 6504, 6700, 6932, 7248, 7684, 8180, 8676, 9268, 9680, 10140, 10624, 11024, 11400

Offset: 1

N. J. A. Sloane, Nov 22 2013, based on a posting by V.J. Pohjola to the Sequence Fans Mailing List, Nov 22 2013

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
Negative stretches: terms 941-1031 and 13197-1431205. - Hans Havermann, Nov 23 2013
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
All terms are multiples of 4, cf. A008586. - Reinhard Zumkeller, Nov 22 2013
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

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

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
OEIS "graph" command, Pin plot of 200 terms, scatter plot of 10000 terms
Hans Havermann, Four graphs with an increasing number of terms, personal web page.
Veikko Pohjola, Re: Sister sequences, SeqFan list, Nov 22 2013. (Early observation on how the sequence evolves.)

Cf. A077065, A077068, A232359, A232361.

Partial sums of A232342.

a232221 n = a232221_list !! (n-1)
a232221_list = scanl1 (+) a232342_list
-- Reinhard Zumkeller, Dec 16 2013, Nov 22 2013

A232361 Peak values in A232221.

Original entry on oeis.org

590868, 586496, 392956, 392612, 369532, 371872, 348760, 92820, 98000, 97424, 268428, 7855172, 11351516, 12763068, 12778820, 12778872, 12778940, 14485860, 16971756, 16976632, 17062548, 17065428, 16934020, 33084456, 33100460, 33116544, 32916668, 32921404

Offset: 1

Reinhard Zumkeller, Nov 24 2013

sign

max{A232221(A232359(n)-1),A232221(A232359(n)+1)} < a(n).

First entry < 0: a(77) = -279411084.

			n = 17: A232359(17) = 3628, a(17) = 12778940:
A232221(3628) = 12778940, whereas A232221(3628-1) = 12778756 and A232221(3628+1) = 12778920 are both less than a(17), with altitude differences 184 and 20. Therefore a(17) is the altitude of a peak in A232221. It can be identified in oeis.org/A232221/graph as top of a foothill.
n = 461: A232359(461) = 373250, a(461) = -2419822248:
A232221(373250) = -2419822248, whereas A232221(373250-1) = -2419823140 and A232221(373250+1) = -2419823192 are both less than a(461), with altitude differences 892 and 944. Therefore a(461) is the altitude of a peak in A232221. It can be identified in Hans Havermann's third plot (< 1.5 million) as the below-axis peak under the 4 of 400000.

Reinhard Zumkeller, Table of n, a(n) for n = 1..890

a232361 n = a232361_list !! (n-1)  -- a232361_list is defined in A232359.

a(n) = A232221(A232359(n)).

cp's OEIS Frontend

A232221 a(n) = Sum_{i=1..n} (A077068(i) - A077065(i)).

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