A002063 -id:A002063 - cp's OEIS frontend

A166976 Array of A002450 in the top row and higher-order differences in subsequent rows, read by antidiagonals.

0, 1, 1, 3, 4, 5, 9, 12, 16, 21, 27, 36, 48, 64, 85, 81, 108, 144, 192, 256, 341, 243, 324, 432, 576, 768, 1024, 1365, 729, 972, 1296, 1728, 2304, 3072, 4096, 5461, 2187, 2916, 3888, 5184, 6912, 9216, 12288, 16384, 21845, 6561

Offset: 0

Paul Curtz, Oct 26 2009

			The array starts:
0,   1,   5,  21,  85, 341,1365,5461,21845,87381,349525,    A002450
1,   4,  16,  64, 256,1024,4096,16384,65536,262144,1048576, A000302
3,  12,  48, 192, 768,3072,12288,49152,196608,786432,       A002001, A164346, A110594
9,  36, 144, 576,2304,9216,36864,147456                     A002063, A055841

A002450 := proc(n) (4^n-1)/3 ; end proc:
A166976 := proc(n,k) option remember; if n = 0 then A002450(k) else procname(n-1,k+1)-procname(n-1,k) ; end if; end proc: # R. J. Mathar, Jul 02 2011

T(0,k) = A002450(k). T(n,k) = T(n-1,k+1) - T(n-1,k), n > 0.

A183354 One quarter the number of nX2 1..4 arrays with no two neighbors of any element equal to each other.

Original entry on oeis.org

4, 36, 144, 576, 2304, 9216, 36864, 147456, 589824, 2359296, 9437184, 37748736, 150994944, 603979776, 2415919104, 9663676416, 38654705664, 154618822656, 618475290624, 2473901162496, 9895604649984, 39582418599936, 158329674399744

Offset: 1

R. H. Hardin Jan 04 2011

nonn

Column 2 of A183362

			Some solutions for 5X2 with a(1,1)=1
..1..1....1..3....1..2....1..3....1..3....1..2....1..4....1..2....1..3....1..3
..4..2....4..2....4..4....4..4....4..2....1..2....3..3....4..3....2..2....4..3
..3..3....4..2....3..3....3..2....4..1....4..4....2..2....2..1....4..1....4..2
..1..1....3..1....1..1....1..2....2..3....2..3....1..1....2..1....4..1....3..2
..2..4....2..1....2..2....1..3....1..3....1..1....4..4....3..4....2..3....3..1

R. H. Hardin, Table of n, a(n) for n = 1..200

Empirical: a(n)=4*a(n-1) = A002063(n-1) for n>2

A272342 a(n) = 27*8^n.

Original entry on oeis.org

27, 216, 1728, 13824, 110592, 884736, 7077888, 56623104, 452984832, 3623878656, 28991029248, 231928233984, 1855425871872, 14843406974976, 118747255799808, 949978046398464, 7599824371187712, 60798594969501696

Offset: 0

Andres Cicuttin, Apr 26 2016

a(n) are cubes that can be expressed as sum of exactly four distinct powers of two: a(n)=2^3n + 2^(3n+1) + 2^(3n+3) + 2^(3n+4). For example a(0) = 2^0 + 2^1 + 2^3 + 2^4 = 1 + 2 + 8 + 16 = 27. It is conjectured the a(n) are the only cubes that can be expressed as sum of exactly four distinct nonnegative powers of two (tested on cubes up to (10^7)^3).

Index entries for linear recurrences with constant coefficients, signature (8).

Cf. A001018, A002063.

Mathematica
```
nmax=120; 27*8^Range[0, nmax]
```

a(n) = 27*8^n; \\ Michel Marcus, Apr 27 2016

a(n) = 27*8^n = 2^3n + 2^(3n+1) + 2^(3n+3) + 2^(3n+4).
a(n) = 8*a(n-1), n>0; a(0)=27.
G.f.: 27/(1-8*x).
E.g.f.: 27*exp(8*x).
a(n) = 27*A001018(n). - Michel Marcus, Apr 26 2016

A356036 Triangle read by rows, giving in the first column the powers of 3 (A000244) and in the next columns 4/3 times the previous row entry.

Original entry on oeis.org

1, 3, 4, 9, 12, 16, 27, 36, 48, 64, 81, 108, 144, 192, 256, 243, 324, 432, 576, 768, 1024, 729, 972, 1296, 1728, 2304, 3072, 4096, 2187, 2916, 3888, 5184, 6912, 9216, 12288, 16384, 6561, 8748, 11664, 15552, 20736, 27648, 36864, 49152, 65536, 19683, 26244, 34992, 46656, 62208, 82944, 110592, 147456, 196608, 262144

Offset: 0

Wolfdieter Lang, Aug 01 2022

This is Boethius's triangle, with rows read as columns. See the link and reference.

			The triangle T begins:
n\k     0     1      2      3      4      5      6      7      8      9  ...
0:      1
1:      3     4
2:      9    12     16
3:     27    36     48     64
4:     81   108    144    192    256
5:    243   324    432    576    768   1024
6:    729   972   1296   1728   2304   3072   4096
7:   2187  2916   3888   5184   6912   9216  12288  16384
8:   6561  8748  11664  15552  20736  27648  36864  49152  65536
9:  19683 26244  34992  46656  62208  82944 110592 147456 196608 262144
...

Thomas Sonar, 3000 Jahre Analysis, 2. Auflage, Springer Spektrum, 2016, p.94, Abb. 3.1.2 und Abb. 3.1.3.

Anicius Manlius Severinus Boethius, De Institutione Arithmetica, 1488, p. 55 of 104, top of left column.

Columns: A000244, A003946, A257970, ...
Diagonals: A000302, A002001(n+1), A002063, A002063(n+3), A118265(n+4), ...
Row sums: A005061(n+1).

T[n_, k_] := 3^(n - k) * 4^k; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Aug 05 2022 *)

T(n, k) = 3^(n-k)*4^k, for n >= 0, and k = 1, 2, ..., n.

G.f. of row polynomials R(n, y) = Sum_{k=0..n} T(n, k)*y^k: G(x, y) = 1/((1 - 3*x)*(1 - 4*x*y)).

cp's OEIS Frontend

A166976 Array of A002450 in the top row and higher-order differences in subsequent rows, read by antidiagonals.

Views

Author

Keywords

Examples

Programs

Maple

Formula

A183354 One quarter the number of nX2 1..4 arrays with no two neighbors of any element equal to each other.

Views

Author

Keywords

Comments

Examples

Links

Formula

A272342 a(n) = 27*8^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A356036 Triangle read by rows, giving in the first column the powers of 3 (A000244) and in the next columns 4/3 times the previous row entry.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula