A007283 -id:A007283 - cp's OEIS frontend

A164346 a(n) = 3 * 4^n.

3, 12, 48, 192, 768, 3072, 12288, 49152, 196608, 786432, 3145728, 12582912, 50331648, 201326592, 805306368, 3221225472, 12884901888, 51539607552, 206158430208, 824633720832, 3298534883328, 13194139533312, 52776558133248, 211106232532992, 844424930131968

Offset: 0

Klaus Brockhaus, Aug 13 2009

Binomial transform of A000244 without initial 1.
Second binomial transform of A007283.
Third binomial transform of A010701.
Inverse binomial transform of A005053 without initial 1.
First differences of A024036. - Omar E. Pol, Feb 16 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (4).

Cf. A000302 (powers of 4), A000244 (powers of 3), A007283 (3*2^n), A010701 (all 3's), A005053, A002001, A096045, A140660 (3*4^n+1), A002023 (6*4^n), A002063(9*4^n), A056120, A084509.

Magma
```
[ 3*4^n: n in [0..22] ];
```

3 4^Range[0,30]  (* Harvey P. Dale, Mar 11 2011 *)

A164346(n)=3*4^n \\ M. F. Hasler, Jul 28 2015

def A164346(n): return 3<<(n<<1) # Chai Wah Wu, Aug 30 2024

a(n) = 4*a(n-1) for n > 1; a(0) = 3.
G.f.: 3/(1-4*x).
a(n) = A002001(n+1). a(n) = A096045(n)+2. a(n) = A140660(n)-1.
a(n) = A002023(n)/2. a(n) = A002063(n)/3. a(n) = A056120(n+3)/9.
Apparently a(n) = A084509(n+3)/2.
a(n) = A110594(n+1), n>1. - R. J. Mathar, Aug 17 2009
a(n) = 3*A000302(n). - Omar E. Pol, Feb 18 2013
a(n) = A000079(2*n) + A000079(2*n+1). - M. F. Hasler, Jul 28 2015
E.g.f.: 3*exp(4*x). - G. C. Greubel, Sep 15 2017

A082505 a(n) = sum of (n-1)-th row terms of triangle A134059.

Original entry on oeis.org

0, 1, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368, 1610612736, 3221225472

Offset: 0

Labos Elemer, Apr 28 2003

a(n) is the least number x such that gcd(2^x, x-phi(x)) = 2^n. If cototient is replaced by totient, analogous values are different: A053576.

			G.f. = x + 6*x^2 + 12*x^3 + 24*x^4 + 48*x^5 + 96*x^6 + 192*x^7 + 384*x^8 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (2).

Cf. A000010, A007283, A009195, A050339, A134059.

Essentially the same as A003945 (and perhaps also A058764).

[0, 1] cat [ &+[ 3*Binomial(n,k): k in [0..n] ]: n in [1..30] ]; // Klaus Brockhaus, Dec 02 2009

0,1,seq(3*2^(n-1), n=2..40); # G. C. Greubel, Apr 27 2021

{0}~Join~Map[Total, {{1}}~Join~Table[3 Binomial[n, k], {n, 30}, {k, 0, n}]] (* Michael De Vlieger, Jul 03 2016, after Harvey P. Dale at A134059 *)
Table[3*2^(n-1) -(3/2)*Boole[n==0] -2*Boole[n==1], {n,0,40}] (* G. C. Greubel, Apr 27 2021 *)
Join[{0,1},NestList[2#&,6,30]] (* Harvey P. Dale, Jan 22 2024 *)

{a(n) = local(A); if( n<1, 0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (-6*k + 16) * A[k-1] + 2 * sum( j=1, k-1, A[j] * A[k-j])); A[n])} /* Michael Somos, Jul 23 2011 */

a(n)=if(n<2,n,3<<(n-1)) \\ Charles R Greathouse IV, Jun 16 2012

[0,1]+[3*2^(n-1) for n in (2..40)] # G. C. Greubel, Apr 27 2021

a(n) = A007283(n-1) for n>1, with a(0) = 0 and a(1) = 1.
G.f.: x * (1 + 4*x) / (1 - 2*x) = x / (1 - 6*x / (1 + 4*x)). - Michael Somos, Jun 15 2012
Starting (1, 6, 12, 24, 48, ...) = binomial transform of [1, 5, 1, 5, 1, 5, ...]. - Gary W. Adamson, Nov 18 2007
a(n+1) = Sum_{k=0..n} A109466(n,k)*A144706(k). - Philippe Deléham, Oct 30 2008
a(n) = (-6*n + 16) * a(n-1) + 2 * Sum_{k=1..n-1} a(k) * a(n-k) if n>1. - Michael Somos, Jul 23 2011
E.g.f.: (-3 - 4*x + 3*exp(2*x))/2. - Ilya Gutkovskiy, Jul 04 2016
a(n) = 3*2^(n-1) - (3/2)*[n=0] - 2*[n=1]. - G. C. Greubel, Apr 27 2021

More terms from Klaus Brockhaus, Dec 02 2009

A164090 a(n) = 2*a(n-2) for n > 2; a(1) = 2, a(2) = 3.

Original entry on oeis.org

2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 512, 768, 1024, 1536, 2048, 3072, 4096, 6144, 8192, 12288, 16384, 24576, 32768, 49152, 65536, 98304, 131072, 196608, 262144, 393216, 524288, 786432, 1048576, 1572864, 2097152, 3145728

Offset: 1

Klaus Brockhaus, Aug 09 2009

Interleaving of A000079 without initial 1 and A007283.
Agrees from a(2) onward with A145751 for all terms listed there (up to 65536). Apparently equal to 2, 3 followed by A090989. Equals 2 followed by A163978.
Binomial transform is A000129 without first two terms, second binomial transform is A020727, third binomial transform is A164033, fourth binomial transform is A164034, fifth binomial transform is A164035.
Number of achiral necklaces or bracelets with n beads using up to 2 colors. For n=5, the eight achiral necklaces or bracelets are AAAAA, AAAAB, AAABB, AABAB, AABBB, ABABB, ABBBB, and BBBBB. - Robert A. Russell, Sep 22 2018

Vincenzo Librandi, Table of n, a(n) for n = 1..2000
Index entries for linear recurrences with constant coefficients, signature (0,2).

Cf. A000079 (powers of 2), A007283 (3*2^n), A029744, A145751, A090989, A163978, A000129, A020727, A164033, A164034, A164035, A052955, A027383, A060482, A070875.

Second column of A284855.

[ n le 2 select n+1 else 2*Self(n-2): n in [1..42] ];

a[n_] := If[EvenQ[n], 3*2^(n/2 - 1), 2^((n + 1)/2)]; Array[a, 42] (* Jean-François Alcover, Oct 12 2017 *)
RecurrenceTable[{a[1]==2,a[2]==3,a[n]==2a[n-2]},a,{n,50}] (* or *) LinearRecurrence[{0,2},{2,3},50] (* Harvey P. Dale, Mar 01 2018 *)

a(n) = if(n%2,2,3) * 2^((n-1)\2); \\ Andrew Howroyd, Oct 07 2017

a(n) = A029744(n+1).
a(n) = A052955(n-1) + 1.
a(n) = A027383(n-2) + 2 for n > 1.
a(n) = A060482(n-1) + 3 for n > 3.
a(n) = A070875(n) - A070875(n-1).
a(n) = (7 - (-1)^n)*2^((1/4)*(2*n - 1 + (-1)^n))/4.
G.f.: x*(2+3*x)/(1-2*x^2).
a(n) = A063759(n-1), n>1. - R. J. Mathar, Aug 17 2009
Sum_{n>=1} 1/a(n) = 5/3. - Amiram Eldar, Mar 28 2022

A169634 a(n) = 3*7^n.

Original entry on oeis.org

3, 21, 147, 1029, 7203, 50421, 352947, 2470629, 17294403, 121060821, 847425747, 5931980229, 41523861603, 290667031221, 2034669218547, 14242684529829, 99698791708803, 697891541961621, 4885240793731347, 34196685556119429, 239376798892836003, 1675637592249852021

Offset: 0

Klaus Brockhaus, Apr 04 2010

Essentially first differences of A120741.
Binomial transform of A169604.
Second binomial transform of A005053 without initial term 1.
Inverse binomial transform of A103333 without initial term 1.
Second inverse binomial transform of A013708.
Except for first term 3, these are the integers that satisfy phi(n) = 4*n/7. - Michel Marcus, Jul 14 2015
Number of distinct quadratic residues (QR) over Z_7^n such that gcd(QR, 7^n) = 1 where n >= 1. - Param Mayurkumar Parekh, Feb 11 2023

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Shalosh B. Ekhad and Doron Zeilberger, A Bijective Proof of Richard Stanley's Observation that the sum of the cubes of the n-th row of Stern's Diatomic array equals 3 times 7 to the power n-1, arXiv:2103.12852 [math.CO], 2021.
Richard P. Stanley, Some Linear Recurrences Motivated by Stern's Diatomic Array, arXiv:1901.04647 [math.CO], 2019. Also American Mathematical Monthly 127.2 (2020): 99-111.
Index entries for linear recurrences with constant coefficients, signature (7).

Cf. A120741, A169604 (3*6^n), A005053 (expand (1-2x)/(1-5x)), A103333 (expand (1-5x)/(1-8x)), A013708 (3^(2*n+1)), A007283 (3*2^n), A164346 (3*4^n).

Magma
```
[ 3*7^n: n in [0..19] ];
```

3*7^Range[0, 25] (* Paolo Xausa, Jan 17 2025 *)

a(n) = 7*a(n-1) for n > 0; a(0) = 3.

G.f.: 3/(1-7*x).

A182633 Number of toothpicks added at n-th stage in the toothpick structure of A182632.

Original entry on oeis.org

0, 3, 6, 12, 12, 12, 24, 36, 24, 12, 24, 48, 60, 48, 60, 84, 48, 12, 24, 48, 60, 60, 84, 132, 132, 72, 60, 120, 168, 144, 156, 192, 96, 12, 24, 48, 60, 60, 84, 132, 132, 84, 84, 156, 228, 228, 228

Offset: 0

Omar E. Pol, Dec 07 2010

First differences of A182632.

a(n) is also the number of components added at n-th stage in the toothpick structure formed by V-toothpicks with an initial Y-toothpick, since a V-toothpick has two components and a Y-toothpick has three components (For more information see A161206, A160120, A161644).

			From _Omar E. Pol_, Feb 08 2013 (Start):
When written as a triangle:
0;
3;
6;
12,12;
12,24,36,24;
12,24,48,60,48,60, 84, 48;
12,24,48,60,60,84,132,132,72,60,120,168,144,156,192,96;
12,24,48,60,60,84,132,132,84,84,156,228,228,228,...
...
It appears that positive terms of the right border are A007283.
(End)

David Applegate, The movie version
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.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences
Index entries for sequences related to cellular automata

A139250, A139251, A160120, A160121, A161206, A161207, A161644, A161645, A182632.

It appears that a(n) = 2*A161645(n) but with a(1)=3.

a(n) = 3*A182635(n). - Omar E. Pol, Feb 09 2013

A334101 Numbers of the form q*(2^k), where q is one of the Fermat primes and k >= 0; Numbers n for which A329697(n) == 1.

Original entry on oeis.org

3, 5, 6, 10, 12, 17, 20, 24, 34, 40, 48, 68, 80, 96, 136, 160, 192, 257, 272, 320, 384, 514, 544, 640, 768, 1028, 1088, 1280, 1536, 2056, 2176, 2560, 3072, 4112, 4352, 5120, 6144, 8224, 8704, 10240, 12288, 16448, 17408, 20480, 24576, 32896, 34816, 40960, 49152, 65537, 65792, 69632, 81920, 98304, 131074, 131584, 139264

Offset: 1

Antti Karttunen, Apr 14 2020

nonn

Numbers k that themselves are not powers of two, but for which A171462(k) = k-A052126(k) is [a power of 2].
Numbers k such that A000265(k) is in A019434.
Squares of these numbers can be found (as a subset) in A334102, and the cubes (as a subset) in A334103.

Row 1 of A334100.
Cf. A000120, A000265, A052126, A171462, A209229, A329697, A334102, A334103.
Cf. A019434 (primes present), A007283, A020714, A110287 (other subsequences).
Subsequence of A018900.

A000265(n) = (n>>valuation(n,2));
isA019434(n) = ((n>2)&&isprime(n)&&!bitand(n-2,n-1));
isA334101(n) = isA019434(A000265(n));

A052126(n) = if(1==n,n,n/vecmax(factor(n)[, 1]));
A209229(n) = (n && !bitand(n,n-1));
isA334101(n) = ((!A209229(n))&&A209229(n-A052126(n)));

For all n, A000120(a(n)) = 2.

A181490 Numbers k such that 32^k-1 and 32^k+1 are twin primes (A001097).

Original entry on oeis.org

1, 2, 6, 18

Offset: 1

M. F. Hasler, Oct 30 2010

Sequences A181491 and A181492 list the corresponding primes.
No more terms below three million. - Charles R Greathouse IV, Mar 14 2011
Intersection of A002235 and A002253. - Jeppe Stig Nielsen, Mar 05 2018

Cf. A001097, A001359, A002235, A002253, A006512, A014574, A294730.

Filtered([1..300],k->IsPrime(3*2^k-1) and IsPrime(3*2^k+1)); # Muniru A Asiru, Mar 11 2018

a:=k->`if`(isprime(3*2^k-1) and isprime(3*2^k+1),k,NULL); seq(a(k),k=1..1000); # Muniru A Asiru, Mar 11 2018

fQ[n_] := PrimeQ[3*2^n - 1] && PrimeQ[3*2^n + 1]; k = 1; lst= {}; While[k < 15001, If[fQ@k, AppendTo[lst, k]; Print@k]; k++ ] (* Robert G. Wilson v, Nov 05 2010 *)
Select[Range[20],AllTrue[3*2^#+{1,-1},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 24 2014 *)

PARI
```
for( k=1,999, ispseudoprime(3<
				
```

Equals { k | A007283(k) in A014574 } = { k | A153893(k) in A001359 }.

Pari program repaired by Charles R Greathouse IV, Mar 14 2011

A336101 Numbers divisible by exactly one odd prime.

Original entry on oeis.org

3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 31, 34, 36, 37, 38, 40, 41, 43, 44, 46, 47, 48, 49, 50, 52, 53, 54, 56, 58, 59, 61, 62, 67, 68, 71, 72, 73, 74, 76, 79, 80, 81, 82, 83, 86, 88, 89, 92, 94, 96, 97, 98, 100, 101, 103, 104

Offset: 1

Peter Munn, Jul 08 2020

Numbers k for which A001221(A000265(k)) = 1. - Antti Karttunen, Jul 08 2020
Numbers whose odd part is a prime power (A246655). - Amiram Eldar, Jul 08 2020
Numbers of the form 2^r * p^q  with p an odd prime (A065091), r >= 0, q >= 1. - Bernard Schott, Dec 14 2020

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000265, A001221, A246655, A340373 (characteristic function).
Positions of ones in A005087.
Subsequence of A267895.
Subsequences: A007283 (3*2^n), A020714 (5*2^n), A005009 (7*2^n), A005015 (11*2^n), A005029 (13*2^n), A038550 (p*2^n, p odd prime), A065091 (odd primes), A061345 \ {1} (odd prime powers).

Select[Range[104], PrimePowerQ[#/2^IntegerExponent[#, 2]] &] (* Amiram Eldar, Jul 08 2020 *)

isA336101(n) = (1==omega(n>>valuation(n,2))); \\ Antti Karttunen, Jul 08 2020

A364258 a(n) = A163511(n) - n.

Original entry on oeis.org

1, 1, 2, 0, 4, 4, 0, -2, 8, 18, 8, 14, 0, 2, -4, -8, 16, 64, 36, 106, 16, 54, 28, 26, 0, 20, 4, 8, -8, -8, -16, -20, 32, 210, 128, 590, 72, 338, 212, 304, 32, 184, 108, 202, 56, 102, 52, 74, 0, 86, 40, 124, 8, 52, 16, 22, -16, 6, -16, -4, -32, -28, -40, -50, 64, 664, 420, 3058, 256, 1806, 1180, 2330, 144, 1052, 676

Offset: 0

Antti Karttunen, Jul 25 2023

Compare also to the scatter plot of A364294.

Antti Karttunen, Table of n, a(n) for n = 0..12288

Cf. A007283, A163511, A364255 [= gcd(n,a(n))], A364287 (positions of negative terms), A364292 (of terms <= 0), A364288, A364294 [= -a(A364293(n))].

f[n_] := Reverse@ Map[Ceiling[(Length@ # - 1)/2] &, DeleteCases[Split@ Join[Riffle[IntegerDigits[n, 2], 0], {0}], {k__} /; k == 1]]; {1}~Join~Table[-n + Function[t, Prime[t] Product[Prime[m]^(f[n][[m]]), {m, t}] ][DigitCount[n, 2, 1]], {n, 120}] (* Michael De Vlieger, Jul 25 2023 *)

from sympy import nextprime
def A364258(n):
    c, p, k = 1, 1, n
    while k:
        c *= (p:=nextprime(p))**(s:=(~k&k-1).bit_length())
        k >>= s+1
    return c*p-n # Chai Wah Wu, Jul 25 2023

a(n) = A364288(A163511(n)).
For n >= 1, a(2*n) = 2*a(n).
For n >= 0, a(A007283(n)) = 0.

A371100 Array A read by upward antidiagonals in which the entry A(n,k) in row n and column k is defined by A(n, k) = 4^n(6k - 3 - 2*(-1)^n) + (4^n - 1)/3, n,k >= 1.

Original entry on oeis.org

21, 21, 45, 341, 117, 69, 341, 725, 213, 93, 5461, 1877, 1109, 309, 117, 5461, 11605, 3413, 1493, 405, 141, 87381, 30037, 17749, 4949, 1877, 501, 165, 87381, 185685, 54613, 23893, 6485, 2261, 597, 189, 1398101, 480597, 283989, 79189, 30037, 8021, 2645, 693, 213, 1398101, 2970965, 873813, 382293, 103765, 36181, 9557, 3029, 789, 237

Offset: 1

Antti Karttunen and Ali Sada, Apr 18 2024

			The top left corner of the array:
n\k|      1       2       3        4        5        6        7        8
---+--------------------------------------------------------------------------
1  |     21,     45,     69,      93,     117,     141,     165,     189, ...
2  |     21,    117,    213,     309,     405,     501,     597,     693, ...
3  |    341,    725,   1109,    1493,    1877,    2261,    2645,    3029, ...
4  |    341,   1877,   3413,    4949,    6485,    8021,    9557,   11093, ...
5  |   5461,  11605,  17749,   23893,   30037,   36181,   42325,   48469, ...
6  |   5461,  30037,  54613,   79189,  103765,  128341,  152917,  177493, ...
7  |  87381, 185685, 283989,  382293,  480597,  578901,  677205,  775509, ...
8  |  87381, 480597, 873813, 1267029, 1660245, 2053461, 2446677, 2839893, ...
...

Paolo Xausa, Table of n, a(n) for n = 1..11325 (first 150 antidiagonals, flattened).

Cf. A007283, A079319, A257852, A371094, A371101.
Cf. A372351 (same terms, in different order), A372290 (sorted into ascending order, without duplicates), A372293 (odd numbers that do not occur here).
Leftmost column is A144864 duplicated, without its initial 1.
Row 1: A102603.

A371100[n_, k_] := 4^n*(6*k - 3 - 2*(-1)^n) + (4^n - 1)/3;
Table[A371100[n - k + 1, k], {n, 10}, {k, n}] (* Paolo Xausa, Apr 21 2024 *)

up_to = 55;
A371100sq(n,k) = 4^n*(6*k - 3 - 2*(-1)^n) + (4^n - 1)/3;
A371100list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A371100sq((a-(col-1)),col))); (v); };
v371100 = A371100list(up_to);
A371100(n) = v371100[n];

A(n, k) = A007283(n)*A257852(n,k) + A079319(n).
A(n, k) = A371094(A257852(n,k)).
A(n+2, k) = 5 + 16*A(n,k).

cp's OEIS Frontend

A164346 a(n) = 3 * 4^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

Formula

A082505 a(n) = sum of (n-1)-th row terms of triangle A134059.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Sage

Formula

Extensions

A164090 a(n) = 2*a(n-2) for n > 2; a(1) = 2, a(2) = 3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A169634 a(n) = 3*7^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A182633 Number of toothpicks added at n-th stage in the toothpick structure of A182632.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A334101 Numbers of the form q*(2^k), where q is one of the Fermat primes and k >= 0; Numbers n for which A329697(n) == 1.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

PARI

Formula

A181490 Numbers k such that 3*2^k-1 and 3*2^k+1 are twin primes (A001097).

Views

Author

Keywords

Comments

Crossrefs

Programs

A181490 Numbers k such that 32^k-1 and 32^k+1 are twin primes (A001097).

A371100 Array A read by upward antidiagonals in which the entry A(n,k) in row n and column k is defined by A(n, k) = 4^n(6k - 3 - 2*(-1)^n) + (4^n - 1)/3, n,k >= 1.