A368296 -id:A368296 - 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.

A097137 Convolution of 3^n and floor(n/2).

Original entry on oeis.org

0, 0, 1, 4, 14, 44, 135, 408, 1228, 3688, 11069, 33212, 99642, 298932, 896803, 2690416, 8071256, 24213776, 72641337, 217924020, 653772070, 1961316220, 5883948671, 17651846024, 52955538084, 158866614264, 476599842805

Offset: 0

Paul Barry, Jul 29 2004

nonn

a(n+1) gives partial sums of A033113 and second partial sums of A015518(n+1). Binomial transform of {0,0,1,1,4,4,16,16,...}.

Partial sums of floor(3^n/8) = round(3^n/8). - Mircea Merca, Dec 28 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-2,-4,3).

Column k=3 of A368296.

Cf. A033113.

a:=[0,0,1,4];; for n in [5..30] do a[n]:=4*a[n-1]-2*a[n-2]-4*a[n-3] +3*a[n-4]; od; a; # G. C. Greubel, Jul 14 2019

[Round((3*3^n-4*n-4)/16): n in [0..30]]; // Vincenzo Librandi, Jun 25 2011

A097137 := proc(n) add( floor(3^i/8),i=0..n) ; end proc:

CoefficientList[Series[x^2/((1-x)^2(1-3x)(1+x)),{x,0,30}],x]  (* Harvey P. Dale, Mar 11 2011 *)

my(x='x+O('x^30)); concat([0,0], Vec(x^2/((1-x)^2*(1-3*x)*(1+x)))) \\ G. C. Greubel, Jul 14 2019

(x^2/((1-x)^2*(1-3*x)*(1+x))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jul 14 2019

G.f.: x^2/((1-x)^2*(1-3*x)*(1+x)).
a(n) = 4*a(n-1) - 2*a(n-2) - 4*a(n-3) + 3*a(n-4).
a(n) = Sum_{k=0..n} floor((n-k)/2)*3^k = Sum_{k=0..n} floor(k/2)*3^(n-k).
From Mircea Merca, Dec 26 2010: (Start)
a(n) = round((3*3^n - 4*n - 4)/16) = floor((3*3^n - 4*n - 3)/16) = ceiling((3*3^n - 4*n - 5)/16) = round((3*3^n - 4*n - 3)/16).
a(n) = a(n-2) + (3^(n-1)-1)/2, n > 2. (End)
a(n) = (floor(3^(n+1)/8) - floor((n+1)/2))/2. - Seiichi Manyama, Dec 22 2023

A097138 Convolution of 4^n and floor(n/2).

Original entry on oeis.org

0, 0, 1, 5, 22, 90, 363, 1455, 5824, 23300, 93205, 372825, 1491306, 5965230, 23860927, 95443715, 381774868, 1527099480, 6108397929, 24433591725, 97734366910, 390937467650, 1563749870611, 6254999482455, 25019997929832

Offset: 0

Paul Barry, Jul 29 2004

a(n+1) gives partial sums of A033114 and second partial sums of A015521.

Partial sums of 1/3*floor(4^n/5). - Mircea Merca, Dec 26 2010

			a(3) = 1/3*floor(4^0/5)+1/3*floor(4^1/5)+1/3*floor(4^2/5) +1/3*floor(4^3/5) = 0 + 0 + 1 + 4 = 5.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (5,-3,-5,4).

Column k=4 of A368296.

Cf. A015521, A033114.

[(4^(n+2)-30*n+9*(-1)^n-25)/180: n in [0..30]]; // Vincenzo Librandi, May 31 2011

A097138 := proc(n) (4^(n+2)-30*n+9*(-1)^n-25)/180 ; end proc: # R. J. Mathar, Jan 08 2011

LinearRecurrence[{5,-3,-5,4},{0,0,1,5},30] (* Harvey P. Dale, Sep 17 2017 *)

G.f.: x^2/((1-x)*(1-4*x)*(1-x^2)).
a(n) = Sum_{k=0..n} floor((n-k)/2)4^k = Sum_{k=0..n} floor(k/2)*4^(n-k).
a(n) = 5*a(n-1) - 3*a(n-2) - 5*a(n-3) + 4*a(n-4).
From Mircea Merca, Dec 26 2010: (Start)
3*a(n) = round((16*4^n-30*n-25)/60) = floor((8*4^n-15*n-8)/30) = ceiling((8*4^n-15*n-17)/30) = round((8*4^n-15*n-8)/30).
a(n) = a(n-2)+(4^(n-1)-1)/3, n>1. (End)
a(n) = (4^(n+2)-30*n+9*(-1)^n-25)/180. - Bruno Berselli, Dec 27 2010
a(n) = (floor(4^(n+1)/15) - floor((n+1)/2))/3. - Seiichi Manyama, Dec 22 2023

A097139 Convolution of 5^n and floor(n/2).

Original entry on oeis.org

0, 0, 1, 6, 32, 162, 813, 4068, 20344, 101724, 508625, 2543130, 12715656, 63578286, 317891437, 1589457192, 7947285968, 39736429848, 198682149249, 993410746254, 4967053731280, 24835268656410, 124176343282061, 620881716410316

Offset: 0

Paul Barry, Jul 29 2004

a(n+1) gives partial sums of A033115 and second partial sums of A015531.

Partial sums of (1/4)*floor(5^n/6) = (1/3)*floor(5^n/8). - Mircea Merca, Dec 27 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-4,-6,5).

Column k=5 of A368296.

[5^(n+1)/96 -n/8 -3/32 +(-1)^n/24: n in [0..30]]; // Vincenzo Librandi, Jun 25 2011

A097139 := proc(n) 5^(n+1)/96 -n/8 -3/32 +(-1)^n/24 ; end proc: # R. J. Mathar, Jan 08 2011

f[n_] := Floor[5^n/6]/4; Accumulate@ Array[f, 24, 0]
a[n_] := a[n] = 6 a[n - 1] - 4 a[n - 2] - 6 a[n - 3] + 5 a[n - 4]; a[0] = a[1] = 0; a[2] = 1; a[3] = 6; Array[a, 24, 0]
CoefficientList[ Series[x^2/((1 - x) (1 - 5 x) (1 - x^2)), {x, 0, 23}], x] (* Robert G. Wilson v, Jan 02 2011 *)
LinearRecurrence[{6,-4,-6,5},{0,0,1,6},30] (* Harvey P. Dale, Mar 16 2019 *)

a(n) = 5^(n+1)/96 -n/8 -3/32 +(-1)^n/24. - R. J. Mathar, Jan 08 2011
G.f.: x^2/((1-x)*(1-5*x)*(1-x^2)).
a(n) = 6*a(n-1) - 4*a(n-2) - 6*a(n-3) + 5*a(n-4).
a(n) = Sum_{k=0..n} floor((n-k)/2)*4^k = Sum_{k=0..n} floor(k/2)*4^(n-k).
From Mircea Merca, Dec 27 2010: (Start)
4*a(n) = round((5*5^n-12*n-9)/24) = floor((5*5^n-12*n-5)/24) = ceiling((5*5^n-12*n-13)/24) = round((5*5^n-12*n-5)/24).
a(n) = a(n-2) + (5^(n-1)-1)/4, n>1. (End)
a(n) = (floor(5^(n+1)/24) - floor((n+1)/2))/4. - Seiichi Manyama, Dec 22 2023

A178719 Partial sums of (1/5)*floor(6^n/7).

Original entry on oeis.org

0, 0, 1, 7, 44, 266, 1599, 9597, 57586, 345520, 2073125, 12438755, 74632536, 447795222, 2686771339, 16120628041, 96723768254, 580342609532, 3482055657201, 20892333943215, 125354003659300, 752124021955810, 4512744131734871, 27076464790409237, 162458788742455434, 974752732454732616

Offset: 0

Mircea Merca, Dec 26 2010

Partial sums of A033116.

			a(3) = (1/5)*(floor(6^1/7) + floor(6^2/7) + floor(6^3/7)) = (1/5)*(0+5+30) = (1/5)*35 = 7.

Vincenzo Librandi, Table of n, a(n) for n = 0..500
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 (7,-5,-7,6).

Column k=6 of A368296.

Cf. A033116.

[(1/5)*Floor((12*6^n-35*n-12)/70): n in [0..30]]; // Vincenzo Librandi, Jun 21 2011

A178719 := proc(n) add( floor(6^i/7)/5,i=0..n) ; end proc:

f[n_] := Floor[6^n/7]/5; Accumulate@ Array[f, 22]
CoefficientList[Series[x^2/((1+x)(1-6x)(1-x)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Mar 26 2014 *)

vector(30, n, n--; (((12*6^n-35*n-12)/70)\1)/5) \\ G. C. Greubel, Jan 24 2019

[floor((12*6^n-35*n-12)/70)/5 for n in (0..30)] # G. C. Greubel, Jan 24 2019

a(n) = (1/5)*round((24*6^n - 70*n - 49)/140).
a(n) = (1/5)*floor((12*6^n - 35*n - 12)/70).
a(n) = (1/5)*ceiling((12*6^n - 35*n - 37)/70).
a(n) = (1/5)*round((12*6^n - 35*n - 12)/70).
a(n) = a(n-2) + (6^(n-1) - 1)/5, n > 1.
a(n) = 7*a(n-1) - 5*a(n-2) - 7*a(n-3) + 6*a(n-4), n > 3.
G.f.: x^2 / ( (1+x)*(1-6*x)*(1-x)^2 ).
a(n) = (24*6^n - 70*n + 25*(-1)^n - 49)/700. - Bruno Berselli, Feb 18 2011
a(n) = (floor(6^(n+1)/35) - floor((n+1)/2))/5. - Seiichi Manyama, Dec 22 2023

A178730 Partial sums of floor(7^n/8)/6.

Original entry on oeis.org

0, 1, 8, 58, 408, 2859, 20016, 140116, 980816, 6865717, 48060024, 336420174, 2354941224, 16484588575, 115392120032, 807744840232, 5654213881632, 39579497171433, 277056480200040, 1939395361400290, 13575767529802040, 95030372708614291, 665212608960300048, 4656488262722100348, 32595417839054702448

Offset: 1

Mircea Merca, Dec 26 2010

Partial sums of A033117.

			a(3) = (1/6)*(floor(7/8) + floor(7^2/8) + floor(7^3/8)) = (1/6)*(0+6+42) = 8.

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 (8,-6,-8,7).

Column k=7 of A368296.

Cf. A033117.

[Floor((7*7^n-24*n-7)/48)/6: n in [1..30]]; // Vincenzo Librandi, Jun 21 2011

A178730 := proc(n) add( floor(7^i/8)/6,i=0..n) ; end proc:

CoefficientList[Series[x/((1+x)(1-7x)(1-x)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Mar 26 2014 *)

vector(30, n, (((7^(n+1)-24*n-7)/48)\1)/6) \\ G. C. Greubel, Jan 24 2019

[floor((7^(n+1)-24*n-7)/48)/6 for n in (1..30)] # G. C. Greubel, Jan 24 2019

6*a(n) = round((7*7^n - 24*n - 16)/48).
6*a(n) = floor((7*7^n - 24*n - 7)/48).
6*a(n) = ceiling((7*7^n - 24*n - 25)/48).
6*a(n) = round((7*7^n - 24*n - 7)/48).
a(n) = a(n-2) + (7^(n-1) - 1)/6, n > 2.
a(n) = 8*a(n-1) - 6*a(n-2) - 8*a(n-3) + 7*a(n-4), n > 4.
G.f.: x^2/((1+x)*(1-7*x)*(1-x)^2).
a(n) = (7^(n+1) - 24*n + 9*(-1)^n - 16)/288. - Bruno Berselli, Jan 11 2011
a(n) = (floor(7^(n+1)/48) - floor((n+1)/2))/6. - Seiichi Manyama, Dec 22 2023

A178827 Partial sums of floor(8^n/9)/7.

Original entry on oeis.org

0, 1, 9, 74, 594, 4755, 38043, 304348, 2434788, 19478309, 155826477, 1246611822, 9972894582, 79783156663, 638265253311, 5106122026496, 40848976211976, 326791809695817, 2614334477566545, 20914675820532370, 167317406564258970, 1338539252514071771, 10708314020112574179, 85666512160900593444, 685332097287204747564

Offset: 1

Mircea Merca, Dec 27 2010

			a(3) = (1/7)*(floor(8/9) + floor(64/9) + floor(512/9)) = (1/7)*(0 + 7 + 56) = (1/7)*63 = 9.

Bruno Berselli, 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 (9,-7,-9,8).

Column k=8 of A368296.

Cf. A033118.

a:=[0,1,9,74];; for n in [5..30] do a[n]:=9*a[n-1]-7*a[n-2]-9*a[n-3] +8*a[n-4]; od; a; # G. C. Greubel, Jan 22 2019

[ &+[Floor(8^k/9)/7: k in [1..n]]: n in [1..25] ]; // Bruno Berselli, Apr 28 2011
(Decimal BASIC)
FOR n=1 TO 1000
   PRINT n; (32*8^n-126*n-81+49*(-1)^n)/1764
NEXT n
END   ! Bruno Berselli, Apr 28 2011

A178827 := proc(n) add( floor(8^i/9)/7,i=0..n) ; end proc:

Rest[CoefficientList[Series[x^2/((1+x)*(1-8*x)*(1-x)^2), {x,0,30}],x]] (* G. C. Greubel, Jan 22 2019 *)

my(x='x+O('x^30)); concat([0], Vec(x^2/((1+x)*(1-8*x)*(1-x)^2))) \\ G. C. Greubel, Jan 22 2019

a=(x^2/((1+x)*(1-8*x)*(1-x)^2)).series(x, 30).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Jan 22 2019

7*a(n) = round((32*8^n - 126*n - 81)/252).
7*a(n) = floor((16*8^n - 63*n - 16)/126).
7*a(n) = ceiling((16*8^n - 63*n - 65)/126).
7*a(n) = round((16*8^n - 63*n - 16)/126).
a(n) = a(n-2) + (8^(n-1) - 1)/7, n > 2.
a(n) = 9*a(n-1) - 7*a(n-2) - 9*a(n-3) + 8*a(n-4), n > 4.
G.f.: x^2/((1+x)*(1-8*x)*(1-x)^2).
7*a(n) = (32*8^n - 126*n - 81 + 49*(-1)^n)/252. - Bruno Berselli, Jan 19 2011
a(n) = (floor(8^(n+1)/63) - floor((n+1)/2))/7. - Seiichi Manyama, Dec 22 2023

cp's OEIS Frontend

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

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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).

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A097137 Convolution of 3^n and floor(n/2).

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A097138 Convolution of 4^n and floor(n/2).

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A097139 Convolution of 5^n and floor(n/2).

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A178719 Partial sums of (1/5)*floor(6^n/7).

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