A008588 -id:A008588 - cp's OEIS frontend

A089357 a(n) = 2^(6*n).

1, 64, 4096, 262144, 16777216, 1073741824, 68719476736, 4398046511104, 281474976710656, 18014398509481984, 1152921504606846976, 73786976294838206464, 4722366482869645213696, 302231454903657293676544, 19342813113834066795298816

Offset: 0

Douglas Winston (douglas.winston(AT)srupc.com), Dec 26 2003

For n > 0, numbers M such that a(n) is the highest power of 2 in the Collatz (3x+1) iteration are given by 2^k*(a(n)-1)/3 for any k >= 0. Example: For n = 1, the numbers such that 64 is the highest power of 2 in the Collatz (3x+1) iteration are given by 2^k*(64-1)/3 = 21*2^k for any k >= 0. See A008908 for more information on the Collatz (3x+1) iteration. - Derek Orr, Sep 22 2014

Powers of 64. - Alexander Fraebel, Aug 29 2020

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (64).

Cf. A000079, A000302, A001018, A001025, A008588, A008908, A009976.

[2^(6*n): n in [0..20]]; // Vincenzo Librandi, Jun 07 2011

seq(2^(6*n), n=0..14); # Nathaniel Johnston, Jun 26 2011

2^(6 Range[0,20]) (* or *) NestList[64#&,1,20] (* Harvey P. Dale, Sep 28 2011 *)

makelist(2^(6*n),n,0,20); /* Martin Ettl, Nov 12 2012 */

a(n)=64^n \\ Charles R Greathouse IV, Jul 02 2016

G.f.: 1/(1-64*x). - Philippe Deléham, Nov 24 2008
a(n) = 63*a(n-1) + 64^(n-1), a(0)=1. - Vincenzo Librandi, Jun 07 2011
E.g.f.: exp(64*x). - Ilya Gutkovskiy, Jul 02 2016
a(n) = A000079(A008588(n)). - Wesley Ivan Hurt, Jul 02 2016
a(n) = 64*a(n-1). - Miquel Cerda, Oct 27 2016

A092260 Permutation of natural numbers generated by 6-rowed array shown below.

Original entry on oeis.org

1, 11, 2, 13, 10, 3, 23, 14, 9, 4, 25, 22, 15, 8, 5, 35, 26, 21, 16, 7, 6, 37, 34, 27, 20, 17, 12, 47, 38, 33, 28, 19, 18, 49, 46, 39, 32, 29, 24, 59, 50, 45, 40, 31, 30, 61, 58, 51, 44, 41, 36, 71, 62, 57, 52, 43, 42, 73, 70, 63, 56, 53, 48, 83, 74, 69, 64, 55, 54, 85, 82, 75

Offset: 1

Giovanni Teofilatto, Feb 19 2004

nonn

11 13 23 25 35 37 47 49 59... (A091998)
10 14 22 26 34 38 46 50 58... (A091999)
9 15 21 27 33 39 45 51 57... (A016945)
8 16 20 28 32 40 44 52 56... (A092259)
7 17 19 29 31 41 43 53 55... (A092242)
12 18 24 30 36 42 48 54 60... (A008588, excluding initial term)
For such arrays A_k, here A_6, see a W. Lang comment on A113807, the A_7 case. However, to get the array A_6 one should take the last line as the first one and add a 0 in front (thus obtaining a permutation of the nonnegative integers). - Wolfdieter Lang, Feb 02 2012

Cf. A088520, A090243, A090298, A113807.

Edited and extended by Ray Chandler, Feb 21 2004

A228081 a(n) = 64^n + 1.

Original entry on oeis.org

2, 65, 4097, 262145, 16777217, 1073741825, 68719476737, 4398046511105, 281474976710657, 18014398509481985, 1152921504606846977, 73786976294838206465, 4722366482869645213697, 302231454903657293676545, 19342813113834066795298817, 1237940039285380274899124225

Offset: 0

Arkadiusz Wesolowski, Aug 09 2013

These numbers can be written as the sum of two relatively prime squares and also as the sum of two relatively prime cubes (i.e., 2^(6*n) + 1 = (2^(3*n))^2 + 1^2 = (2^(2*n))^3 + 1^3).

			a(2) = 64^2 + 1 = 4097.

Arkadiusz Wesolowski, Table of n, a(n) for n = 0..100
Index entries for sequences related to sums of cubes
Index entries for sequences related to sums of squares
Index entries for linear recurrences with constant coefficients, signature (65,-64).

Cf. A000051 (2^n + 1), A052539 (4^n + 1), A062395 (8^n + 1).

Magma
```
[64^n+1 : n in [0..15]];
```

Table[64^n + 1, {n, 0, 15}]
LinearRecurrence[{65,-64},{2,65},20] (* Harvey P. Dale, Jul 17 2020 *)

PARI
```
for(n=0, 15, print1(64^n+1, ", "))
```

a(n) = 64*a(n-1) - 63.
a(n) = A089357(n) + 1 = 2^A008588(n) + 1.
G.f.: (2 - 65*x)/((1 - x)*(1 - 64*x)).
E.g.f.: e^x + e^(64*x).

A355200 Numbers k that can be written as the sum of 3 divisors of k (not necessarily distinct).

Original entry on oeis.org

3, 4, 6, 8, 9, 12, 15, 16, 18, 20, 21, 24, 27, 28, 30, 32, 33, 36, 39, 40, 42, 44, 45, 48, 51, 52, 54, 56, 57, 60, 63, 64, 66, 68, 69, 72, 75, 76, 78, 80, 81, 84, 87, 88, 90, 92, 93, 96, 99, 100, 102, 104, 105, 108, 111, 112, 114, 116, 117, 120, 123, 124, 126, 128, 129, 132, 135

Offset: 1

Wesley Ivan Hurt, Jun 23 2022

From Bernard Schott, Aug 06 2023: (Start)
Equivalently: positive numbers that are divisible by 3 or by 4.
Proof (similar to the proof proposed by Robert Israel in A355641).
If k is divisible by 3, then k is in the sequence because k = k/3 + k/3 + k/3.
If k is divisible by 4, then k is in the sequence because k = k/2 + k/4 + k/4.
Moreover, if k is positive and divisible by 6 (A008588), then k = k/3 + k/3 + k/3, but k is also in the sequence because k = k/2 + k/3 + k/6.
Conversely, to show that every term of this sequence is divisible by 3 or by 4, we consider all positive integer solutions of the equation 1 = 1/a + 1/b + 1/c. Without loss of generality, we may assume a <= b <= c, then 3/a >= 1/a + 1/b + 1/c = 1. So a <= 3. Similarly, given a, we have 2/b >= 1/b + 1/c = 1 - 1/a, so b <= 2/(1 - 1/a).
-> if a = 1, then 1 = 1 + 1/b + 1/c; this equation has clearly no solution.
-> if a = 2, then 1/2 = 1/b + 1/c with b <= 2/(1 - 1/2) = 4; in this case, there are two solutions: (a,b,c) = (2,3,6) or (a,b,c) = (2,4,4).
-> if a = 3, then 2/3 = 1/b + 1/c with b <= 2/(1 - 1/3) = 3; in this case, there is one solution: (a,b,c) = (3,3,3).
It turns out that there are only 3 solutions with a <= b <= c. Each corresponds to a possible pattern k = k/a + k/b + k/c for writing k as the sum of 3 of its divisors, which works when k is divisible by 3 or by 4. (End)
From David A. Corneth, Aug 07 2023: (Start)
Proof that a(n + 6) = a(n) + 12.
As k is in the sequence, k = k/d1 + k/d2 + k/d3 where d1, d2 and d3 | k and they are not necessarily distinct. By discussion above from Bernard Schott, Aug 06 2023, (d1, d2, d3) are in {(2, 3, 6), (2, 4, 4), (3, 3, 3)}. The lcm of these tuples are 6, 4 and 3 respectively. So any number k in the sequence is divisible by 3, 4 or 6.
This tells us that if k is in the sequence then k + 12 is in the sequence since k + 12 is divisible by one of 3, 4 or 6 since lcm(3, 4, 6) = 12.
So we can write a(n + m) = a(n) + 12 for some m. Inspection gives m = 6 so a(n + 6) = a(n) + 12. (End)

			6 is in the sequence since it can be written as the sum of 3 of its (not necessarily distinct) divisors: 6 = 1+2+3 = 2+2+2 with 1|6, 2|6, and 3|6.

Ray Chandler, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-2,2,-1).

Cf. A002966, A008588, A348536.
Numbers k that can be written as the sum of j divisors of k (not necessarily distinct) for j=1..10: A000027 (j=1), A299174 (j=2), this sequence (j=3), A354591 (j=4), A355641 (j=5), A356609 (j=6), A356635 (j=7), A356657 (j=8), A356659 (j=9), A356660 (j=10).
Equals positive terms of A008585 union A008586.
Cf. A005843, A134667.

q[n_, k_] := AnyTrue[Tuples[Divisors[n], k], Total[#] == n &]; Select[Range[135], q[#, 3] &] (* Amiram Eldar, Aug 21 2022 *)
Table[2n + (Sin[Pi*n/3] + Sin[2*Pi*n/3])/Sqrt[3], {n, 100}] (* Wesley Ivan Hurt, Oct 30 2023 *)
CoefficientList[Series[(3 - 2*x + 4*x^2 - 2*x^3 + 3*x^4)/((x - 1)^2*(1 + x^2 + x^4)), {x, 0, 80}], x] (* Wesley Ivan Hurt, Jul 17 2025 *)

isok(k) = my(d=divisors(k)); forpart(p=k, if (setintersect(d, Set(p)) == Set(p), return(1)), , [3,3]); \\ Michel Marcus, Aug 21 2022

is(n) = my(v = [3,4,6]); sum(i = 1, 3, n%v[i] == 0) > 0 \\ David A. Corneth, Oct 08 2022

a(n + 6) = a(n) + 12. - David A. Corneth, Oct 08 2022
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/3 - log(3)/4. - Amiram Eldar, Sep 10 2023
From Wesley Ivan Hurt, Oct 30 2023: (Start)
a(n) = 2*n + (sin(Pi*n/3) + sin(2*Pi*n/3))/sqrt(3).
a(n) = A005843(n) + A134667(n). (End)
From Wesley Ivan Hurt, Jul 17 2025: (Start)
G.f.: x*(3-2*x+4*x^2-2*x^3+3*x^4)/((x-1)^2*(1+x^2+x^4)).
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - 2*a(n-4) + 2*a(n-5) - a(n-6). (End)

A046954 Numbers k such that 6*k + 1 is nonprime.

Original entry on oeis.org

0, 4, 8, 9, 14, 15, 19, 20, 22, 24, 28, 29, 31, 34, 36, 39, 41, 42, 43, 44, 48, 49, 50, 53, 54, 57, 59, 60, 64, 65, 67, 69, 71, 74, 75, 78, 79, 80, 82, 84, 85, 86, 88, 89, 92, 93, 94, 97, 98, 99, 104, 106, 108, 109, 111, 113, 114, 116, 117, 119, 120, 124, 127, 129, 130, 132, 133, 134, 136, 139, 140

Offset: 1

Felice Russo

nonn

Equals A171696 U A121763; A121765 U A171696 = A046953; A121763 U A121765 = A067611 where A067611 U A002822 U A171696 = A001477. - Juri-Stepan Gerasimov, Feb 13 2010, Feb 15 2010

These numbers (except 0) can be written as 6xy +-(x+y) for x > 0, y > 0. - Ron R Spencer, Aug 01 2016

			a(2)=8 because 6*8 + 1 = 49, which is composite.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A047845 (2n+1), A045751 (4n+1), A127260 (8n+1).

Cf. A046953, A008588, A016921, subsequence of A067611, complement of A024899.

Filtered([0..250], k-> not IsPrime(6*k+1)) # G. C. Greubel, Feb 21 2019

a046954 n = a046954_list !! (n-1)
a046954_list = map (`div` 6) $ filter ((== 0) . a010051' . (+ 1)) [0,6..]
-- Reinhard Zumkeller, Jul 13 2014

[n: n in [0..250] | not IsPrime(6*n+1)]; // G. C. Greubel, Feb 21 2019

remove(k-> isprime(6*k+1), [$0..140])[]; # Muniru A Asiru, Feb 22 2019

a = Flatten[Table[If[PrimeQ[6*n + 1] == False, n, {}], {n, 0, 50}]] (* Roger L. Bagula, May 17 2007 *)
Select[Range[0, 200], !PrimeQ[6 # + 1] &] (* Vincenzo Librandi, Sep 27 2013 *)

is(n)=!isprime(6*n+1) \\ Charles R Greathouse IV, Aug 01 2016

[n for n in (0..250) if not is_prime(6*n+1)] # G. C. Greubel, Feb 21 2019

Edited by N. J. A. Sloane, Aug 08 2008 at the suggestion of R. J. Mathar
Corrected by Juri-Stepan Gerasimov, Feb 13 2010, Feb 15 2010
Corrected by Vincenzo Librandi, Sep 27 2013

A088723 Numbers k with at least one divisor d>1 such that d+1 also divides k.

Original entry on oeis.org

6, 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 72, 78, 80, 84, 90, 96, 100, 102, 108, 110, 112, 114, 120, 126, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 180, 182, 186, 192, 198, 200, 204, 210, 216, 220, 222, 224, 228, 234, 240, 246, 252, 258, 260

Offset: 1

Reinhard Zumkeller, Oct 12 2003

nonn

Complement of A088725.
Complement of A132895 relative to A005843, the even numbers. - Ray Chandler, May 29 2008
The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 1, 21, 222, 2218, 22187, 221945, 2219452, 22194766, 221947862, 2219480686, ... . Apparently, the asymptotic density of this sequence exists and equals 0.22194... . - Amiram Eldar, Apr 20 2025

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

Cf. A088722, A088724, A088725, A088726.

Cf. A027750, A008588 (subsequence).

a088723 n = a088723_list !! (n-1)
a088723_list = filter f [2..] where
   f x = 1 `elem` (zipWith (-) (tail divs) divs)
         where divs = tail $ a027750_row x
-- Reinhard Zumkeller, Dec 16 2013

Select[Range[300],MemberQ[Differences[Select[Divisors[#], #>1&]], 1]&]  (* Harvey P. Dale, Apr 03 2011 *)

isok(k) = if(k%2, 0, if(!(k%3), 1, fordiv(k, d, if(d > 1 && !(k % (d+1)), return(1))); 0)); \\ Amiram Eldar, Apr 20 2025

A088722(a(n)) > 0.

Extended by Ray Chandler, May 29 2008

A082108 a(n) = 4n^2 + 6n + 1.

Original entry on oeis.org

1, 11, 29, 55, 89, 131, 181, 239, 305, 379, 461, 551, 649, 755, 869, 991, 1121, 1259, 1405, 1559, 1721, 1891, 2069, 2255, 2449, 2651, 2861, 3079, 3305, 3539, 3781, 4031, 4289, 4555, 4829, 5111, 5401, 5699, 6005, 6319, 6641, 6971, 7309, 7655, 8009, 8371

Offset: 0

Paul Barry, Apr 03 2003

a(n) is the sum of the numerator and denominator of (n+1)/(2*n) + (n+2)/(2*(n+1)); all fractions are reduced and n > 0. - J. M. Bergot, Jun 14 2017

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A028387, A069131.

[4*n^2+6*n+1: n in [0..60]]; // G. C. Greubel, Dec 22 2022

(* Programs from Michael De Vlieger, Jun 15 2017 *)
Table[4n^2 +6n +1, {n,0,50}]
LinearRecurrence[{3,-3,1}, {1,11,29}, 51]
CoefficientList[Series[(1+8*x-x^2)/(1-x)^3, {x,0,50}], x] (* End *)

a(n)=4*n^2+6*n+1 \\ Charles R Greathouse IV, Oct 07 2015

[4*n^2+6*n+1 for n in range(61)] # G. C. Greubel, Dec 22 2022

a(n) = a(n-1) + 8*n + 2. - Vincenzo Librandi, Aug 08 2010
From Michael De Vlieger, Jun 15 2017: (Start)
G.f.: (1 + 8*x - x^2)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
a(n) = A016742(n) + A008588(n) + 1. - Felix Fröhlich, Jun 16 2017
Sum_{k=1..n} a(k-1)/(2*k)! = 1 - 1/(2*n)!. - Robert Israel, Jul 19 2017
E.g.f.: (1 + 10*x + 4*x^2)*exp(x). - G. C. Greubel, Dec 22 2022

Incorrect formula and useless examples deleted by R. J. Mathar, Aug 31 2010

A330853 First occurrences of gaps between primes 6k+1: gap sizes.

Original entry on oeis.org

6, 12, 18, 30, 24, 54, 42, 36, 48, 60, 78, 66, 72, 84, 90, 96, 114, 102, 162, 108, 126, 120, 132, 150, 138, 144, 174, 168, 156, 192, 204, 180, 198, 252, 270, 216, 222, 186, 228, 210, 240, 282, 246, 234, 276, 264, 258, 312, 330, 318, 288, 306, 294, 336, 300, 378

Offset: 1

Alexei Kourbatov, Apr 27 2020

nonn

Contains A268925 as a subsequence.
Conjecture: the sequence is a permutation of all positive multiples of 6, i.e., all positive terms of A008588.
Conjecture: a(n) = O(n). See arXiv:2002.02115 (sect.7) for discussion.

			The first primes of the form 6k+1 are 7 and 13, so a(1)=13-7=6. The next prime 6k+1 is 19, and the gap 19-13=6 already occurred, so a new term is not added to the sequence. The next prime 6k+1 is 31, and the gap 31-19=12 is occurring for the first time; therefore a(2)=12.

Alexei Kourbatov, Table of n, a(n) for n = 1..135
Alexei Kourbatov and Marek Wolf, On the first occurrences of gaps between primes in a residue class, arXiv preprint arXiv:2002.02115 [math.NT], 2020.

Cf. A002476, A014320, A058320, A330854 (primes 6k+1 preceding the first-occurrence gaps), A330855 (primes 6k+1 at the end of the first-occurrence gaps).

isFirstOcc=vector(9999,j,1); s=7; forprime(p=13,1e8, if(p%6!=1,next); g=p-s; if(isFirstOcc[g/6], print1(g", "); isFirstOcc[g/6]=0); s=p)

a(n) = A330855(n) - A330854(n).

A037917 Numbers n such that the Fibonacci number F(n) is divisible by a square.

Original entry on oeis.org

6, 12, 18, 24, 25, 30, 36, 42, 48, 50, 54, 56, 60, 66, 72, 75, 78, 84, 90, 91, 96, 100, 102, 108, 110, 112, 114, 120, 125, 126, 132, 138, 144, 150, 153, 156, 162, 168, 174, 175, 180, 182, 186, 192, 198, 200, 204, 210, 216, 220, 222, 224, 225, 228, 234, 240

Offset: 1

Marc LeBrun

nonn

Is a(n) asymptotic to C*n with 4 < C < 4.5 ? - Benoit Cloitre, Sep 04 2002
Numbers are a superset of the multiples of 6 (A008588), because 8 divides Fibonacci(6m) = A134492(m). Sequence apparently also contains the multiples of 25. Are all a(n) composite? Members not divisible by 6 or 25 are 56, 91, 110, 112, 153, 182, 220, 224, 273, 280, ... - Ralf Stephan, Jan 26 2014
These numbers are the positive multiples of A065069. - Charles R Greathouse IV, Feb 02 2014
To address Cloitre's question, if such C exists it must be less than 4.3 using the known terms of A065069. - Charles R Greathouse IV, Feb 04 2014

Amiram Eldar, Table of n, a(n) for n = 1..328 (terms 1..235 from T. D. Noe, based on Kelly's table)
Blair Kelly, Fibonacci Factorizations
Eric Weisstein's World of Mathematics, Fibonacci Number.

Cf. A000045, A037918, A001177, A030426.

Select[Range[100], MoebiusMu[Fibonacci[#]] == 0 &] (* Alonso del Arte, Jan 26 2014 *)

is(n)=!issquarefree(fibonacci(n)) \\ Charles R Greathouse IV, Feb 02 2014

More terms from Eric W. Weisstein

A083039 Number of divisors of n that are <= 3.

Original entry on oeis.org

1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3

Offset: 1

Daniele A. Gewurz (gewurz(AT)mat.uniroma1.it) and Francesca Merola (merola(AT)mat.uniroma1.it), May 06 2003

Periodic of period 6. Parker vector of the wreath product of S_3 and S, the symmetric group of a countable set.

			The divisors of 6 are 1, 2, 3 and 6. Of those divisors, 1, 2 and 3 are <= 3. That's three divisors, therefore, a(6) = 3. - _David A. Corneth_, Sep 30 2017

Antti Karttunen, Table of n, a(n) for n = 1..12000
D. A. Gewurz and F. Merola, Sequences realized as Parker vectors of oligomorphic permutation groups, J. Integer Seq., 6 (2003), 03.1.6
M. D. Hirschhorn, J. A. Sellers, Enumeration of unigraphical partitions, JIS 11 (2008) 08.4.6
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (-1, 0, 1, 1).

Cf. A000005, A007310, A008588, A047228, A083040, A138553.

LinearRecurrence[{-1, 0, 1, 1},{1, 2, 2, 2},90] (* Ray Chandler, Aug 26 2015 *)

PARI
```
a(n)=[3,1,2,2,2,1][n%6+1];
```

G.f.: x/(1-x) + x^2/(1-x^2) + x^3/(1-x^3).
a(n) = a(n-6) = a(-n).
a(n) = 11/6 - (1/2)*(-1)^n - (1/3)*cos(2*Pi*n/3) - (1/3)*3^(1/2)*sin(2*Pi*n/3). - Richard Choulet, Dec 12 2008
a(n) = Sum_{k=1..1} cos(n*(k - 1)/1*2*Pi)/1 + Sum_{k=1..2} cos(n*(k - 1)/2*2*Pi)/2 + Sum_{k=1..3} cos(n*(k - 1)/3*2*Pi)/3. - Mats Granvik, Sep 09 2012
a(n) = log_2(gcd(n,2) + gcd(n,6)). - Gary Detlefs, Feb 15 2014
a(n) = Sum_{d|n, d<=3} 1. - Wesley Ivan Hurt, Oct 30 2023

cp's OEIS Frontend

A089357 a(n) = 2^(6*n).

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A092260 Permutation of natural numbers generated by 6-rowed array shown below.

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A228081 a(n) = 64^n + 1.

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A355200 Numbers k that can be written as the sum of 3 divisors of k (not necessarily distinct).

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A046954 Numbers k such that 6*k + 1 is nonprime.

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A088723 Numbers k with at least one divisor d>1 such that d+1 also divides k.

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A082108 a(n) = 4*n^2 + 6*n + 1.

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A082108 a(n) = 4n^2 + 6n + 1.