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A077071
Row sums of A077070.
2
0, 2, 8, 16, 30, 46, 66, 88, 118, 150, 186, 224, 268, 314, 364, 416, 478, 542, 610, 680, 756, 834, 916, 1000, 1092, 1186, 1284, 1384, 1490, 1598, 1710, 1824, 1950, 2078, 2210, 2344, 2484, 2626, 2772, 2920, 3076, 3234, 3396, 3560, 3730, 3902, 4078, 4256
OFFSET
0,2
COMMENTS
Conjecture: given n balls, all of which are initially in the first of n numbered boxes, a(n-1) is the number of steps of the following process required to move them all to the last box. A step consists of first identifying j, the lowest numbered box which has at least one ball. If it has only one ball then move it to box j+1; otherwise move half its balls rounded down to box j+1 and (unless it's the first box) half its balls rounded down to box j-1. See also A356254. - Mikhail Kurkov, Nov 24 2022
FORMULA
a(n) is asymptotic to 2*n^2 and it seems that a(n) = 2*n^2 + O(n^(3/2)) (where O(n^(3/2))/n^(3/2) is bounded and O(n^(3/2)) < 0). - Benoit Cloitre, Oct 30 2002
G.f.: (1/(1-x)^2) * Sum_{k>=0} t/(1-t) where t = x^2^k. Twice the value of the partial sum of A005187. a(0) = 0, a(2n) = a(n) + a(n-1) + 4*n^2 + 2*n, a(2n+1) = 2*a(n) + 4*n^2 + 6*n + 2. - Ralf Stephan, Sep 12 2003
a(n) = 2*n*(n+1) - 2*A000788(n) and therefore asymptotically a(n) = 2*n^2 - n*log_2(n) + O(n). - Peter J. Taylor, Dec 21 2022
PROG
(PARI) {a(n) = sum( k=0, n, -valuation( polcoeff( pollegendre(2*n), 2*k), 2))}
(PARI) a(n)=my(P=pollegendre(2*n)); -sum(k=0, n, valuation(polcoeff(P, 2*k), 2)) \\ Charles R Greathouse IV, Apr 12 2012
CROSSREFS
Sequence in context: A194643 A327329 A136514 * A187216 A210729 A294534
KEYWORD
nonn
AUTHOR
Michael Somos, Oct 25 2002
STATUS
approved