A178455 -id:A178455 - cp's OEIS frontend

A178420 Partial sums of floor(2^n/3).

0, 1, 3, 8, 18, 39, 81, 166, 336, 677, 1359, 2724, 5454, 10915, 21837, 43682, 87372, 174753, 349515, 699040, 1398090, 2796191, 5592393, 11184798, 22369608, 44739229, 89478471, 178956956, 357913926, 715827867, 1431655749, 2863311514

Offset: 1

Mircea Merca, Dec 21 2010

Essentially the same as A011377: 0 followed by the terms of A011377. - Joerg Arndt, Apr 22 2016

Partial sums of A000975(n-1).

			a(5) = 0 + 1 + 2 + 5 + 10 = 18.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Mircea Merca, Inequalities and Identities Involving Sums of Integer Functions J. Integer Sequences, Vol. 14 (2011), Article 11.9.1.
Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,2).

Column k=2 of A368296.

Cf. A000975, A178455.

[Floor((4*2^n-3*n-4)/6): n in [1..30]]; // Vincenzo Librandi, Jun 23 2011

Maple
```
seq(round((4*2^n-3*n-4)/6),n=1..50)
```

f[n_] := Floor[(4 2^n - 3 n - 4)/6]; f[Range[60]] (* Vladimir Joseph Stephan Orlovsky, Jan 29 2011 *)
CoefficientList[Series[x / ((1 + x) (1 - 2 x) (1 - x)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Mar 26 2014 *)
LinearRecurrence[{3,-1,-3,2},{0,1,3,8},40] (* or *) Accumulate[ Table[ Floor[ 2^n/3],{n,40}]] (* Harvey P. Dale, Dec 24 2015 *)

a(n)=(4<Charles R Greathouse IV, Jul 31 2013

a(n) = A011377(n-1) for n >= 1. - Joerg Arndt, Apr 22 2016
a(n) = round((8*2^n - 6*n - 9)/12).
a(n) = floor((4*2^n - 3*n - 4)/6).
a(n) = ceiling((4*2^n - 3*n - 5)/6).
a(n) = round((4*2^n - 3*n - 4)/6).
a(n) = a(n-2) + 2^(n-1) - 1, n > 2.
From Bruno Berselli, Jan 15 2011: (Start)
a(n) = (8*2^n - 6*n - 9 + (-1)^n)/12.
G.f.: x^2/((1+x)*(1-2*x)*(1-x)^2). (End)
G.f.: Q(0)/(3*(1-x)^2), where Q(k) = 1 - 1/(4^k - 2*x*16^k/(2*x*4^k - 1/(1 + 1/(2*4^k - 8*x*16^k/(4*x*4^k + 1/Q(k+1)))))); (continued fraction). - Sergei N. Gladkovskii, May 21 2013
a(n) = 2*a(n-1) + floor(n/2) for n > 1. - Bruno Berselli, Apr 30 2014
a(n) = floor(2^(n+1)/3) - floor((n+1)/2). - Seiichi Manyama, Dec 22 2023

A368343 Square array T(n,k), n >= 3, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} k^(n-j) * floor(j/3).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 2, 1, 4, 7, 5, 2, 1, 5, 13, 16, 7, 2, 1, 6, 21, 41, 34, 9, 3, 1, 7, 31, 86, 125, 70, 12, 3, 1, 8, 43, 157, 346, 377, 143, 15, 3, 1, 9, 57, 260, 787, 1386, 1134, 289, 18, 4, 1, 10, 73, 401, 1562, 3937, 5547, 3405, 581, 22, 4

Offset: 3

Seiichi Manyama, Dec 22 2023

			Square array begins:
  1,  1,   1,    1,    1,     1,     1, ...
  1,  2,   3,    4,    5,     6,     7, ...
  1,  3,   7,   13,   21,    31,    43, ...
  2,  5,  16,   41,   86,   157,   260, ...
  2,  7,  34,  125,  346,   787,  1562, ...
  2,  9,  70,  377, 1386,  3937,  9374, ...
  3, 12, 143, 1134, 5547, 19688, 56247, ...

Seiichi Manyama, Antidiagonals n = 3..142, flattened

Columns k=0..4 give A002264, A130518, A178455, A368344, A368345.

Cf. A055129, A368296.

PARI
```
T(n, k) = sum(j=0, n, k^(n-j)*(j\3));
```

T(n,k) = T(n-3,k) + Sum_{j=0..n-3} k^j.
T(n,k) = 1/(k-1) * Sum_{j=0..n} floor(k^j/(k^2+k+1)) = Sum_{j=0..n} floor(k^j/(k^3-1)) for k > 1.
T(n,k) = (k+1)*T(n-1,k) - k*T(n-2,k) + T(n-3,k) - (k+1)*T(n-4,k) + k*T(n-5,k).
G.f. of column k: x^3/((1-x) * (1-k*x) * (1-x^3)).
T(n,k) = 1/(k-1) * (floor(k^(n+1)/(k^3-1)) - floor((n+1)/3)) for k > 1.

A368346 a(n) = Sum_{k=0..n} 2^(n-k) * floor(k/4).

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 7, 15, 32, 66, 134, 270, 543, 1089, 2181, 4365, 8734, 17472, 34948, 69900, 139805, 279615, 559235, 1118475, 2236956, 4473918, 8947842, 17895690, 35791387, 71582781, 143165569, 286331145, 572662298, 1145324604, 2290649216, 4581298440

Offset: 0

Seiichi Manyama, Dec 22 2023

Index entries for linear recurrences with constant coefficients, signature (3,-2,0,1,-3,2).

Partial sums of A083593.

Cf. A178420, A178452, A178455, A178457, A178459, A178460, A178742.

a(n, m=4, k=2) = (k^(n+1)\(k^m-1)-(n+1)\m)/(k-1);

def A368346(n): return (1<>2) # Chai Wah Wu, Dec 22 2023

a(n) = a(n-4) + 2^(n-3) - 1.
a(n) = Sum_{k=0..n} floor(2^k/15).
a(n) = 3*a(n-1) - 2*a(n-2) + a(n-4) - 3*a(n-5) + 2*a(n-6).
G.f.: x^4/((1-x) * (1-2*x) * (1-x^4)).
a(n) = floor(2^(n+1)/15) - floor((n+1)/4).

cp's OEIS Frontend

A178420 Partial sums of floor(2^n/3).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A368343 Square array T(n,k), n >= 3, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} k^(n-j) * floor(j/3).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A368346 a(n) = Sum_{k=0..n} 2^(n-k) * floor(k/4).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Python

Formula