A002063 -id:A002063 - cp's OEIS frontend

A356880 Squares that can be expressed as the sum of two powers of two (2^x + 2^y).

4, 9, 16, 36, 64, 144, 256, 576, 1024, 2304, 4096, 9216, 16384, 36864, 65536, 147456, 262144, 589824, 1048576, 2359296, 4194304, 9437184, 16777216, 37748736, 67108864, 150994944, 268435456, 603979776, 1073741824, 2415919104, 4294967296, 9663676416, 17179869184

Offset: 1

Karl-Heinz Hofmann, Sep 02 2022

If x is even, y = x + 3; if x is odd, y = x.
Proof for odd x: (2^odd + 2^odd) = 2^(odd + 1) = 2^even --> must be a square.
Proof for even x: 2^even + 2^(even + 3) = 1*(2^even) + (2^even * 2^3) = 1*(2^even) + (2^even * 8) = 1*(2^even) + 8*(2^even) = 9*(2^even); since 9 is a square and 2^even is a square, the multiplication result must be a square too.
And 9 is the only square that can be written as 1 + a power of 2.
Note that a(n) = A272711(n+1) for n=1..23, but beyond it differs more and more.

			2^4 + 2^7 = 144, a square, thus 144 is a term.

Index entries for linear recurrences with constant coefficients, signature (0,4).

Cf. A029744, A000302, A002063.
Intersection of A000290 and A048645\{1}.
Cf. A272711, A270473 (squares that can be expressed as 3^x + 3^y).
Cf. A220221.

seq(`if`(n::even, 9*2^(n-2), 2^(n+1)),n=1..50); # Robert Israel, Sep 15 2022

Select[Range[2, 2^17]^2, DigitCount[#, 2, 1] <= 2 &] (* Amiram Eldar, Sep 03 2022 *)

a(n) = if (n%2, 2^(n+1), 9*2^(n-2)); \\ Michel Marcus, Sep 15 2022

def A356880(n):
    if n % 2 == 0: return 9*2**(n-2)
    else: return 2**(n+1)

a(n) = A029744(n+1)^2.
a(n) = 9 * 2^(n-2) if n is even (see A002063).
a(n) = 2^(n+1) if n is odd (see A000302).
From Stefano Spezia, Sep 09 2022: (Start)
G.f.: x*(4 + 9*x)/(1 - 4*x^2).
E.g.f.: (9*(cosh(2*x) - 1) + 8*sinh(2*x))/4. (End)

A056120 a(n) = (3^3)*4^(n-3) with a(0)=1, a(1)=1 and a(2)=7.

Original entry on oeis.org

1, 1, 7, 27, 108, 432, 1728, 6912, 27648, 110592, 442368, 1769472, 7077888, 28311552, 113246208, 452984832, 1811939328, 7247757312, 28991029248, 115964116992, 463856467968, 1855425871872

Offset: 0

Barry E. Williams, Jul 05 2000

For n>=3, a(n) is equal to the number of functions f:{1,2,...,n}->{1,2,3,4} such that for fixed, different x_1, x_2, x_3 in {1,2,...,n} and fixed y_1, y_2, y_3 in {1,2,3,4} we have f(x_i)<>y_i, (i=1,2,...,n). - Milan Janjic, May 13 2007

G. C. Greubel, Table of n, a(n) for n = 0..1000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index entries for linear recurrences with constant coefficients, signature (4).

Cf. A055841.

First differences of A002063.

Concatenation([1,1,7], List([3..25], n-> 27*4^(n-3) )); # G. C. Greubel, Jan 18 2020

[1,1,7] cat [27*4^(n-3): n in [3..25]]; // G. C. Greubel, Jan 18 2020

1,1,7, seq( 27*4^(n-3), n=3..25); # G. C. Greubel, Jan 18 2020

Table[If[n<2, 1, If[n==2, 7, 27*4^(n-3)]], {n,0,25}] (* G. C. Greubel, Jan 18 2020 *)

vector(26, n, if(n<2, 1, if(n==2, 7, 27*4^(n-3))) ) \\ G. C. Greubel, Jan 18 2020

[1,1,7]+[27*4^(n-3) for n in (3..25)] # G. C. Greubel, Jan 18 2020

a(n) = 4*a(n-1) + (-1)^n*binomial(3, 3-n).
G.f.: (1-x)^3/(1-4*x).
E.g.f.: (37 - 44*x + 8*x^2 + 27*exp(4*x))/64. - G. C. Greubel, Jan 18 2020

a(21) corrected by R. J. Mathar, Dec 03 2014

A084431 Expansion of g.f. (1 + 6x + 5x^2)/((1-2x)(1+2*x)).

Original entry on oeis.org

1, 6, 9, 24, 36, 96, 144, 384, 576, 1536, 2304, 6144, 9216, 24576, 36864, 98304, 147456, 393216, 589824, 1572864, 2359296, 6291456, 9437184, 25165824, 37748736, 100663296, 150994944, 402653184, 603979776, 1610612736, 2415919104

Offset: 0

Paul Barry, Jun 26 2003

Binomial transform is A085287.

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

Bisections are A002023 and A002063.

Cf. A085287.

[(-10*0^n-3*(-2)^n+21*2^n)/8: n in [0..30]]; // Vincenzo Librandi, Nov 16 2011

CoefficientList[Series[(1+6x+5x^2)/((1-2x)(1+2x)),{x,0,30}],x] (* or *) Join[{1},Flatten[NestList[4#&,{6,9},15]]] (* Harvey P. Dale, Nov 05 2011 *)

a(n) = (-10*0^n - 3*(-2)^n + 21*2^n)/8.
a(n) = 4*a(n-2), n > 1. - Harvey P. Dale, Nov 05 2011
E.g.f.: (9*cosh(2*x) + 12*sinh(2*x) - 5)/4. - Stefano Spezia, Sep 20 2023

A153465 a(n) = 9*4^n - 2.

Original entry on oeis.org

34, 142, 574, 2302, 9214, 36862, 147454, 589822, 2359294, 9437182, 37748734, 150994942, 603979774, 2415919102, 9663676414, 38654705662, 154618822654, 618475290622, 2473901162494, 9895604649982, 39582418599934, 158329674399742, 633318697598974, 2533274790395902

Offset: 1

Vincenzo Librandi, Dec 27 2008

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A002063.

I:=[34, 142]; [n le 2 select I[n] else 5*Self(n-1)-4*Self(n-2): n in [1..50]]; // Vincenzo Librandi, Feb 22 2012

LinearRecurrence[{5, -4}, {34, 142}, 50] (* Vincenzo Librandi, Feb 22 2012 *)
CoefficientList[Series[(34 - 28 x )/((1 - x) (1 - 4 x)), {x, 0, 30}], x] (* Vincenzo Librandi, May 03 2014 *)

a(n)=9*4^n-2; \\ Charles R Greathouse IV, Jan 11 2012

a(n) = A002063(n) - 2. - R. J. Mathar, Jan 03 2009
a(n) = 5*a(n-1) - 4*a(n-2).
G.f.: x*(34-28*x)/((1-x)*(1-4*x)). - Vincenzo Librandi, May 03 2014
E.g.f.: 9*exp(4*x) - 2*exp(x) - 7. - Stefano Spezia, Sep 14 2024

A175880 a(1)=1, a(2)=2. If n >= 3: if n/2 is in the sequence, a(n)=0, otherwise a(n)=n.

Original entry on oeis.org

1, 2, 3, 0, 5, 0, 7, 8, 9, 0, 11, 12, 13, 0, 15, 0, 17, 0, 19, 20, 21, 0, 23, 0, 25, 0, 27, 28, 29, 0, 31, 32, 33, 0, 35, 36, 37, 0, 39, 0, 41, 0, 43, 44, 45, 0, 47, 48, 49, 0, 51, 52, 53, 0, 55, 0, 57, 0, 59, 60, 61, 0, 63, 0, 65, 0, 67, 68, 69, 0, 71, 0, 73, 0, 75, 76, 77, 0, 79, 80

Offset: 1

Adriano Caroli, Dec 05 2010

If n > 0 and n is in the sequence, then a(2*n) = 0. Example: 5 is in the sequence, so a(2*5) = a(10) = 0.

Is this a(n) = n*A039982(n-1), n > 1? [R. J. Mathar, Dec 07 2010]

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A053661.

A000040, A001749, A002001, A002042, A002063, A002089, A003947, A004171 and A081294 are subsequences.

import Data.List (delete)
a175880 n = a175880_list !! (n-1)
a175880_list = 1 : f [2..] [2..] where
   f (x:xs) (y:ys) | x == y    = x : (f xs $ delete (2*x) ys)
                   | otherwise = 0 : (f xs (y:ys))
for_bFile = take 10000 a175880_list
-- Reinhard Zumkeller, Feb 09 2011

A110654 := proc(n) 2*n+1-(-1)^n ; %/4 ;end proc:
A175880 := proc(n) if n <=2 then n; else if type(n,'even') then n-2*procname(A110654(n)) ; else n; end if; end if; end proc:
seq(A175880(n),n=1..40) ; # R. J. Mathar, Dec 07 2010

a(n) = n - (1 + (-1)^n) * a((2*n + 1 - (-1)^n)/4), n >= 3.

a(n) = n - A010673(n+1)*a(A110654(n)).

A204106 T(n,k)=Number of (n+1)X(k+1) 0..2 arrays with column and row pair sums b(i,j)=a(i,j)+a(i,j-1) and c(i,j)=a(i,j)+a(i-1,j) such that b(i,j)b(i-1,j)-c(i,j)c(i,j-1) is nonzero.

Original entry on oeis.org

36, 144, 144, 576, 864, 576, 2304, 5184, 5184, 2304, 9216, 31104, 46656, 31104, 9216, 36864, 186624, 419904, 419904, 186624, 36864, 147456, 1119744, 3779136, 5738688, 3779136, 1119744, 147456, 589824, 6718464, 34012224, 78428736, 78428736

Offset: 1

R. H. Hardin Jan 10 2012

Also 0..2 arrays with no 2X2 subblock having equal diagonal elements or equal antidiagonal elements
Table starts
.....36......144........576.........2304...........9216............36864
....144......864.......5184........31104.........186624..........1119744
....576.....5184......46656.......419904........3779136.........34012224
...2304....31104.....419904......5738688.......78428736.......1073134656
...9216...186624....3779136.....78428736.....1631513664......34026967296
..36864..1119744...34012224...1073134656....34026967296....1084257353088
.147456..6718464..306110016..14683622976...710001723456...34589078037504
.589824.40310784.2754990144.200937920832.14819050600704.1104253773912576

			Some solutions for n=5 k=3
..0..1..0..1....1..2..0..1....0..1..2..1....2..2..0..1....2..2..2..1
..2..1..2..1....1..2..0..1....2..1..0..0....0..1..0..2....0..0..0..1
..0..0..0..1....1..2..0..2....2..1..2..1....2..1..0..1....2..2..2..2
..1..1..2..1....1..2..0..1....2..0..2..0....2..1..2..2....0..0..0..1
..0..0..2..0....1..2..0..1....1..0..2..0....0..0..0..0....2..2..2..2
..1..1..1..1....0..2..0..1....1..0..1..1....1..2..1..2....1..0..0..1

R. H. Hardin, Table of n, a(n) for n = 1..285

Column 1 is A002063
Column 2 is A067411(n+2)
Column 3 is A055995(n+2)

Empirical for column k:
k=1: T(n,k)=4*T(n-1,k)
k=2: T(n,k)=6*T(n-1,k)
k=3: T(n,k)=9*T(n-1,k)
k=4: T(n,k)=15*T(n-1,k)-270*T(n-3,k)+324*T(n-4,k)
k=5: T(n,k)=25*T(n-1,k)-45*T(n-2,k)-963*T(n-3,k)+2025*T(n-4,k)+3645*T(n-5,k)-6561*T(n-6,k)
k=6: (order 15)
k=7: (order 45)

A365327 Triangle read by rows: T(n,k) is the number of spanning subgraphs of the n-cycle graph with domination number k.

Original entry on oeis.org

2, 3, 1, 4, 3, 1, 0, 11, 4, 1, 0, 11, 15, 5, 1, 0, 10, 26, 21, 6, 1, 0, 0, 43, 49, 28, 7, 1, 0, 0, 33, 98, 80, 36, 8, 1, 0, 0, 22, 126, 189, 120, 45, 9, 1, 0, 0, 0, 141, 322, 325, 170, 55, 10, 1, 0, 0, 0, 89, 462, 671, 517, 231, 66, 11, 1, 0, 0, 0, 46, 480, 1162, 1236, 777, 304, 78, 12, 1, 0, 0, 0, 0, 417, 1586, 2483, 2093, 1118, 390, 91, 13, 1

Offset: 1

Roman Hros, Sep 01 2023

For n >= 3 the n-cycle graph is a simple graph. In the case of n=1 the graph is a loop and for n=2 a double edge.
The number of spanning subgraphs of the n-cycle graph is given by 2^n which is also the sum of the n-th row Sum_{k=1..n} T(n,k).
The average domination number is given by (Sum_{k=1..n} k*T(n,k))/2^n.
The relative average domination number is given by ((Sum_{k=1..n} k*T(n,k))/2^n)/n.

			Example of spanning subgraphs of cycle with 2 vertices:
Domination number: 2      1      1      1
                          /\            /\
                  .  .   .  .   .  .   .  .
                                 \/     \/
The triangle T(n,k) begins:
n\k 1   2   3    4    5     6     7    8    9  10  11  12 ...
1:  2
2:  3   1
3:  4   3   1
4:  0  11   4    1
5:  0  11  15    5    1
6:  0  10  26   21    6     1
7:  0   0  43   49   28     7     1
8:  0   0  33   98   80    36     8    1
9:  0   0  22  126  189   120    45    9    1
10: 0   0   0  141  322   325   170   55   10   1
11: 0   0   0   89  462   671   517  231   66  11   1
12: 0   0   0   46  480  1162  1236  777  304  78  12   1

Roman Hros, Table of n, a(n) for n = 1..253 (Rows n = 1..22)
Roman Hros, Effective Enumeration of Selected Graph Characteristics, IIT.SRC 2020: 16th Student Research Conference in Informatics and Information Technologies.

Row sums are A000079.
Column sums are A002063(k-1).
Cf. A373436.

T(n,n) = 1 for n > 1.
T(n,n-1) = T(n-1, n-2) + 1 for n > 3.
T(n,n-2) = T(n-1, n-3) + T(n, n-1) for n > 5.
T(n,n-3) = T(n-1, n-4) + T(n, n-2) - 5 for n > 6.
T(n,n-4) = T(n-1, n-5) + T(n-1, n-4) + 11 + Sum_{i=1..n-9} (i+4) for n > 8.
G.f.:
For n > 3; G(n) = x*(G(n-1) + G(n-2) + 2*G(n-3)) + g(n); where
         2*(1-x)*x^(n/3)   for n mod 3 = 0.
g(n) = { 0                 for n mod 3 = 1.
         (1-x)*x^((n+1)/3) for n mod 3 = 2.
For n mod 3 = 0:
T(n,k) = 2*T(n-3,k-1) + T(n-2,k-1) + T(n-1,k-1) + 2 for k =  n/3.
T(n,k) = 2*T(n-3,k-1) + T(n-2,k-1) + T(n-1,k-1) - 2 for k =  n/3 + 1.
T(n,k) = 2*T(n-3,k-1) + T(n-2,k-1) + T(n-1,k-1)     for k >= n/3 + 2.
For n mod 3 = 1:
T(n,k) = 2*T(n-3,k-1) + T(n-2,k-1) + T(n-1,k-1)     for k >= (n+2)/3.
For n mod 3 = 2:
T(n,k) = 2*T(n-3,k-1) + T(n-2,k-1) + T(n-1,k-1) + 1 for k =  (n+1)/3.
T(n,k) = 2*T(n-3,k-1) + T(n-2,k-1) + T(n-1,k-1) - 1 for k =  (n+1)/3 + 1.
T(n,k) = 2*T(n-3,k-1) + T(n-2,k-1) + T(n-1,k-1)     for k >= (n+1)/3 + 2.

A159018 a(0)=5; a(n) = a(n-1) + floor(sqrt(a(n-1))), n > 0.

Original entry on oeis.org

5, 7, 9, 12, 15, 18, 22, 26, 31, 36, 42, 48, 54, 61, 68, 76, 84, 93, 102, 112, 122, 133, 144, 156, 168, 180, 193, 206, 220, 234, 249, 264, 280, 296, 313, 330, 348, 366, 385, 404, 424, 444, 465, 486, 508, 530, 553, 576, 600, 624, 648, 673, 698, 724, 750, 777, 804, 832, 860, 889, 918, 948, 978

Offset: 0

Philippe Deléham, Apr 02 2009

nonn

Row 1 in square array A159016.
This sequence contains an infinity of squares. - Philippe Deléham, Apr 04 2009
Intersection of the sequence with A000290 generates A002063. - Vincenzo Librandi, Apr 10 2009, clarified by R. J. Mathar, Dec 03 2010

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A028392.

A:= Array(0..100):
A[0]:= 5:
for n from 1 to 100 do A[n]:= A[n-1]+floor(sqrt(A[n-1])) od:
convert(A,list); # Robert Israel, Nov 26 2020

A164032 Number of "ON" cells in a certain 2-dimensional cellular automaton.

Original entry on oeis.org

1, 9, 4, 36, 4, 36, 16, 144, 4, 36, 16, 144, 16, 144, 64, 576, 4, 36, 16, 144, 16, 144, 64, 576, 16, 144, 64, 576, 64, 576, 256, 2304, 4, 36, 16, 144, 16, 144, 64, 576, 16, 144, 64, 576, 64, 576, 256, 2304, 16, 144, 64, 576, 64, 576, 256, 2304, 64, 576, 256, 2304, 256

Offset: 1

John W. Layman, Aug 08 2009

nonn

This automaton starts with one ON cell and evolves according to the rule that a cell is ON in a given generation if and only if the number of ON cells, among the cell itself and its eight nearest neighbors, was exactly one in the preceding generation.

			Can be arranged into blocks of length 2^k:
1,
9,
4, 36,
4, 36, 16, 144,
4, 36, 16, 144, 16, 144, 64, 576,
4, 36, 16, 144, 16, 144, 64, 576, 16, 144, 64, 576, 64, 576, 256, 2304,
4, 36, 16, 144, 16, 144, 64, 576, 16, 144, 64, 576, 64, 576, 256, 2304, 16, 144, 64, 576, 64, 576, 256, 2304, 64, 576, 256, 2304, 256, ...
...

David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.], which is also available at arXiv:1004.3036v2
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A000120, A048883, A079315, A122108, A160239, A002063 (last entry in each block)

wt[i_] := DigitCount[i, 2, 1];
a[n_] := If[OddQ[n], 1, 9] 4^wt[Floor[(n-1)/2]];
Array[a, 61] (* Jean-François Alcover, Oct 08 2018, after N. J. A. Sloane *)

a(n) = 4^hammingweight((n-1)\2) * if(n%2, 1, 9); \\ Michel Marcus, Oct 08 2018

It appears that this is the self-generating sequence defined by the following process: start with s={1,9} and repeatedly extend by concatenating s with 4*s, thus obtaining {1,9} -> {1,9,4,36} -> {1,9,4,36,4,36,16,144},... , etc.
Also, it appears that if n=2^k+j, with n>2 and 1<=j<=2^k, then a(n)=4a(j), with a(1)=1, a(2)=9.
From N. J. A. Sloane, Jul 21 2014: (Start)
Both of these assertions are not difficult to prove. At generation G = 2^k (k>=1) the ON cells are bounded by a box of edge 2G-1, and in that box there are (G/2)^2 3X3 blocks each containing 9 ON cells (separated by rows of OFF cells of width 1), so a total of a(2^k) = 9*2^(2k-2) ON cells (cf. A002063).
This box is full (more precisely, every cell in it has more than one ON neighbor), and at generation G+1 we have just 4 ON cells which are now at the corners of a box of edge 2G+1. Until the next power of 2 there is no interaction between the configurations that grow at the four corners, and so a(2^k+j) = 4a(j), as conjectured.
In fact this implies an explicit formula for a(n):
a(n) = c*4^wt(floor((n-1)/2)),
where c=1 if n is odd, c=9 if n is even, and wt(i) = A000120(i) is the binary weight function. For example, if n=20, [(n-1)/2]=9 which has weight 2, so a(20) = 9*4^2 = 144. (End)

A164093 a(n) = 9*4^n + 2.

Original entry on oeis.org

38, 146, 578, 2306, 9218, 36866, 147458, 589826, 2359298, 9437186, 37748738, 150994946, 603979778, 2415919106, 9663676418, 38654705666, 154618822658, 618475290626, 2473901162498, 9895604649986, 39582418599938, 158329674399746, 633318697598978, 2533274790395906

Offset: 1

Vincenzo Librandi, Aug 10 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Cf. A002063, A153465.

LinearRecurrence[{5,-4},{38,146},50] (* Vincenzo Librandi, Mar 06 2012 *)

a(n) = A002063(n) + 2.
a(n) = 5*a(n-1) - 4*a(n-2).
G.f.: 2*x*(19-22*x)/((4*x-1)*(x-1)).
E.g.f.: exp(x)*(9*exp(3*x) + 2) - 11. - Elmo R. Oliveira, Mar 08 2025

Edited by R. J. Mathar, Aug 21 2009

cp's OEIS Frontend

A356880 Squares that can be expressed as the sum of two powers of two (2^x + 2^y).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A056120 a(n) = (3^3)*4^(n-3) with a(0)=1, a(1)=1 and a(2)=7.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A084431 Expansion of g.f. (1 + 6*x + 5*x^2)/((1-2*x)*(1+2*x)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A153465 a(n) = 9*4^n - 2.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A175880 a(1)=1, a(2)=2. If n >= 3: if n/2 is in the sequence, a(n)=0, otherwise a(n)=n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Maple

Formula

A204106 T(n,k)=Number of (n+1)X(k+1) 0..2 arrays with column and row pair sums b(i,j)=a(i,j)+a(i,j-1) and c(i,j)=a(i,j)+a(i-1,j) such that b(i,j)*b(i-1,j)-c(i,j)*c(i,j-1) is nonzero.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A365327 Triangle read by rows: T(n,k) is the number of spanning subgraphs of the n-cycle graph with domination number k.

Views

Author

Keywords

Comments

Examples

Links

A084431 Expansion of g.f. (1 + 6x + 5x^2)/((1-2x)(1+2*x)).

A204106 T(n,k)=Number of (n+1)X(k+1) 0..2 arrays with column and row pair sums b(i,j)=a(i,j)+a(i,j-1) and c(i,j)=a(i,j)+a(i-1,j) such that b(i,j)b(i-1,j)-c(i,j)c(i,j-1) is nonzero.