A052548 -id:A052548 - cp's OEIS frontend

A252574 T(n,k)=Number of (n+2)X(k+2) 0..3 arrays with every 3X3 subblock row and diagonal sum equal to 1 2 5 6 or 7 and every 3X3 column and antidiagonal sum not equal to 1 2 5 6 or 7.

702, 843, 742, 1069, 868, 890, 1694, 1795, 1558, 1469, 2985, 3441, 4168, 3286, 2637, 5401, 8980, 9051, 10885, 7610, 4583, 9936, 23007, 30532, 25882, 34532, 17261, 8279, 18972, 47737, 92725, 107651, 88844, 96099, 39419, 15476, 36144, 133142, 208375

Offset: 1

R. H. Hardin, Dec 18 2014

Table starts
...702....843....1069.....1694......2985.......5401........9936........18972
...742....868....1795.....3441......8980......23007.......47737.......133142
...890...1558....4168.....9051.....30532......92725......208375.......715437
..1469...3286...10885....25882....107651.....395500......959944......4006901
..2637...7610...34532....88844....498696....2538669.....6590930.....37656158
..4583..17261...96099...263046...1887578...11285381....31059674....225598956
..8279..39419..275252...802760...7115253...53055996...155864925...1380518004
.15476..94224..896803..2655311..31894631..337833222...994640586..12185963853
.28007.218717.2561903..7860183.122743127.1537313257..4728064653..74805119002
.51488.504824.7521424.24273030.470091119.7468646942.24277561799.469290271873

			Some solutions for n=4 k=4
..3..2..2..3..2..1....0..2..0..0..2..0....0..0..2..0..0..2....0..0..2..0..0..2
..0..2..0..0..2..0....1..1..3..1..1..0....3..2..1..3..1..2....3..2..2..3..2..1
..0..0..2..0..0..2....2..0..0..2..0..0....0..2..0..0..2..0....0..2..0..0..2..0
..3..1..1..3..2..1....0..2..0..0..2..0....0..0..2..0..0..2....0..0..2..0..0..2
..0..2..0..0..2..0....2..1..3..2..2..3....3..1..2..3..1..2....3..1..2..3..2..1
..0..0..2..0..0..2....2..0..0..1..0..0....0..2..0..0..2..0....0..2..0..0..1..0

R. H. Hardin, Table of n, a(n) for n = 1..392

Column 1 is A005010(n-1)
Column 2 is A052548(n+3)
Row 1 is A083706(n+1)

Empirical for column k:
k=1: [linear recurrence of order 54] for n>60
k=2: [order 45] for n>50
k=3: [order 39] for n>46
k=4: [order 54] for n>60
k=5: [order 84] for n>89
Empirical for row n:
n=1: [linear recurrence of order 33] for n>43
n=2: [order 27] for n>34
n=3: [order 24] for n>32
n=4: [order 24] for n>31
n=5: [order 24] for n>32
n=6: [order 42] for n>50
n=7: [order 36] for n>45

A250769 T(n,k)=Number of (n+1)X(k+1) 0..1 arrays with nondecreasing x(i,j)-x(i,j-1) in the i direction and nondecreasing absolute value of x(i,j)-x(i-1,j) in the j direction.

Original entry on oeis.org

9, 18, 18, 35, 34, 36, 68, 62, 66, 72, 133, 114, 114, 130, 144, 262, 214, 196, 216, 258, 288, 519, 410, 344, 350, 418, 514, 576, 1032, 798, 622, 572, 648, 820, 1026, 1152, 2057, 1570, 1158, 962, 996, 1234, 1622, 2050, 2304, 4106, 3110, 2208, 1680, 1558, 1812

Offset: 1

R. H. Hardin, Nov 27 2014

Table starts
....9...18....35....68...133...262...519..1032..2057..4106...8203..16396..32781
...18...34....62...114...214...410...798..1570..3110..6186..12334..24626..49206
...36...66...114...196...344...622..1158..2208..4284..8410..16634..33052..65856
...72..130...216...350...572...962..1680..3046..5700.10922..21272..41870..82956
..144..258...418...648...996..1558..2526..4284..7600.14010..26586..51472.100956
..288..514...820..1234..1812..2666..4020..6322.10468.18250..33252..62642.120756
..576.1026..1622..2396..3412..4798..6810..9960.15272.24794..42622..76948.144156
.1152.2050..3224..4710..6580..8978.12192.16798.23948.35946..57400..97526.174756
.2304.4098..6426..9328.12884.17254.22758.30036.40368.56314..82994.130648.219756
.4608.8194.12828.18554.25460.33722.43692.56074.72276.95114.130220.188858.293556

			Some solutions for n=4 k=4
..1..1..1..1..0....1..0..0..0..0....1..0..1..1..0....1..1..0..1..1
..1..1..1..1..0....1..1..1..1..1....1..0..1..1..0....1..1..0..1..1
..1..1..1..1..0....0..0..0..0..0....1..0..1..1..1....1..1..0..1..1
..1..1..1..1..0....0..0..0..0..1....1..0..1..1..1....1..1..0..1..1
..0..0..0..0..1....0..0..0..0..1....1..0..1..1..1....1..1..0..1..1

R. H. Hardin, Table of n, a(n) for n = 1..611

Column 1 is A005010(n-1)
Column 2 is A052548(n+3)
Row 1 is A083706(n+1)

Empirical for column k: (k+2)^2*2^(n-1) plus a linear polynomial in n
k=1: a(n) = 2*a(n-1); a(n) = 9*2^(n-1)
k=2: a(n) = 3*a(n-1) -2*a(n-2); a(n) = 16*2^(n-1) + 2
k=3: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3); a(n) = 25*2^(n-1) + 2*n + 8
k=4: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3); a(n) = 36*2^(n-1) + 10*n + 22
k=5: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3); a(n) = 49*2^(n-1) + 32*n + 52
k=6: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3); a(n) = 64*2^(n-1) + 84*n + 114
k=7: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3); a(n) = 81*2^(n-1) + 198*n + 240
k=8: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3); a(n) = 100*2^(n-1) + 438*n + 494
k=9: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3); a(n) = 121*2^(n-1) + 932*n + 1004
Empirical for row n: (4*n+4)*2^(k-1) plus a quadratic polynomial in k
n=1: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3); a(n) = 8*2^(n-1) + n
n=2: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3); a(n) = 12*2^(n-1) + 4*n + 2
n=3: a(n) = 5*a(n-1) -9*a(n-2) +7*a(n-3) -2*a(n-4); a(n) = 16*2^(n-1) + n^2 + 11*n + 8
n=4: a(n) = 5*a(n-1) -9*a(n-2) +7*a(n-3) -2*a(n-4); a(n) = 20*2^(n-1) + 4*n^2 + 26*n + 22
n=5: a(n) = 5*a(n-1) -9*a(n-2) +7*a(n-3) -2*a(n-4); a(n) = 24*2^(n-1) + 11*n^2 + 57*n + 52
n=6: a(n) = 5*a(n-1) -9*a(n-2) +7*a(n-3) -2*a(n-4); a(n) = 28*2^(n-1) + 26*n^2 + 120*n + 114
n=7: a(n) = 5*a(n-1) -9*a(n-2) +7*a(n-3) -2*a(n-4); a(n) = 32*2^(n-1) + 57*n^2 + 247*n + 240
n=8: a(n) = 5*a(n-1) -9*a(n-2) +7*a(n-3) -2*a(n-4); a(n) = 36*2^(n-1) + 120*n^2 + 502*n + 494
n=9: a(n) = 5*a(n-1) -9*a(n-2) +7*a(n-3) -2*a(n-4); a(n) = 40*2^(n-1) + 247*n^2 + 1013*n + 1004

A100585 a(n+1) = a(n)+floor(a(n)/3), a(1) = 3.

Original entry on oeis.org

3, 4, 5, 6, 8, 10, 13, 17, 22, 29, 38, 50, 66, 88, 117, 156, 208, 277, 369, 492, 656, 874, 1165, 1553, 2070, 2760, 3680, 4906, 6541, 8721, 11628, 15504, 20672, 27562, 36749, 48998, 65330, 87106, 116141, 154854, 206472, 275296, 367061, 489414, 652552

Offset: 1

N. J. A. Sloane, Dec 01 2004

nonn

Original definition: Write down the numbers from 3 to infinity. Take next number, M say, that has not been crossed off. Counting through the numbers that have not yet been crossed off after that M, cross off every 4th term. Repeat, always crossing off every 4th term of those that remain. The numbers that are left form the sequence.

Can be stated as the number of animals starting from a single trio if any trio of animals can produce a single offspring. See A061418 for the equivalent sequence for pairs of animals. - Luca Khan, Sep 05 2024

Robert Israel, Table of n, a(n) for n = 1..7987
Popular Computing (Calabasas, CA), Sieves: Problem 43, Vol. 2 (No. 13, Apr 1974), pp. 6-7. This is Sieve #6 with K=4. [Annotated and scanned copy]
Index entries for sequences generated by sieves

Cf. A003309, A003310, A100464, A100562, A003312, A003311, A052548, A100586, A061418.

R:= 3: x:= 3:
for i from 2 to 100 do x:= x + floor(x/3); R:= R,x od:
R; # Robert Israel, Sep 09 2024

t = Range[3, 2500000]; r = {}; While[Length[t] > 0, AppendTo[r, First[t]]; t = Drop[t, {1, -1, 4}];]; r (* Ray Chandler, Dec 02 2004 *)
NestList[#+Floor[#/3]&,3,50] (* Harvey P. Dale, Jan 14 2019 *)

a(n,s=3)=for(i=2,n,s+=s\3);s \\ M. F. Hasler, Oct 06 2014

a(1)=3, a(n+1) = a(n) + floor(a(n)/3). - Ben Paul Thurston, Jan 09 2008

More terms from Ray Chandler, Dec 02 2004

Simpler definition from M. F. Hasler, Oct 06 2014

A229001 Total sum A(n,k) of the k-th powers of lengths of ascending runs in all permutations of [n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 1, 3, 0, 1, 4, 12, 0, 1, 6, 18, 60, 0, 1, 10, 32, 96, 360, 0, 1, 18, 66, 186, 600, 2520, 0, 1, 34, 152, 426, 1222, 4320, 20160, 0, 1, 66, 378, 1110, 2964, 9086, 35280, 181440, 0, 1, 130, 992, 3186, 8254, 22818, 75882, 322560, 1814400

Offset: 0

Alois P. Heinz, Sep 10 2013

			A(3,2) = 32 = 9+5+5+5+5+3 = 3^2+4*(2^2+1^2)+3*1^2: (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), (3,2,1).
Square array A(n,k) begins:
:    0,    0,    0,     0,     0,      0,      0, ...
:    1,    1,    1,     1,     1,      1,      1, ...
:    3,    4,    6,    10,    18,     34,     66, ...
:   12,   18,   32,    66,   152,    378,    992, ...
:   60,   96,  186,   426,  1110,   3186,   9846, ...
:  360,  600, 1222,  2964,  8254,  25620,  86782, ...
: 2520, 4320, 9086, 22818, 66050, 214410, 765506, ...

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

Columns k=0-10 give: A001710(n+1) for n>0, A001563, A228959, A229003, A228994, A228995, A228996, A228997, A228998, A228999, A229000.
Rows n=0-2 give: A000004, A000012, A052548.
Main diagonal gives: A229002.

A:= (n, k)-> add(`if`(n=t, 1, n!/(t+1)!*(t*(n-t+1)+1
             -((t+1)*(n-t)+1)/(t+2)))*t^k, t=1..n):
seq(seq(A(n, d-n), n=0..d), d=0..12);

A[n_, k_] := Sum[If[n == t, 1, n!/(t + 1)!*(t*(n - t + 1) + 1 - ((t + 1)*(n - t) + 1)/(t + 2))]* t^k, {t, 1, n}]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Dec 27 2013, translated from Maple *)

A(n,k) = Sum_{t=1..n} t^k * A122843(n,t).

For fixed k, A(n,k) ~ n! * n * sum(t>=1, t^k*(t^2+t-1)/(t+2)!) = n! * n * ((Bell(k) - Bell(k+1) + sum(j=0..k, (-1)^j*(2^j*((2*k-j+1)/(j+1))-1) *Bell(k-j)*C(k,j)))*exp(1) - (-1)^k*(2^k-1)), where Bell(k) are Bell numbers A000110. - Vaclav Kotesovec, Sep 12 2013

A245397 A(n,k) is the sum of k-th powers of coefficients in full expansion of (z_1+z_2+...+z_n)^n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 4, 10, 1, 1, 6, 27, 35, 1, 1, 10, 93, 256, 126, 1, 1, 18, 381, 2716, 3125, 462, 1, 1, 34, 1785, 36628, 127905, 46656, 1716, 1, 1, 66, 9237, 591460, 7120505, 8848236, 823543, 6435, 1, 1, 130, 51033, 11007556, 495872505, 2443835736, 844691407, 16777216, 24310

Offset: 0

Alois P. Heinz, Jul 21 2014

			A(3,2) = 93: (z1+z2+z3)^3 = z1^3 +3*z1^2*z2 +3*z1^2*z3 +3*z1*z2^2 +6*z1*z2*z3 +3*z1*z3^2 +z2^3 +3*z2^2*z3 +3*z2*z3^2 +z3^3 => 1^2+3^2+3^2+3^2+6^2+3^2+1^2+3^2+3^2+1^2 = 93.
Square array A(n,k) begins:
0 :    1,    1,      1,       1,         1,           1, ...
1 :    1,    1,      1,       1,         1,           1, ...
2 :    3,    4,      6,      10,        18,          34, ...
3 :   10,   27,     93,     381,      1785,        9237, ...
4 :   35,  256,   2716,   36628,    591460,    11007556, ...
5 :  126, 3125, 127905, 7120505, 495872505, 41262262505, ...

Alois P. Heinz, Antidiagonals n = 0..50, flattened

Columns k=0-10 give: A001700(n-1) for n>0, A000312, A033935, A055733, A055740, A246240, A246241, A246242, A246243, A246244, A246245.
Rows n=0+1, 2 give: A000012, A052548.
Main diagonal gives A245398.

b:= proc(n, i, k) option remember; `if`(n=0 or i=1, 1,
      add(b(n-j, i-1, k)*binomial(n, j)^k, j=0..n))
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..10);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1, 0, Sum[b[n-j, i-1, k] * Binomial[n, j]^(k-1)/j!, {j, 0, n}]]]; A[n_, k_] := n!*b[n, n, k]; Table[ Table[A[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Jan 30 2015, after Alois P. Heinz *)

A(n,k) = [x^n] (n!)^k * (Sum_{j=0..n} x^j/(j!)^k)^n.

A309010 Square array A(n, k) = Sum_{j=0..n} binomial(n,j)^k, n >= 0, k >= 0, read by antidiagonals.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 4, 4, 1, 2, 6, 8, 5, 1, 2, 10, 20, 16, 6, 1, 2, 18, 56, 70, 32, 7, 1, 2, 34, 164, 346, 252, 64, 8, 1, 2, 66, 488, 1810, 2252, 924, 128, 9, 1, 2, 130, 1460, 9826, 21252, 15184, 3432, 256, 10, 1, 2, 258, 4376, 54850, 206252, 263844, 104960, 12870, 512, 11

Offset: 0

Seiichi Manyama, Jul 06 2019

A(n,k) is the constant term in the expansion of (Product_{j=1..k-1} (1 + x_j) + Product_{j=1..k-1} (1 + 1/x_j))^n for k > 0. - Seiichi Manyama, Oct 27 2019
Let B_k be the binomial poset containing all k-tuples of equinumerous subsets of {1,2,...} ordered by inclusion componentwise (described in Stanley reference below).  Then A(k,n) is the number of elements in any n-interval of B_k. - Geoffrey Critzer, Apr 16 2020
Column k is the diagonal of the rational function 1 / (Product_{j=1..k} (1-x_j) - Product_{j=1..k} x_j) for k>0. - Seiichi Manyama, Jul 11 2020

			Square array, A(n, k), begins:
   1,  1,   1,    1,     1,      1, ... A000012;
   2,  2,   2,    2,     2,      2, ... A007395;
   3,  4,   6,   10,    18,     34, ... A052548;
   4,  8,  20,   56,   164,    488, ... A115099;
   5, 16,  70,  346,  1810,   9826, ...
   6, 32, 252, 2252, 21252, 206252, ...
Antidiagonals, T(n, k), begin:
  1;
  1,  2;
  1,  2,   3;
  1,  2,   4,    4;
  1,  2,   6,    8,    5;
  1,  2,  10,   20,   16,     6;
  1,  2,  18,   56,   70,    32,     7;
  1,  2,  34,  164,  346,   252,    64,    8;
  1,  2,  66,  488, 1810,  2252,   924,  128,   9;
  1,  2, 130, 1460, 9826, 21252, 15184, 3432, 256,  10;

R. P. Stanley, Enumerative Combinatorics Vol I, Second Edition, Cambridge, 2011, Example 3.18.3 d, page 366.

Seiichi Manyama, Antidiagonals n = 0..100, flattened

Columns k=0..12 give A000027(n+1), A000079, A000984, A000172, A005260, A005261, A069865, A182421, A182422, A182446, A182447, A342294, A342295.
Main diagonal gives A167010.
Cf. A328747, A328748, A328807, A328812.
Cf. A000012, A007395, A052548, A115099.

[(&+[Binomial(k,j)^(n-k): j in [0..k]]): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 26 2022

nn = 8; Table[ek[x_] := Sum[x^n/n!^k, {n, 0, nn}];Range[0, nn]!^k CoefficientList[Series[ek[x]^2, {x, 0, nn}],x], {k, 0, nn}] // Transpose // Grid (* Geoffrey Critzer, Apr 17 2020 *)

A(n, k) = sum(j=0, n, binomial(n, j)^k); \\ Seiichi Manyama, Jan 08 2022

flatten([[sum(binomial(k,j)^(n-k) for j in (0..k)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Aug 26 2022

A(n, k) = Sum_{j=0..n} binomial(n,j)^k (array).
A(n, n+1) = A328812(n).
A(n, n) = A167010(n).
T(n, k) = A(k, n-k) (antidiagonals).
T(n, n) = A000027(n+1).
T(n, n-1) = A000079(n-1).
T(n, n-2) = A000984(n-2).
T(n, n-3) = A000172(n-3).
T(n, n-4) = A005260(n-4).
T(n, n-5) = A005261(n-5).
T(n, n-6) = A069865(n-6).
T(n, n-7) = A182421(n-7).
T(n, n-8) = A182422(n-8).
T(n, n-9) = A182446(n-9).
T(n, n-10) = A182447(n-10).
T(n, n-11) = A342294(n-11).
T(n, n-12) = A342295(n-12).
Sum_{n>=0} A(n,k) x^n/(n!^k) = (Sum_{n>=0} x^n/(n!^k))^2. - Geoffrey Critzer, Apr 17 2020

A356784 Inventory of positions as an irregular table; row 0 contains 0, subsequent rows contain the 0-based positions of 0's, followed by the position of 1's, of 2's, etc. in prior rows flattened.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 2, 3, 0, 1, 2, 4, 3, 5, 6, 7, 0, 1, 2, 4, 8, 3, 5, 9, 6, 10, 7, 12, 11, 13, 14, 15, 0, 1, 2, 4, 8, 16, 3, 5, 9, 17, 6, 10, 18, 7, 12, 21, 11, 19, 13, 22, 14, 24, 15, 26, 20, 23, 25, 28, 27, 29, 30, 31, 0, 1, 2, 4, 8, 16, 32, 3, 5, 9, 17, 33

Offset: 0

Rémy Sigrist, Oct 01 2022

The n-th row contains A011782(n) terms, and is a permutation of 0..A011782(n)-1.

The leading term of each row is 0, and is followed by powers of 2, and then by positive nonpowers of 2.

			Table begins:
   0,
   0,
   0, 1,
   0, 1, 2, 3,
   0, 1, 2, 4, 3, 5, 6, 7,
   0, 1, 2, 4, 8, 3, 5, 9, 6, 10, 7, 12, 11, 13, 14, 15,
   ...
For n = 5:
- the terms in rows 0..4 are: 0, 0, 0, 1, 0, 1, 2, 3, 0, 1, 2, 4, 3, 5, 6, 7,
- we have 0's at positions 0, 1, 2, 4, 8,
- we have 1's at positions 3, 5, 9,
- we have 2's at positions 6, 10,
- we have 3's at positions 7, 12,
- we have one 4 at position 11,
- we have one 5 at position 13,
- we have one 6 at position 14,
- we have one 7 at position 15,
- so row 5 is: 0, 1, 2, 4, 8, 3, 5, 9, 6, 10, 7, 12, 11, 13, 14, 15.

Rémy Sigrist, Table of n, a(n) for n = 0..8191
Rémy Sigrist, Scatterplot of first 2^22 terms
Rémy Sigrist, C++ program

Cf. A000051, A005126, A011782 (row lengths), A052548, A131577, A342585, A357317, A357491.

terms = [0,]
for i in range(1,10):
    new_terms = []
    for j in range(max(terms)+1):
        for k in range(len(terms)):
            if terms[k] == j: new_terms.append(k)
    terms.extend(new_terms)
print(terms) # Gleb Ivanov, Nov 01 2022

a(n) = 0 iff n belongs to A131577.
a(n) = 1 iff n belongs to A000051 \ {2}.
a(n) = 2 iff n belongs to A052548 \ {3, 4}.
a(n) = 3 iff n belongs to A005126 \ {2, 4}.
T(n, 0) = 0.
T(n, k) = 2^(k-1) for k = 1..n-1.

A100314 Number of 2 X n 0-1 matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01;0), (10;0) and (01;1).

Original entry on oeis.org

1, 4, 8, 14, 24, 42, 76, 142, 272, 530, 1044, 2070, 4120, 8218, 16412, 32798, 65568, 131106, 262180, 524326, 1048616, 2097194, 4194348, 8388654, 16777264, 33554482, 67108916, 134217782, 268435512, 536870970, 1073741884, 2147483710, 4294967360, 8589934658

Offset: 0

Sergey Kitaev, Nov 13 2004

An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1 < i2, j1 < j2 and these elements are in the same relative order as those in the triple (x,y,z). In general, the number of m X n 0-1 matrices in question is given by 2^m + 2^n + 2*(n*m-n-m).

Arthur H. Stroud, Approximate calculation of multiple integrals, Prentice-Hall, 1971.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Ronald Cools, Encyclopaedia of Cubature Formulas
S. Kitaev, On multi-avoidance of right angled numbered polyomino patterns, Integers: Electronic Journal of Combinatorial Number Theory 4 (2004), A21, 20pp.
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. this sequence (m=2), A100315 (m=3), A100316 (m=4).
Cf. A000051, A000079, A001787, A002064, A005126, A005843.
Cf. A052548, A058331, A099003, A132750, A176691.
Row sums of A131830.

List([0..40],n->2^n+2*n); # Muniru A Asiru, Dec 21 2018

[2^n+2*n: n in [1..40]]; // Vincenzo Librandi, Oct 22 2011

a:= proc(n) 2^n + 2*n: end: seq(a(n),n=0..50); # Gary W. Adamson, Jul 20 2007

LinearRecurrence[{4,-5,2}, {1,4,8}, 34] (* Jean-François Alcover, Mar 19 2020 *)

makelist(2^n + 2*n, n, 0, 50); /* Franck Maminirina Ramaharo, Dec 19 2018 */

[2^n +2*n for n in range(41)] # G. C. Greubel, Feb 01 2023

a(n) = 2^n + 2*n.
From Gary W. Adamson, Jul 20 2007: (Start)
Binomial transform of (1, 3, 1, 1, 1, ...).
For n > 0, a(n) = 2*A005126(n-1). (End)
From R. J. Mathar, Jun 13 2008: (Start)
G.f.: 1 + 2*x*(2 -4*x +x^2)/((1-x)^2*(1-2*x)).
a(n+1)-a(n) = A052548(n). (End)
From Colin Barker, Oct 16 2013: (Start)
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3).
G.f.: (1 - 3*x^2)/((1-x)^2*(1-2*x)). (End)
E.g.f.: exp(2*x) + 2*x*exp(x). - Franck Maminirina Ramaharo, Dec 19 2018
a(n) = A000079(n) + A005843(n). - Muniru A Asiru, Dec 21 2018

a(0)=1 prepended by Alois P. Heinz, Dec 21 2018

A058896 a(n) = 4^n - 4.

Original entry on oeis.org

-3, 0, 12, 60, 252, 1020, 4092, 16380, 65532, 262140, 1048572, 4194300, 16777212, 67108860, 268435452, 1073741820, 4294967292, 17179869180, 68719476732, 274877906940, 1099511627772, 4398046511100, 17592186044412, 70368744177660, 281474976710652, 1125899906842620

Offset: 0

Henry Bottomley, Jan 08 2001

Harry J. Smith, Table of n, a(n) for n = 0..500
Mattia Fregola, Elementary Cellular Automata Rule 1 generating OEIS sequence A277799, A058896, A141725, A002450
Index entries for linear recurrences with constant coefficients, signature (5, -4).

Cf. A000302, A000918, A002450, A024036, A052548, A058809, A141725, A277799.

List([0..25],n->4^n-4); # Muniru A Asiru, Mar 09 2018

seq(4^n-4,n=0..25); # Muniru A Asiru, Mar 09 2018

Array[4^# - 4 &, 26, 0]  (* Michael De Vlieger, Feb 18 2018 *)

a(n) = { 4^n - 4 } \\ Harry J. Smith, Jun 23 2009

a(n) = A000302(n) - 4 = 4*a(n-1) + 12 = 4*A024036(n-1) = 12*A002450(n-1).
G.f.: 3*(5*x - 1)/(1 - x)/(1 - 4*x).
a(n) = A000918(n)*A052548(n). - Reinhard Zumkeller, Feb 14 2009
From Elmo R. Oliveira, Nov 16 2023 (Start)
a(n) = 5*a(n-1) - 4*a(n-2) for n > 1.
E.g.f.: exp(4*x) - 4*exp(x). (End)

A175161 a(n) = 8*(2^n + 1).

Original entry on oeis.org

16, 24, 40, 72, 136, 264, 520, 1032, 2056, 4104, 8200, 16392, 32776, 65544, 131080, 262152, 524296, 1048584, 2097160, 4194312, 8388616, 16777224, 33554440, 67108872, 134217736, 268435464, 536870920, 1073741832, 2147483656, 4294967304, 8589934600, 17179869192

Offset: 0

Reinhard Zumkeller, Feb 28 2010

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Sequences of the form m*(2^n + 1): A000051 (m=1), A052548 (m=2), A140504 (m=4), A153973 (m=6), A231643 (m=5), this sequence (m=8), A175162 (m=16), A175163 (m=32).

Cf. A159741, A173786, A175166.

I:=[16,24]; [n le 2 select I[n] else 3*Self(n-1) - 2*Self(n-2): n in [1..41]]; // G. C. Greubel, Jul 08 2021

8*(2^Range[0, 40] + 1) (* G. C. Greubel, Jul 08 2021 *)
LinearRecurrence[{3,-2},{16,24},40] (* Harvey P. Dale, Feb 10 2022 *)

[8*(2^n +1) for n in (0..40)] # G. C. Greubel, Jul 08 2021

a(n) = A173786(n+3, 3).
a(n) = A175166(2*n)/A159741(n) for n > 0.
a(n) = 3*a(n-1) -2*a(n-2) with a(0)=16, a(1)=24. - Vincenzo Librandi, Dec 28 2010
G.f.: 8*(2 - 3*x)/((1-x)*(1-2*x)). - Chai Wah Wu, Jun 20 2020
a(n) = 8 * A000051(n). - Alois P. Heinz, Jun 20 2020
E.g.f.: 8*(exp(2*x) + exp(x)). - G. C. Greubel, Jul 08 2021

cp's OEIS Frontend

A252574 T(n,k)=Number of (n+2)X(k+2) 0..3 arrays with every 3X3 subblock row and diagonal sum equal to 1 2 5 6 or 7 and every 3X3 column and antidiagonal sum not equal to 1 2 5 6 or 7.

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A250769 T(n,k)=Number of (n+1)X(k+1) 0..1 arrays with nondecreasing x(i,j)-x(i,j-1) in the i direction and nondecreasing absolute value of x(i,j)-x(i-1,j) in the j direction.

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A100585 a(n+1) = a(n)+floor(a(n)/3), a(1) = 3.

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A229001 Total sum A(n,k) of the k-th powers of lengths of ascending runs in all permutations of [n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A245397 A(n,k) is the sum of k-th powers of coefficients in full expansion of (z_1+z_2+...+z_n)^n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A309010 Square array A(n, k) = Sum_{j=0..n} binomial(n,j)^k, n >= 0, k >= 0, read by antidiagonals.

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A356784 Inventory of positions as an irregular table; row 0 contains 0, subsequent rows contain the 0-based positions of 0's, followed by the position of 1's, of 2's, etc. in prior rows flattened.

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