A128135 -id:A128135 - cp's OEIS frontend

A128134 A128132 * A007318.

1, 1, 2, 2, 5, 3, 3, 10, 11, 4, 4, 17, 27, 19, 5, 5, 26, 54, 56, 29, 6, 6, 37, 95, 130, 100, 41, 7, 7, 50, 153, 260, 265, 162, 55, 8, 8, 65, 231, 469, 595, 483, 245, 71, 9, 9, 82, 332, 784, 1190, 1204, 812, 352, 89, 10

Offset: 1

Gary W. Adamson, Feb 15 2007

A007318 * A128132 = A128133. Row sums = A128135: (1, 3, 10, 28, 72, 176, ...).

			Triangle T(n,k) (with rows n >= 1 and columns k = 1..n) begins:
  1;
  1,  2;
  2,  5,  3;
  3, 10, 11,   4;
  4, 17, 27,  19,   5;
  5, 26, 54,  56,  29,  6;
  6, 37, 95, 130, 100, 41, 7;
  ...

Cf. A002522, A007318, A028387, A128132, A128133, A128135, A164845, A332697.

A128132 * A007318 as infinite lower triangular matrices (assuming the top of the Pascal triangle A007318 is shifted from (0,0) to (1,1)).
From Petros Hadjicostas, Jul 26 2020: (Start)
T(n,k) = n*binomial(n-1, k-1) - binomial(n-2, k-1)*[n <> k] for 1 <= k <= n, where [ ] is the Iverson bracket.
Bivariate o.g.f.:  x*y*(1 - x + x^2*(1 + y))/(1 - x*(1 + y))^2.
T(n,k) = T(n-1,k) + T(n-1,k-1) + binomial(n-1,k-1) for 2 <= k <= n with (n,k) <> (2,2).
T(n,k) = 2*T(n-1,k) - T(n-2,k) - T(n-2,k-2) + 2*T(n-1,k-1) - 2*T(n-2,k-1) for 3 <= k <= n with (n,k) <> (3,3).
T(n,1) = n - 1 for n >= 2.
T(n,2) = A002522(n-1) for n >= 2.
T(n,3) = A164845(n-3) for n >= 3.
T(n,4) = A332697(n-3) for n >= 4.
T(n,n) = n for n >= 1.
T(n,n-1) = A028387(n-2) for n >= 2. (End)

A134315 A134309 * A097806.

Original entry on oeis.org

1, 1, 1, 0, 2, 2, 0, 0, 4, 4, 0, 0, 0, 8, 8, 0, 0, 0, 0, 16, 16, 0, 0, 0, 0, 0, 32, 32, 0, 0, 0, 0, 0, 0, 64, 64

Offset: 1

Gary W. Adamson, Oct 19 2007

A134315 * [1,2,3,...] = A128135: (1, 3, 10 28, 72, 176, 416, ...).

Triangle read by rows given by [1,-1,0,0,0,0,0,0,...] DELTA [1,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 20 2007

			First few rows of the triangle are:
  1;
  1, 1;
  0, 2, 2;
  0, 0, 4, 4;
  0, 0, 0, 8, 8;
  ...

Cf. A134309, A097806, A128135.

A134309 * A134315 as infinite lower triangular matrices. Triangle read by rows, for n>1, (n-1) zeros followed by 2^(n-1), 2^(n-1). As an infinite lower triangular matrix, (1, 1, 2, 4, 8, ...) in the main diagonal and (1, 2, 4, 8, ...) in the subdiagonal.

G.f.: (-1-x+x*y)/(-1+2*x*y). - R. J. Mathar, Aug 11 2015

A204204 Triangle based on (0,3/4,1) averaging array.

Original entry on oeis.org

3, 3, 7, 3, 10, 15, 3, 13, 25, 31, 3, 16, 38, 56, 63, 3, 19, 54, 94, 119, 127, 3, 22, 73, 148, 213, 246, 255, 3, 25, 95, 221, 361, 459, 501, 511, 3, 28, 120, 316, 582, 820, 960, 1012, 1023, 3, 31, 148, 436, 898, 1402, 1780, 1972, 2035, 2047, 3, 34, 179

Offset: 1

Clark Kimberling, Jan 12 2012

See A204201 for a discussion and guide to other averaging arrays.

			First six rows:
3
3...7
3...10...15
3...13...25...31
3...16...38...56...63
3...19...54...94...119..127

Cf. A204201.

a = 0; r = 3/4; b = 1;
t[1, 1] = r;
t[n_, 1] := (a + t[n - 1, 1])/2;
t[n_, n_] := (b + t[n - 1, n - 1])/2;
t[n_, k_] := (t[n - 1, k - 1] + t[n - 1, k])/2;
u[n_] := Table[t[n, k], {k, 1, n}]
Table[u[n], {n, 1, 5}]   (* averaging array *)
u = Table[3 (1/2) (1/r) 2^n*u[n], {n, 1, 12}];
TableForm[u]  (* A204204 triangle *)
Flatten[u]    (* A204204 sequence *)

From Philippe Deléham, Dec 24 2013: (Start)
T(n,n) = A000225(n+1).
Sum_{k=1..n} T(n,k) = A128135(n+1).
T(n,k)=T(n-1,k)+3*T(n-1,k-1)-2*T(n-2,k-1)-2*T(n-2,k-2), T(1,1)=3, T(2,1)=3, T(2,2)=7, T(n,k)=0 if k<1 or if k>n. (End)

A132344 a(n) = n*2^(floor(n/2)).

Original entry on oeis.org

0, 1, 4, 6, 16, 20, 48, 56, 128, 144, 320, 352, 768, 832, 1792, 1920, 4096, 4352, 9216, 9728, 20480, 21504, 45056, 47104, 98304, 102400, 212992, 221184, 458752, 475136, 983040, 1015808, 2097152, 2162688, 4456448, 4587520, 9437184, 9699328, 19922944, 20447232, 41943040, 42991616

Offset: 0

Simon Plouffe, Nov 19 2007

This sequence is related to a greedily and recursively defined sequence (see links). - Sela Fried, Aug 30 2024

Sela Fried, On integer sequence A128135, 2024.
Sela Fried, Proofs of some Conjectures from the OEIS, arXiv:2410.07237 [math.NT], 2024. See p. 11.
Simon Plouffe, Illustration. [broken link]
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-4).

Cf. A004526, A016116.

Maple
```
seq(n*2^(floor(n/2)),n=1..120);
```

Table[n*2^Floor[n/2], {n, 0, 100}] (* Wesley Ivan Hurt, Dec 12 2013 *)
LinearRecurrence[{0,4,0,-4},{0,1,4,6},50] (* Harvey P. Dale, Aug 27 2022 *)

a(n) = n*2^(n\2); \\ Michel Marcus, Feb 17 2018

G.f.: x*(1 + 4*x + 2*x^2)/(1 - 2*x^2)^2. - Ilya Gutkovskiy, Feb 24 2017
a(n) = n*A016116(n). - Michel Marcus, Feb 17 2018
Sum{n>=1} 1/a(n) = sqrt(2)*arcsinh(1) + log(2)/2. - Amiram Eldar, Sep 15 2024

More terms from Michel Marcus, Feb 17 2018

A292767 Square array read by antidiagonals downwards: T(k,n) = sum of the site-perimeters of words of length n >= 1 over an alphabet of size k >= 1.

Original entry on oeis.org

4, 6, 10, 8, 28, 18, 10, 72, 74, 28, 12, 176, 281, 152, 40, 14, 416, 1020, 762, 270, 54, 16, 960, 3591, 3664, 1680, 436, 70, 18, 2176, 12366, 17120, 10050, 3238, 658, 88, 20, 4864, 41877, 78336, 58500, 23160, 5677, 944, 108, 22, 10752, 139968, 352768, 333750, 161352, 47236, 9276, 1302, 130

Offset: 1

N. J. A. Sloane, Sep 27 2017

			Array begins (rows are indexed by k = 1,2,3,4,...; columns by n = 1,2,3,4,...):
   4,   6,    8,    10,     12,      14,       16, ...
  10,  28,   72,   176,    416,     960,     2176, ...
  18,  74,  281,  1020,   3591,   12366,    41877, ...
  28, 152,  762,  3664,  17120,   78336,   352768, ...
  40, 270, 1680, 10050,  58500,  333750,  1875000, ...
  54, 436, 3238, 23160, 161352, 1102464,  7420896, ...
  70, 658, 5677, 47236, 383131, 3049270, 23916361, ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275
Aubrey Blecher, Charlotte Brennan, Arnold Knopfmacher, and Toufik Mansour, The Site-Perimeter of Words, Transactions on Combinatorics, Vol. 6 No. 2 (2017), pp. 37-48. ISSN (print): 2251-8657, ISSN (on-line): 2251-8665.

Row k=2 is A128135.

RowGf[k_] := k x (36 + 12k + (8 - 24k - 8k^2) x + (2 - 5k + 4k^2 - k^3) x^2)/(12(1 - k x)^2);
T[k_, n_] := SeriesCoefficient[RowGf[k], {x, 0, n}];
Table[T[k - n + 1, n], {k, 1, 10}, {n, k, 1, -1}] // Flatten (* Jean-François Alcover, Aug 27 2019, from PARI *)

RowGf(k) = {k*x*(36 + 12*k + (8 - 24*k - 8*k^2)*x + (2 - 5*k + 4*k^2 - k^3)*x^2)/(12*(1 - k*x)^2)}
M(k,n)={Mat(vectorv(k,k,Vec(RowGf(k) + O(x*x^n))))}
{ M(10,8) } \\ Andrew Howroyd, Oct 27 2018

G.f. of row k: k*x*(36 + 12*k + (8 - 24*k - 8*k^2)*x + (2 - 5*k + 4*k^2 - k^3)*x^2)/(12*(1 - k*x)^2). - Andrew Howroyd, Oct 27 2018

Terms a(16) and beyond from Andrew Howroyd, Oct 27 2018

A266491 a(n) = n*A130658(n).

Original entry on oeis.org

0, 1, 4, 6, 4, 5, 12, 14, 8, 9, 20, 22, 12, 13, 28, 30, 16, 17, 36, 38, 20, 21, 44, 46, 24, 25, 52, 54, 28, 29, 60, 62, 32, 33, 68, 70, 36, 37, 76, 78, 40, 41, 84, 86, 44, 45, 92, 94, 48, 49, 100, 102, 52, 53, 108, 110, 56, 57, 116, 118, 60, 61, 124, 126, 64

Offset: 0

Paul Curtz, Dec 30 2015

Successive differences:
r(0):    0,   1,    4,    6,   4,    5,   12,   14, ...
r(1):    1,   3,    2,   -2,   1,    7,    2,   -6, ...
r(2):    2,  -1,   -4,    3,   6,   -5,   -8,    7, ... (see A103889)
r(3):   -3,  -3,    7,    3, -11,   -3,   15,    3, ...
r(4):    0,  10,   -4,  -14,   8,   18,  -12,  -22, ...
r(5):   10, -14,  -10,   22,  10,  -30,  -10,   38, ...
r(6):  -24,   4,   32,  -12, -40,   20,   48,  -28, ...
r(7):   28,  28,  -44,  -28,  60,   28,  -76,  -28, ...
r(8):    0, -72,   16,   88, -32, -104,   48,  120, ...
r(9):  -72,  88,   72, -120, -72,  152,   72, -184, ...
r(10): 160, -16, -192,   48, 224,  -80, -256,  112, ...
etc.
Let b(n) = 1, 1, 1, 1, 2, 2, 4, 4, 8, 8, 16, 16, ..., with n>=0, which is formed from the terms of A011782 repeated twice.
Conjecture: all terms of the row r(i) are divisible by b(i).
Conjecture: the terms of the first column divided by b(n) provide 0, 1, 2, -3, 0, 5, -6, 7, 0, -9, 10, -11, ..., the absolute values of which are listed in A190621.

Cf. A005408, A011782, A042963, A058962, A103889, A128135, A130658, A190621, A227380.

[n*(3-(-1)^((n-1)*n div 2))/2: n in [0..70]]; // Vincenzo Librandi, Jan 08 2016

Table[n (3 - (-1)^((n - 1) n/2))/2, {n, 0, 55}]
Table[n (Boole@ OddQ@ Floor[n/2] + 1), {n, 0, 55}] (* or *) Table[SeriesCoefficient[x (3/(1 - x)^2 + 2 x/(1 + x^2)^2 - (1 - x^2)/(1 + x^2)^2)/2, {x, 0, n}], {n, 0, 55}] (* Michael De Vlieger, Jan 04 2016 *)

vector(60, n, n--; n*(3-(-1)^((n-1)*n/2))/2) \\ Altug Alkan, Jan 04 2016

a(n) = n*(3 - (-1)^((n-1)*n/2))/2.
a(n) = a(n-4) + 4*A130658(n) for n>3.
a(n) = 2*a(n-1) -3*a(n-2) +4*a(n-3) -3*(n-4) +2*a(n-5) -a(n-6) for n>5.
G.f.: x*(3/(1 - x)^2 + 2*x/(1 + x^2)^2 - (1 - x^2)/(1 + x^2)^2)/2. - Michael De Vlieger, Jan 04 2016

Edited by Bruno Berselli, Jan 07 2016

A386250 Total number of ones in runs of 1's of length >= 4 over all binary strings of length n.

Original entry on oeis.org

0, 0, 0, 0, 4, 13, 36, 92, 224, 528, 1216, 2752, 6144, 13568, 29696, 64512, 139264, 299008, 638976, 1359872, 2883584, 6094848, 12845056, 27000832, 56623104, 118489088, 247463936, 515899392, 1073741824, 2231369728, 4630511616, 9596567552, 19864223744, 41070624768, 84825604096, 175019917312

Offset: 0

Félix Balado, Aug 14 2025

			For n=6 there are eight binary strings that contain runs of 1s of length >= 4: 001111, 011110, 011111, 101111, 111100, 111101, 111110 and 111111; the runs of length >= 4 in these strings contain a(6) = 36 ones.

Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A001787, A066373, A128135, A053220.

LinearRecurrence [{4,-4}, {4,13}, 30] (* Hugo Pfoertner, Aug 14 2025 *)

For n>=4, a(n) = (5*n-12)*2^(n-5).

G.f.: -x^4*(3*x-4)/(2*x-1)^2. - Alois P. Heinz, Aug 14 2025

cp's OEIS Frontend

A128134 A128132 * A007318.

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A134315 A134309 * A097806.

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A204204 Triangle based on (0,3/4,1) averaging array.

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A132344 a(n) = n*2^(floor(n/2)).

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Maple

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A292767 Square array read by antidiagonals downwards: T(k,n) = sum of the site-perimeters of words of length n >= 1 over an alphabet of size k >= 1.

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A266491 a(n) = n*A130658(n).

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Magma

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A386250 Total number of ones in runs of 1's of length >= 4 over all binary strings of length n.

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