A017545 -id:A017545 - cp's OEIS frontend

A139267 Twice octagonal numbers: 2n(3*n-2).

0, 2, 16, 42, 80, 130, 192, 266, 352, 450, 560, 682, 816, 962, 1120, 1290, 1472, 1666, 1872, 2090, 2320, 2562, 2816, 3082, 3360, 3650, 3952, 4266, 4592, 4930, 5280, 5642, 6016, 6402, 6800, 7210, 7632, 8066, 8512, 8970, 9440, 9922

Offset: 0

Omar E. Pol, May 14 2008, May 19 2008

Sequence found by reading the line from 0, in the direction 0, 2,..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. Opposite numbers to the members of A033580 in the same spiral. - Omar E. Pol, Sep 09 2011

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. A000567, A017545.

Cf. numbers of the form n*(n*k-k+4)/2 listed in A226488 (this sequence is the case k=12).

List([0..50], n-> 2*n*(3*n-2)); # G. C. Greubel, Sep 18 2019

[2*n*(3*n-2): n in [0..50]]; // G. C. Greubel, Sep 18 2019

seq(2*n*(3*n-2), n=0..50); # G. C. Greubel, Sep 18 2019

Table[2*n*(3*n-2), {n,0,50}] (* G. C. Greubel, Jun 07 2017 *)

a(n)=2*n*(3*n-2) \\ Charles R Greathouse IV, Oct 07 2015

[2*n*(3*n-2) for n in (0..50)] # G. C. Greubel, Sep 18 2019

a(n) = 2*A000567(n) = 6*n^2 - 4*n = 2*n*(3*n - 2).
a(n) = a(n-1) + 12*n - 10, with n>0, a(0)=0. - Vincenzo Librandi, Aug 03 2010
G.f.: x*(2+10*x)/(1-3*x+3*x^2-x^3). - Colin Barker, Jan 06 2012
After 0, a(n) = Sum_{i=0..n-1} (12*i + 2). - Bruno Berselli, Sep 11 2013
E.g.f.: 2*x*(1 + 3*x)*exp(x). - G. C. Greubel, Sep 18 2019

A017581 a(n) = 12*n + 5.

Original entry on oeis.org

5, 17, 29, 41, 53, 65, 77, 89, 101, 113, 125, 137, 149, 161, 173, 185, 197, 209, 221, 233, 245, 257, 269, 281, 293, 305, 317, 329, 341, 353, 365, 377, 389, 401, 413, 425, 437, 449, 461, 473, 485, 497, 509, 521, 533, 545, 557, 569, 581, 593, 605, 617, 629

Offset: 0

N. J. A. Sloane

Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0(71).
A089911(2*a(n)) = 7. - Reinhard Zumkeller, Jul 05 2013
Equivalently, intersection of A016813 and A016789. - Bruno Berselli, Jan 24 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
Tanya Khovanova, Recursive Sequences.
William A. Stein, Dimensions of the spaces S_k(Gamma_0(N)).
William A. Stein, The modular forms database.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008594, A016789, A016813, A016969, A017533, A017545, A017593, A089911.

a017581 = (+ 5) . (* 12)  -- Reinhard Zumkeller, Jul 05 2013

[12*n+5: n in [0..60]]; // Vincenzo Librandi, Jun 08 2011

12*Range[0,200]+5 (* Vladimir Joseph Stephan Orlovsky, Feb 19 2011 *)

a(n)=12*n+5 \\ Charles R Greathouse IV, Jul 10 2016

a(n) = 2*a(n-1) - a(n-2) for n>1, a(0)=5, a(1)=17. - Vincenzo Librandi, Jun 08 2011
G.f.: x*(5 + 7*x)/(1 - x)^2. - Wolfdieter Lang, Jul 04 2023
E.g.f.: exp(x)*(5 + 12*x). - Stefano Spezia, Feb 21 2024
a(n) = A016969(2*n) = A016789(4*n+1). - Elmo R. Oliveira, Apr 10 2025

A072065 Define a "piece" to consist of 3 mutually touching pennies welded together to form a triangle; sequence gives side lengths of triangles that can be made from such pieces.

Original entry on oeis.org

0, 2, 9, 11, 12, 14, 21, 23, 24, 26, 33, 35, 36, 38, 45, 47, 48, 50, 57, 59, 60, 62, 69, 71, 72, 74, 81, 83, 84, 86, 93, 95, 96, 98, 105, 107, 108, 110, 117, 119, 120, 122, 129, 131, 132, 134, 141, 143, 144, 146, 153, 155, 156, 158, 165, 167, 168, 170, 177, 179, 180

Offset: 1

Jim McCann (jmccann(AT)umich.edu), Aug 04 2002

The "piece" in question is also called a "tribone" [Ardila and Stanley]. - N. J. A. Sloane, Feb 27 2014

			A possible side-9 arrangement:
          A
         A A
        B B C
       D B C C
      D D E E F
     G H H E F F
    G G H I I J J
   K L L M I N J O
  K K L M M N N O O

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
F. Ardila and R. P. Stanley, Tilings, arXiv:math/0501170 [math.CO], 2005.
J. H. Conway and J. C. Lagarias, Tiling with Polyominoes and Combinatorial Group Theory, Journal of Combinatorial Theory, Series A 53 (1990), 183-208. [From _N. J. A. Sloane_, Jul 04 2011]
Jim McCann, Triangle Sequence
N. C. Saldanha and C. Tomei, An overview of domino and lozenge tilings, arXiv:math/9801111 [math.CO], 1998.
Torsten Sillke, A Word Problem
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Union of A008594, A017545, A017629 and A017653.

a072065 n = a072065_list !! n
a072065_list = filter ((`elem` [0,2,9,11]) . (`mod` 12)) [0..]
-- Reinhard Zumkeller, Jan 09 2013

f:=r-> {seq(12*i+r,i=0..100)}; t1:= f(0) union f(2) union f(9) union f(11); t2:=sort(convert(t1,list)); # N. J. A. Sloane, Jul 04 2011

Select[Range[0,200],MemberQ[{0,2,9,11},Mod[#,12]]&] (* Harvey P. Dale, Dec 15 2011 *)
LinearRecurrence[{1,0,0,1,-1},{0,2,9,11,12},70] (* Harvey P. Dale, Jan 30 2015 *)

concat(0, Vec(x^2*(2+7*x+2*x^2+x^3)/((1-x)^2*(1+x)*(1+x^2)) + O(x^100))) \\ Colin Barker, Dec 12 2015

A number n is in the sequence iff n == 0, 2, 9 or 11 (mod 12). See Conway-Lagarias or the Sillke link. - Sascha Kurz, Mar 04 2003
a(1)=0, a(2)=2, a(3)=9, a(4)=11, a(5)=12, a(n) = a(n-1)+a(n-4)-a(n-5). - Harvey P. Dale, Jan 30 2015
From Colin Barker, Dec 12 2015: (Start)
a(n) = (3/4+(3*i)/4)*(i^n-i*(-i)^n)-(-1)^n/2+3*(n+1)-5 where i = sqrt(-1).
G.f.: x^2*(2+7*x+2*x^2+x^3) / ((1-x)^2*(1+x)*(1+x^2)). (End)
E.g.f.: (2 + 3*cos(x) + (6*x - 5)*cosh(x) - 3*sin(x) + (6*x - 3)*sinh(x))/2. - Stefano Spezia, May 05 2022
a(n) = (6*n-4-(-1)^n+3*(-1)^((2*n+1-(-1)^n)/4))/2. - Wesley Ivan Hurt, Nov 09 2023

Offset corrected by Reinhard Zumkeller, Jan 09 2013

A332511 Numbers k such that phi(k) == 2 (mod 12), where phi is the Euler totient function (A000010).

Original entry on oeis.org

3, 4, 6, 121, 242, 529, 1058, 2209, 3481, 4418, 5041, 6889, 6962, 10082, 11449, 13778, 14641, 17161, 22898, 27889, 29282, 32041, 34322, 36481, 51529, 55778, 57121, 63001, 64082, 69169, 72962, 96721, 103058, 114242, 120409, 126002, 128881, 138338, 146689, 175561

Offset: 1

Amiram Eldar, Feb 14 2020

nonn

Dence and Dence noted that the values of phi(k) congruent to 2 (mod 12) are sparse compared to the other possible even values. For example, for k <= 10^4 there only 10 values of phi(k) congruent to 2 (mod 12), compared to 6114, 1650, 511, 1233, and 476 values congruent to 0, 4, 6, 8, and 10 (mod 12), respectively. They proved that the asymptotic density of this sequence is 0 by showing that the only terms above 6 are of the form p^e and 2*p^e with p == 11 (mod 12) a prime and e even.

Dence and Pomerance showed that the asymptotic number of the terms below x is ~ (1/2 + 1/(2*sqrt(2)))*sqrt(x)/log(x).

			121 is a term since phi(121) = 110 == 2 (mod 12).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Joseph B. Dence and Thomas P. Dence, A Surprise Regarding the Equation phi(x) = 2(6n + 1), The College Mathematics Journal, Vol. 26, No. 4 (1995), pp. 297-301.
Thomas Dence and Carl Pomerance, Euler's function in residue classes, in: K. Alladi, P. D. T. A. Elliott, A. Granville and G. Tenenbaum (eds.), Analytic and Elementary Number Theory, Developments in Mathematics, Vol. 1, Springer, Boston, MA, 1998, pp. 7-20, alternative link.

Cf. A000010, A017545, A201488 (coefficient in asymptotic formula), A332512, A332513, A332514, A332515, A332516.

[k:k in [1..180000]| EulerPhi(k) mod 12 eq 2]; // Marius A. Burtea, Feb 14 2020

Select[Range[2*10^5], Mod[EulerPhi[#], 12] == 2 &]

A063667 Number of solutions of phi(x) = 12n + 2.

Original entry on oeis.org

3, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Labos Elemer, Aug 22 2001

nonn

			In the range n=0..10000, only 18 invphi(12n + 2) sets are nonempty, always with 2 terms. E.g., n = 8034, a(8034) = 2 because 12*8034 + 2 = 96410 and invphi(96410) = {96721,193442}. - Original comment corrected by _Antti Karttunen_, Nov 07 2018
In the range n <= 100000, there are 48 nonzero values. - _Antti Karttunen_, Nov 07 2018

Antti Karttunen, Table of n, a(n) for n = 0..100000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000010, A002202, A005277, A014197, A017545.

with(numtheory): [seq(nops(invphi(2+12*j)),j=0..10000)];

A014197(n, m=1) = { n==1 && return(1+(m<2)); my(p, q); sumdiv(n, d, if( d>=m && isprime(d+1), sum( i=0, valuation(q=n\d, p=d+1), A014197(q\p^i, p))))}; \\ From A014197 by M. F. Hasler
A063667(n) = A014197(2+(12*n)); \\ Antti Karttunen, Nov 07 2018

a(n) = invphiNum(12*n+2); \\ Amiram Eldar, Nov 29 2024, using Max Alekseyev's invphi.gp

a(n) = A014197(A017545(n)). - Antti Karttunen, Nov 07 2018

Term a(0) = 3 prepended by Antti Karttunen, Nov 07 2018

A337609 Positive integers m such that A126286^k(m) = m for some positive integer k.

Original entry on oeis.org

2, 3, 14, 21, 26, 34, 38, 39, 50, 57, 62, 74, 75, 85, 86, 93, 94, 98, 110, 111, 118, 122, 129, 134, 142, 146, 147, 154, 158, 165, 170, 182, 183, 194, 201, 202, 206, 214, 218, 219, 230, 235, 237, 242, 254, 255, 266, 273, 274, 278, 286, 290, 291, 298, 302, 309

Offset: 1

Ely Golden, Oct 07 2020

A126286^k(m) means apply A126286 to m k times.
Equivalently, the numbers that belong to a cycle under the map x -> A126286(x).
For any term m in this sequence, A126286(A126286(m)) = m.
Supersequence of A017545. Moreover, this sequence can be represented as an infinite union of arithmetic progressions.
2 and 3 are the only primes in this sequence.

			3 is a term since A126286(A126286(3)) = A126286(2) = 3.

Ely Golden, Table of n, a(n) for n = 1..10000
Ely Golden, Python program for generating terms of this sequence

Cf. A017545, A126286, A337610, A337611, A337612.

A017547 a(n) = (12n + 2)^3.

Original entry on oeis.org

8, 2744, 17576, 54872, 125000, 238328, 405224, 636056, 941192, 1331000, 1815848, 2406104, 3112136, 3944312, 4913000, 6028568, 7301384, 8741816, 10360232, 12167000, 14172488, 16387064, 18821096, 21484952, 24389000, 27543608

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

Table[(12*n+2)^3,{n,0,5!}] (* Vladimir Joseph Stephan Orlovsky, Mar 17 2010 *)
CoefficientList[Series[(8 + 2712 x + 6648 x^2 + 1000 x^3) / (1-x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Dec 28 2014 *)

a(n) = A017545(n)^3. - Michel Marcus, Dec 28 2014

G.f.: (8 + 2712*x + 6648*x^2 + 1000*x^3)/(1-x)^4. - Vincenzo Librandi, Dec 28 2014

More terms from Vladimir Joseph Stephan Orlovsky, Mar 17 2010

cp's OEIS Frontend

A139267 Twice octagonal numbers: 2*n*(3*n-2).

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A017581 a(n) = 12*n + 5.

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A072065 Define a "piece" to consist of 3 mutually touching pennies welded together to form a triangle; sequence gives side lengths of triangles that can be made from such pieces.

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A332511 Numbers k such that phi(k) == 2 (mod 12), where phi is the Euler totient function (A000010).

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Magma

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A063667 Number of solutions of phi(x) = 12n + 2.

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Maple

PARI

PARI

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A337609 Positive integers m such that A126286^k(m) = m for some positive integer k.

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A017547 a(n) = (12n + 2)^3.

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A139267 Twice octagonal numbers: 2n(3*n-2).