A077070 -id:A077070 - cp's OEIS frontend

A077071 Row sums of A077070.

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

Michael Somos, Oct 25 2002

nonn

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

Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, preprint, 2016.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms 13:4 (2017), #47.
Ralf Stephan, Some divide-and-conquer sequences with (relatively) simple generating functions, 2004.
Ralf Stephan, Table of generating functions (ps file).
Ralf Stephan, Table of generating functions (pdf file).

Cf. A077070, A356254.

{a(n) = sum( k=0, n, -valuation( polcoeff( pollegendre(2*n), 2*k), 2))}

a(n)=my(P=pollegendre(2*n)); -sum(k=0,n,valuation(polcoeff(P,2*k), 2)) \\ Charles R Greathouse IV, Apr 12 2012

def A077071(n): return ((n+1)*(n-n.bit_count())<<1)-sum((m:=1<>j)-(r if n<<1>=m*(r:=k<<1|1) else 0)) for j in range(1,n.bit_length()+1)) # Chai Wah Wu, Nov 12 2024

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

A144816 Denominators of triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) is the coefficient of x^(2*k+1) in polynomial t_n(x), used to define continuous and n times differentiable sigmoidal transfer functions.

Original entry on oeis.org

1, 2, 2, 8, 4, 8, 16, 16, 16, 16, 128, 32, 64, 32, 128, 256, 256, 128, 128, 256, 256, 1024, 512, 1024, 256, 1024, 512, 1024, 2048, 2048, 2048, 2048, 2048, 2048, 2048, 2048, 32768, 4096, 8192, 4096, 16384, 4096, 8192, 4096, 32768, 65536, 65536, 16384, 16384, 32768, 32768, 16384, 16384, 65536, 65536

Offset: 0

Alois P. Heinz, Sep 21 2008

			Triangle begins:
    1;
    2,  2;
    8,  4,  8;
   16, 16, 16, 16;
  128, 32, 64, 32, 128;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

See A144815 for more information on T(n,k).
Main diagonal and column k=0 gives A046161.
Column k=1 gives A101926(n-1) = 2^A101925(n-1) = 2^(A005187(n-1)+1).
Cf. A077070.

# Function T(n,k) defined in A144815.
seq(seq(denom(T(n,k)), k=0..n), n=0..10);

row[n_] := Module[{f, a, eq}, f = Function[x, Sum[a[2*k+1]*x^(2*k+1), {k, 0, n}]]; eq = Table[Derivative[k][f][1] == If[k == 0, 1, 0], {k, 0, n}]; Table[a[2*k+1], {k, 0, n}] /. Solve[eq] // First]; Table[row[n] // Denominator, {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 03 2014 *)

A076178 a(n) = 2*n^2 - A077071(n).

Original entry on oeis.org

0, 0, 0, 2, 2, 4, 6, 10, 10, 12, 14, 18, 20, 24, 28, 34, 34, 36, 38, 42, 44, 48, 52, 58, 60, 64, 68, 74, 78, 84, 90, 98, 98, 100, 102, 106, 108, 112, 116, 122, 124, 128, 132, 138, 142, 148, 154, 162, 164, 168, 172, 178, 182, 188, 194, 202, 206, 212, 218, 226, 232

Offset: 0

Benoit Cloitre, Nov 01 2002

nonn

Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, preprint, 2016.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms 13:4 (2017), #47.
Ralf Stephan, Some divide-and-conquer sequences with (relatively) simple generating functions, 2004.
Ralf Stephan, Table of generating functions (ps file).
Ralf Stephan, Table of generating functions (pdf file).

Equals 2 * A078903(n).

Cf. A001105, A067699, A077070, A077071.

a(n)=2*n^2-sum(k=0,n,-valuation(polcoeff(pollegendre(2*n),2*k),2))

def A076178(n): return ((n+1)*n.bit_count()-n<<1)+sum((m:=1<>j)-(r if n<<1>=m*(r:=k<<1|1) else 0)) for j in range(1,n.bit_length()+1)) # Chai Wah Wu, Nov 12 2024

a(n) = A001105(n) - A077071(n). - Omar E. Pol, Nov 13 2024

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A077071 Row sums of A077070.

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A144816 Denominators of triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) is the coefficient of x^(2*k+1) in polynomial t_n(x), used to define continuous and n times differentiable sigmoidal transfer functions.

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A076178 a(n) = 2*n^2 - A077071(n).

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