A095151 -id:A095151 - cp's OEIS frontend

A213387 a(n) = 52^(n-1) - 2 - 3n.

0, 2, 9, 26, 63, 140, 297, 614, 1251, 2528, 5085, 10202, 20439, 40916, 81873, 163790, 327627, 655304, 1310661, 2621378, 5242815, 10485692, 20971449, 41942966, 83886003, 167772080, 335544237, 671088554, 1342177191

Offset: 1

J. M. Bergot, Jun 28 2012

Create an array m(i,j) as follows: m(1,j) = j*(j-1)/2 in the top row, m(i,1) = (i-1)^2 in the left column, and m(i,j) = m(i,j-1) + m(i-1,j) recursively in the main body, j >= 1, i >= 1. The sum of the terms in an antidiagonal is one term in this sequence, a(n) = Sum_{k=1..n} m(n-k+1,k).

			For n=5, m(5,1)=16, m(4,2)=15, m(3,3)=11, m(2,4)=11, m(1,5)=10 gives the sum 63 = 2*A000295(4) + A095151(4) = 2*11 + 41.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Joseph Breen and Emma Copeland, Non-orientable Nurikabe, arXiv:2506.12612 [math.CO], 2025. See pp. 1, 4.
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A000295, A095151, A000217, A000290.

Table[5*2^(n-1)-2-3n,{n,30}] (* or *) LinearRecurrence[{4,-5,2},{0,2,9},30] (* Harvey P. Dale, Sep 28 2012 *)

a(n) = A095151(n-1) + 2*A000295(n-1).
G.f.: x^2*(2+x) / ( (1-2*x)*(1-x)^2 ). - R. J. Mathar, Jun 29 2012
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3); a(1)=0, a(2)=2, a(3)=9. - Harvey P. Dale, Sep 28 2012

A103768 a(n) = (293^n - 18(n + 3)2^n + 6n^2 + 24n + 27)/12.

Original entry on oeis.org

0, 0, 6, 46, 228, 930, 3406, 11682, 38412, 122806, 385182, 1192254, 3656452, 11141178, 33791934, 102161962, 308156748, 928008846, 2791497262, 8390220006, 25203689700, 75680274610, 227185526766, 681858569586, 2046204853708

Offset: 0

Henry Bottomley, Feb 15 2005

Index entries for linear recurrences with constant coefficients, signature (10,-40,82,-91,52,-12).

Table[(29*3^n-18(n+3)2^n+6n^2+24n+27)/12,{n,40}] (* or *) LinearRecurrence[ {10,-40,82,-91,52,-12},{0,0,6,46,228,930},40] (* Harvey P. Dale, Aug 31 2018 *)

concat([0,0], Vec(-2*x^2*(x^3-4*x^2+7*x-3)/((x-1)^3*(2*x-1)^2*(3*x-1)) + O(x^100))) \\ Colin Barker, Sep 13 2014

a(n) = A078341(n, 3) = 3a(n-1)+n*A095151(n-1).

G.f.: -2*x^2*(x^3-4*x^2+7*x-3) / ((x-1)^3*(2*x-1)^2*(3*x-1)). - Colin Barker, Sep 13 2014

A288870 Triangle T from array A(k,n) = (2k+1)2^n + 1, k >=0, n >= 0 read by downwards antidiagonals.

Original entry on oeis.org

2, 3, 4, 5, 7, 6, 9, 13, 11, 8, 17, 25, 21, 15, 10, 33, 49, 41, 29, 19, 12, 65, 97, 81, 57, 37, 23, 14, 129, 193, 161, 113, 73, 45, 27, 16, 257, 385, 321, 225, 145, 89, 53, 31, 18, 513, 769, 641, 449, 289, 177, 105, 61, 35, 20, 1025, 1537, 1281, 897, 577, 353, 209, 121, 69, 39, 22

Offset: 0

Wolfdieter Lang, Jun 21 2017

This entry was motivated by a class work of Ferran D.

			The array A begins:
k\n  0  1  2   3   4   5    6    7    8    9    10 ...
0:   2  3  5   9  17  33   65  129  257  513  1025
1:   4  7 13  25  49  97  193  385  769 1537  3073
2:   6 11 21  41  81 161  321  641 1281 2561  5121
3:   8 15 29  57 113 225  449  897 1793 3585  7169
4:  10 19 37  73 145 289  577 1153 2305 4609  9217
5:  12 23 45  89 177 353  705 1409 2817 5633 11265
6:  14 27 53 105 209 417  833 1665 3329 6657 13313
7:  16 31 61 121 241 481  961 1921 3841 7681 15361
8:  18 35 69 137 273 545 1089 2177 4353 8705 17409
9:  20 39 77 153 305 609 1217 2433 4865 9729 19457
...
The triangle T begins:
m\k    0    1    2   3   4   5   6   7  8  9 10 ...
0:     2
1:     3    4
2:     5    7    6
3:     9   13   11   8
4:    17   25   21  15  10
5:    33   49   41  29  19  12
6:    65   97   81  57  37  23  14
7:   129  193  161 113  73  45  27 16
8:   257  385  321 225 145  89  53 31 18
9:   513  769  641 449 289 177 105 61 35 20
10: 1025 1537 1281 897 577 353 209 121 69 39 22
...

Cf. A288871. Columns of T (no 0's, or rows of A): A000051, A181565, A083575, A083686, A083705, A083683, A168596.

Row sums give A077802(n+1) or A095151(n+1).

Table[(2 k + 1)*2^(m - k) + 1, {m, 0, 10}, {k, 0, m}] // Flatten (* Michael De Vlieger, Jun 25 2017 *)

A(n, k) = (2*n + 1)*2^k + 1;
for(n=0, 10, for(k=0, n, print1(A(k, n - k),", "))) \\ Indranil Ghosh, Jun 22 2017

Array A(k, n) = (2*k+1)*2^n + 1 for k >= 0 and n >= 0.
Triangle T(m, k) = A(k, m-k) = (2*k+1)*2^(m-k) + 1, k >= m >= 0, otherwise T(m, k) = 0.
O.g.f. for column k of T: x^k*(2*(k+1) - (2*k+3)*x)/((1-2*x)*(1-x)), k >= 0.
E.g.f. for column k of T (without leading 0's): (2*k+1)*exp(2*x) + exp(x), k>=0.
E.g.f. for column k of T: 2^(-k)*(2*k+1)*exp(2*x) + exp(x) - S(k,x), with S(k, x) = 2^(-k)* Sum_{m=1..k} A288871(k,m)*x^(m-1)/(m-1)! if k >=1 and S(0,x) = 0.

A368826 Square array T(n,k) = 3*2^k - n read by ascending antidiagonals.

Original entry on oeis.org

3, 2, 6, 1, 5, 12, 0, 4, 11, 24, -1, 3, 10, 23, 48, -2, 2, 9, 22, 47, 96, -3, 1, 8, 21, 46, 95, 192, -4, 0, 7, 20, 45, 94, 191, 384, -5, -1, 6, 19, 44, 93, 190, 383, 768, -6, -2, 5, 18, 43, 92, 189, 382, 767, 1536, -7, -3, 4, 17, 42, 91, 188, 381, 766, 1535, 3072

Offset: 0

Paul Curtz, Jan 07 2024

Similar to A367559.

			Table begins:
   3  6 12 24 48 96 ...
   2  5 11 23 47 95 ...
   1  4 10 22 46 94 ...
   0  3  9 21 45 93 ...
  -1  2  8 20 44 92 ...
  -2  1  7 19 43 91 ...
  ...

Cf. A000225, A000918, A007283, A033484, A048488, A083329, A165751, A367559.

Cf. diagonals: A123720, A079583, A095151, A101945 (after -1).

T[n_, k_] := 3*2^k - n; Table[T[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Jan 15 2024 *)

T(0,k) = 3*2^k     =   A007283(k).
T(1,k) = 3*2^k - 1 =   A083329(k+1).
T(2,k) = 3*2^k - 2 =   A033484(k).
T(3,k) = 3*2^k - 3 = 3*A000225(k).
T(4,k) = 3*2^k - 4 =  -A165751(k).
T(5,k) = 3*2^k - 5 =   A048488(k-1)
T(6,k) = 3*2^k - 6 = 3*A000918(k).

A120933 Triangle read by rows: T(n,k) is the number of binary words of length n for which the length of the maximal leading nondecreasing subword is k (1<=k<=n).

Original entry on oeis.org

2, 1, 3, 2, 2, 4, 4, 4, 3, 5, 8, 8, 6, 4, 6, 16, 16, 12, 8, 5, 7, 32, 32, 24, 16, 10, 6, 8, 64, 64, 48, 32, 20, 12, 7, 9, 128, 128, 96, 64, 40, 24, 14, 8, 10, 256, 256, 192, 128, 80, 48, 28, 16, 9, 11, 512, 512, 384, 256, 160, 96, 56, 32, 18, 10, 12, 1024, 1024, 768, 512, 320

Offset: 1

Emeric Deutsch, Jul 16 2006

			T(4,2)=4 because we have 0100,0101,1100 and 1101.
Triangle starts:
2;
1,3;
2,2,4;
4,4,3,5;
8,8,6,4,6;

Thomas Grubb and Frederick Rajasekaran, Set Partition Patterns and the Dimension Index, arXiv:2009.00650 [math.CO], 2020. Mentions this sequence.

Cf. A000079, A095151.

Maple
```
T:=proc(n,k) if k
				
```

T(n,k) = k*2^(n-k-1) if k

G.f.: G(t,z) = (1-2z+tz^2)/[(1-2z)(1-tz)^2] - 1.

Row sums are the powers of 2 (A000079).

Sum_{k=1..n} k*T(n,k) = 3*2^n-n-3 = A095151(n).

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cp's OEIS Frontend

A213387 a(n) = 5*2^(n-1) - 2 - 3*n.

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A103768 a(n) = (29*3^n - 18(n + 3)*2^n + 6n^2 + 24n + 27)/12.

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A288870 Triangle T from array A(k,n) = (2*k+1)*2^n + 1, k >=0, n >= 0 read by downwards antidiagonals.

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A368826 Square array T(n,k) = 3*2^k - n read by ascending antidiagonals.

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A120933 Triangle read by rows: T(n,k) is the number of binary words of length n for which the length of the maximal leading nondecreasing subword is k (1<=k<=n).

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A213387 a(n) = 52^(n-1) - 2 - 3n.

A103768 a(n) = (293^n - 18(n + 3)2^n + 6n^2 + 24n + 27)/12.

A288870 Triangle T from array A(k,n) = (2k+1)2^n + 1, k >=0, n >= 0 read by downwards antidiagonals.