A092260 -id:A092260 - cp's OEIS frontend

A016945 a(n) = 6*n+3.

3, 9, 15, 21, 27, 33, 39, 45, 51, 57, 63, 69, 75, 81, 87, 93, 99, 105, 111, 117, 123, 129, 135, 141, 147, 153, 159, 165, 171, 177, 183, 189, 195, 201, 207, 213, 219, 225, 231, 237, 243, 249, 255, 261, 267, 273, 279, 285, 291, 297, 303, 309, 315, 321, 327

Offset: 0

N. J. A. Sloane

Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0(37).
Continued fraction expansion of tanh(1/3).
If a 2-set Y and a 3-set Z are disjoint subsets of an n-set X then a(n-4) is the number of 3-subsets of X intersecting both Y and Z. - Milan Janjic, Sep 08 2007
Leaves of the Odd Collatz-Tree: a(n) has no odd predecessors in all '3x+1' trajectories where it occurs: A139391(2*k+1) <> a(n) for all k; A082286(n)=A006370(a(n)). - Reinhard Zumkeller, Apr 17 2008
Let random variable X have a uniform distribution on the interval [0,c] where c is a positive constant. Then, for positive integer n, the coefficient of determination between X and X^n is (6n+3)/(n+2)^2, that is, A016945(n)/A000290(n+2). Note that the result is independent of c. For the derivation of this result, see the link in the Links section below. - Dennis P. Walsh, Aug 20 2013
Positions of 3 in A020639. - Zak Seidov, Apr 29 2015
a(n+2) gives the sum of 6 consecutive terms of A004442 starting with A004442(n). - Wesley Ivan Hurt, Apr 08 2016
Numbers k such that Fibonacci(k) mod 4 = 2. - Bruno Berselli, Oct 17 2017
Also numbers k such that t^k == -1 (mod 7), where t is a member of A047389. - Bruno Berselli, Dec 28 2017

Friedrich L. Bauer, Der (ungerade) Collatz-Baum, Informatik Spektrum 31 (Springer, April 2008), pp. 379-384.
Milan Janjic, Two Enumerative Functions.
Tanya Khovanova, Recursive Sequences.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_0(N)).
William A. Stein, The modular forms database.
Dennis P. Walsh, The correlation for a power curve on nonnegative support.
Eric Weisstein's World of Mathematics, Collatz Problem.
Index entries for sequences related to 3x+1 (or Collatz) problem.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Third row of A092260.
Cf. A004442, A008588, A010503, A016921, A016933, A016957, A016969, A019679, A115754.
Subsequence of A061641; complement of A047263; bisection of A047241.
Cf. A000225. - Loren Pearson, Jul 02 2009
Cf. A020639. - Zak Seidov, Apr 29 2015
Odd numbers in A355200.

List([0..60], n-> 3*(1+2*n)); # G. C. Greubel, Sep 18 2019

a016945 = (+ 3) . (* 6)
a016945_list = [3, 9 ..]
-- Wesley Ivan Hurt, Sep 29 2013

[6*n+3 : n in [0..60]]; // Wesley Ivan Hurt, Sep 29 2013

seq(6*n+3, n=0..60); # Dennis P. Walsh, Aug 20 2013
A016945:=n->6*n+3; # Wesley Ivan Hurt, Sep 29 2013

Range[3, 350, 6] (* Vladimir Joseph Stephan Orlovsky, May 26 2011 *)
Table[6n+3, {n, 0, 60}] (* Wesley Ivan Hurt, Sep 29 2013 *)
LinearRecurrence[{2, -1}, {3, 9}, 55] (* Ray Chandler, Jul 17 2015 *)
CoefficientList[Series[3(1+x)/(1-x)^2, {x, 0, 60}], x] (* Robert G. Wilson v, Dec 14 2016 *)

makelist(6*n+3, n, 0, 60); /* Wesley Ivan Hurt, Sep 29 2013 */

{a(n) = 6*n + 3} \\ Wesley Ivan Hurt, Sep 29 2013

x='x+O('x^60); Vec(3*(1+x)/(1-x)^2) \\ Altug Alkan, Apr 08 2016

[3*(1+2*n) for n in (0..60)] # G. C. Greubel, Sep 18 2019

a(n) = 3*(2*n + 1) = 3*A005408(n), odd multiples of 3.
A008615(a(n)) = n. - Reinhard Zumkeller, Feb 27 2008
A157176(a(n)) = A103333(n+1). - Reinhard Zumkeller, Feb 24 2009
a(n) = 12*n - a(n-1) for n>0, a(0)=3. - Vincenzo Librandi, Nov 20 2010
G.f.: 3*(1+x)/(1-x)^2. - Mario C. Enriquez, Dec 14 2016
E.g.f.: 3*(1 + 2*x)*exp(x). - G. C. Greubel, Sep 18 2019
Sum_{n>=0} (-1)^n/a(n) = Pi/12 (A019679). - Amiram Eldar, Dec 10 2021
From Amiram Eldar, Nov 22 2024: (Start)
Product_{n>=0} (1 - (-1)^n/a(n)) = sqrt(2)/2 (A010503).
Product_{n>=0} (1 + (-1)^n/a(n)) = sqrt(3/2) (A115754). (End)
a(n) = (n+2)^2 - (n-1)^2. - Alexander Yutkin, Mar 15 2025

A203571 Period length 10: [0, 1, 2, 3, 4, 0, 4, 3, 2, 1] repeated.

Original entry on oeis.org

0, 1, 2, 3, 4, 0, 4, 3, 2, 1, 0, 1, 2, 3, 4, 0, 4, 3, 2, 1, 0, 1, 2, 3, 4, 0, 4, 3, 2, 1, 0, 1, 2, 3, 4, 0, 4, 3, 2, 1, 0, 1, 2, 3, 4, 0, 4, 3, 2, 1, 0, 1, 2, 3, 4, 0, 4, 3, 2, 1, 0, 1, 2, 3, 4, 0, 4, 3, 2, 1, 0, 1, 2, 3, 4, 0, 4, 3, 2, 1, 0, 1, 2, 3, 4, 0, 4, 3, 2, 1, 0, 1, 2, 3, 4, 0, 4, 3, 2, 1, 0

Offset: 0

Wolfdieter Lang, Jan 11 2012

This sequence can be continued periodically for negative values of n.
This is the fifth sequence of a k-family of sequences P_k, k>=1, which starts with A000007(n+1), n >= 0 (the 0-sequence), A000035, A193680, A193682, for k = 1, ..., 4, respectively.
In general, the sequence P_k, k >= 1 (periodically continued for negative values of n), is used to define the k equivalence classes [0], [1], ..., [k-1], with [j] := {n integer| P_k(n) = j}. Two integers are equivalent if and only if they are mapped by P_k to the same value. For P_5, P_6 and P_7 see the arrays (not the triangles) A090298, A092260 and A113807, respectively. In each of these cases the class [k] should be replaced by the class [0], and also negative n-values are allowed. Multiplication can be done class-wise. E.g., k = 5: P_5(n) = a(n), 7*12 == 3*2 = 6 == 4; a(7*12) = a(a(7)*a(12)) = a(3*2) = 4. This kind of multiplication could be called multiplication Modd n, in order to distinguish it from multiplication mod n. Addition cannot be done class-wise. E.g., k = 5: 7 + 12 = 19 == 1 is not equivalent to 3 + 2 = 5 == 0; a(7+12) = 1 is not equal to a(a(7) + a(12)) = a(3+2) = 0.
Periodic sequences of this type can be also calculated by a(n) = c + floor(q/(p^m-1)*p^n) mod p, where c is a constant, q is the number representing the periodic digit pattern and m is the period length. c, p and q can be calculated as follows: Let D be the array representing the number pattern to be repeated, m = size of D, max = maximum value of elements in D, min = minimum value of elements in D. Then c := min, p := max - min + 1 and q := p^m * Sum_{i=0..(m-1)} (D(i) - min)/p^i. Example: D = (0, 1, 2, 3, 4, 0, 4, 3, 2, 1), c = 0, m = 10, p = 5 and q = 3034180 for this sequence. - Hieronymus Fischer, Jan 04 2013 [Corrected by Rémi Guillaume, Aug 28 2024]
For periodic sequences with terms < 10 one can use the well-known fact that ab..z/99..9 = 0.ab..zab..zab..z... (infinite periodic decimal fraction), this leads to one of the given formulas. For the general case it is sufficient to shift the terms to nonnegative values and to switch to a sufficiently large basis instead of 10 (there are infinitely many choices). - M. F. Hasler, Jan 13 2013

			a(12) = 12 mod 5 = 2 since 12\5 = floor(12/5) = 2 is even; the sign is +1.
a(7) = -7 mod 5 = 3 since 7\5 = floor(7/5) = 1 is odd; the sign is -1.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,1).

Cf. A000007, A000035, A090298, A092260, A113807, A193680, A193682.

A203571[n_] := {0, 1, 2, 3, 4, 0, 4, 3, 2, 1}[[Mod[n, 10] + 1]]; Table[A203571[n], {n, 0, 59}] (* Jean-François Alcover, Jul 05 2013 *)
PadRight[{},120,{0,1,2,3,4,0,4,3,2,1}] (* Harvey P. Dale, May 21 2021 *)

A203571(n)=[0,1,2,3,4,0,4,3,2,1][n%10+1]  \\  M. F. Hasler, Jan 13 2013

A203571(n)=151709*5^(n%10+1)\2441406%5  \\ M. F. Hasler, Jan 13 2013

a(n) = n mod 5 if (-1)^floor(n/5) = +1 else -n mod 5, n >= 0. (-1)^floor(n/5) is the sign corresponding to the parity of the quotient floor(n/5). This quotient is sometimes denoted by n\5.
O.g.f.: x*(1+2*x+3*x^2+4*x^3+4*x^5+3*x^6+2*x^7+x^8)/(1-x^10) = -x*(1 +2*x +3*x^2 +4*x^3 +4*x^5 +3*x^6 +2*x^7 +x^8) / ( (x-1) *(1+x) *(x^4+x^3+x^2+x+1) *(x^4-x^3+x^2-x+1) ).
a(n) = (2/5)*cos(Pi*n) - cos(4*Pi*n/5) - (1/5)*cos(3*Pi*n/5) + (2/5)*5^(1/2)*cos(3*Pi*n/5) - cos(2*Pi*n/5) - (1/5)*cos(Pi*n/5) - (2/5)*5^(1/2)*cos(Pi*n/5) + 2. - Leonid Bedratyuk, May 13 2012
a(n) = floor(123404321/9999999999*10^(n+1)) mod 10. - Hieronymus Fischer, Jan 04 2013
a(n) = floor(151709/2441406*5^(n+1)) mod 5. - Hieronymus Fischer, Jan 04 2013
a(n) = (5-abs(n-(10*ceiling(n/10)-5)))*(ceiling((n+5)/10)-floor((n+5)/10)). - Wesley Ivan Hurt, Mar 26 2014 [corrected by Jason Yuen, Feb 17 2025]
a(n+10) = a(n) for n in Z; a(-n) = a(n) for n in Z. - Rémi Guillaume, Aug 28 2024

A091999 Numbers that are congruent to {2, 10} mod 12.

Original entry on oeis.org

2, 10, 14, 22, 26, 34, 38, 46, 50, 58, 62, 70, 74, 82, 86, 94, 98, 106, 110, 118, 122, 130, 134, 142, 146, 154, 158, 166, 170, 178, 182, 190, 194, 202, 206, 214, 218, 226, 230, 238, 242, 250, 254, 262, 266, 274, 278, 286, 290, 298, 302, 310, 314, 322, 326, 334

Offset: 1

Ray Chandler, Feb 21 2004

Numbers divisible by 2 but not by 3 or 4. - Robert Israel, Apr 24 2015
For n > 1, a(n) is representable as a sum of four but no fewer consecutive nonnegative integers, i.e., 10 = 1 + 2 + 3 + 4, 14 = 2 + 3 + 4 + 5, 22 = 4 + 5 + 6 + 7, etc. (see A138591). - Martin Renner, Mar 14 2016
Essentially the same as A063221. - Omar E. Pol, Aug 16 2023

David Lovler, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Second row of A092260.
Cf. A109761 (subsequence).
Cf. A002193, A007310, A101263, A138591, A186424.
Cf. A017545, A017641, A063221.

a091999 n = a091999_list !! (n-1)
a091999_list = 2 : 10 : map (+ 12) a091999_list
-- Reinhard Zumkeller, Jan 21 2013

[6*n-3+(-1)^n : n in [1..100]]; // Wesley Ivan Hurt, Apr 23 2015

A091999:=n->6*n-3+(-1)^n: seq(A091999(n), n=1..100); # Wesley Ivan Hurt, Apr 23 2015

Flatten[#+{2,10}&/@(12*Range[0,30])] (* or *) LinearRecurrence[{1,1,-1},{2,10,14},60] (* Harvey P. Dale, Jun 24 2013 *)

a(n) = 6*n - 3 + (-1)^n \\ David Lovler, Jul 16 2022

a(n) = 2*A007310(n).
a(n) = A186424(n) - A186424(n-2), for n > 1.
a(n) = 12*(n-1) - a(n-1), with a(1)=2. - Vincenzo Librandi, Nov 16 2010
G.f.: 2*x*(1+4*x+x^2) / ( (1+x)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011
a(n) = a(n-1) + a(n-2) - a(n-3); a(1)=2, a(2)=10, a(3)=14. - Harvey P. Dale, Jun 24 2013
a(n) = 6*n - 3 + (-1)^n. - Wesley Ivan Hurt, Apr 23 2015
E.g.f.: 2 + (6*x - 2)*cosh(x) + 2*(3*x - 2)*sinh(x). - Stefano Spezia, May 09 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(4*sqrt(3)). - Amiram Eldar, Dec 13 2021
E.g.f.: 2 + (6*x - 3)*exp(x) + exp(-x). - David Lovler, Aug 08 2022
a(n) = A063221(n), n > 1. - Omar E. Pol, Aug 15 2023
From Amiram Eldar, Nov 24 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = sqrt(2) (A002193).
Product_{n>=1} (1 + (-1)^n/a(n)) = 2*sin(Pi/12) (A101263). (End)

A091998 Numbers that are congruent to {1, 11} mod 12.

Original entry on oeis.org

1, 11, 13, 23, 25, 35, 37, 47, 49, 59, 61, 71, 73, 83, 85, 95, 97, 107, 109, 119, 121, 131, 133, 143, 145, 155, 157, 167, 169, 179, 181, 191, 193, 203, 205, 215, 217, 227, 229, 239, 241, 251, 253, 263, 265, 275, 277, 287, 289, 299, 301, 311, 313, 323, 325, 335

Offset: 1

Ray Chandler, Feb 21 2004

Cf. property described by Gary Detlefs in A113801: more generally, these numbers are of the form (2*h*n + (h-4)*(-1)^n-h)/4 (h and n in A000027), then ((2*h*n + (h-4)*(-1)^n - h)/4)^2 - 1 == 0 (mod h); in our case, a(n)^2 - 1 == 0 (mod 12). Also a(n)^2 - 1 == 0 (mod 24).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

First row of A092260.
Cf. A175885 (n == 1 or 10 (mod 11)), A175886 (n == 1 or 12 (mod 13)).
Cf. A097933 (primes), A195143 (partial sums).
Cf. A000217, A005408, A047209, A007310, A047336, A047522, A056020, A090771, A113801, A175887, A188887.

a091998 n = a091998_list !! (n-1)
a091998_list = 1 : 11 : map (+ 12) a091998_list
-- Reinhard Zumkeller, Jan 07 2012

[ n: n in [1..350] | n mod 12 eq 1 or n mod 12 eq 11 ];

LinearRecurrence[{1,1,-1},{1,11,13},100] (* Harvey P. Dale, Jul 26 2017 *)

is(n)=n=n%12;n==11 || n==1 \\ Charles R Greathouse IV, Jul 02 2013

a(n) = 12*n - a(n-1) - 12 (with a(1)=1). - Vincenzo Librandi, Nov 16 2010
a(n) = 6*n + 2*(-1)^n - 3.
G.f.: x*(1+10*x+x^2)/((1+x)*(1-x)^2).
a(n) - a(n-1) - a(n-2) + a(n-3) = 0 for n > 3.
a(n) = a(n-2) + 12 for n > 2.
a(n) = 12*A000217(n-1) + 1 - 2*Sum_{i=1..n-1} a(i) for n > 1.
Sum_{n>=1} (-1)^(n+1)/a(n) = (2 + sqrt(3))*Pi/12. - Amiram Eldar, Dec 04 2021
E.g.f.: 1 + (6*x - 3)*exp(x) + 2*exp(-x). - David Lovler, Sep 04 2022
From Amiram Eldar, Nov 23 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = sqrt(2 + sqrt(3)) = 2*cos(Pi/12) (A188887).
Product_{n>=2} (1 + (-1)^n/a(n)) = (Pi/3)*cos(Pi/12). (End)

Formulae and comment added by Bruno Berselli, Nov 17 2010 - Nov 18 2010

A113807 Permutation of natural numbers generated by 7-rowed array shown below.

Original entry on oeis.org

1, 13, 2, 15, 12, 3, 27, 16, 11, 4, 29, 26, 17, 10, 5, 41, 30, 25, 18, 9, 6, 43, 40, 31, 24, 19, 8, 7, 55, 44, 39, 32, 23, 20, 14, 57, 54, 45, 38, 33, 22, 21, 69, 58, 53, 46, 37, 34, 28, 71, 68, 59, 52, 47, 36, 35, 83, 72, 67, 60, 51, 48, 42, 85, 82, 73, 66, 61, 50, 49

Offset: 1

Giovanni Teofilatto, Jan 22 2006

nonn

For such arrays A_k see the k-family of 2k-periodic sequences P_k defined in a comment on A203571. There the k residue classes mod n have been defined. The present array is A_7 if the last class, starting with 7, is taken as first class [0] after adding a 0 in front. Then one obtains a permutation of the nonnegative integers. However, each complete residue class also includes its negative members. - Wolfdieter Lang, Feb 02 2012

			1 13 15 27 29 41 43 55 57 69 71 83 85 ... (A113801)
2 12 16 26 30 40 44 54 58 68 72 82 86 ... (A113802)
3 11 17 25 31 39 45 53 59 67 73 81 87 ... (A113803)
4 10 18 24 32 38 46 52 60 66 74 80 88 ... (A113804)
5  9 19 23 33 37 47 51 61 65 75 70 89 ... (A113805)
6  8 20 22 34 36 48 50 62 64 76 78 90 ... (A113806)
7 14 21 28 35 42 49 56 63 70 77 84 91 ... (A008589)

Cf. A088520, A090243, A090298, A092260, A204453.

A-numbers added for array rows by Wolfdieter Lang, Dec 15 2011

More terms from Ray Chandler, Dec 15 2011

A090298 Permutation of natural numbers generated by 5-row array shown below.

Original entry on oeis.org

1, 9, 2, 11, 8, 3, 19, 12, 7, 4, 21, 18, 13, 6, 5, 29, 22, 17, 14, 10, 31, 28, 23, 16, 15, 39, 32, 27, 24, 20, 41, 38, 33, 26, 25, 49, 42, 37, 34, 30, 51, 48, 43, 36, 35, 59, 52, 47, 44, 40, 61, 58, 53, 46, 45, 69, 62, 57, 54, 50, 71, 68, 63, 56, 55, 79, 72, 67, 64, 60, 81, 78

Offset: 1

Giovanni Teofilatto, Jan 25 2004

9 11 19 21 29 31 39... (A090771)
8 12 18 22 28 32 38... (A090772)
7 13 17 23 27 33 37... (A063226)
6 14 16 24 26 34 36... (A090773)
10 15 20 25 30 35 40... (A008587, excluding initial term)
-----------------------------------------------------------
For such arrays A_k, here A_5, see a W. Lang comment on A113807, the A_7 case. However, in order to obtain A_5 one should take the last row as the first one after adding a 0 in front (thus getting a permutation of the nonnegative integers). - Wolfdieter Lang, Feb 02 2012

Cf. A088520, A090243, A092260, A113807.

More terms from Ray Chandler, Feb 01 2004

A092259 Numbers that are congruent to {4, 8} mod 12.

Original entry on oeis.org

4, 8, 16, 20, 28, 32, 40, 44, 52, 56, 64, 68, 76, 80, 88, 92, 100, 104, 112, 116, 124, 128, 136, 140, 148, 152, 160, 164, 172, 176, 184, 188, 196, 200, 208, 212, 220, 224, 232, 236, 244, 248, 256, 260, 268, 272, 280, 284, 292, 296, 304, 308, 316, 320, 328, 332

Offset: 1

Giovanni Teofilatto, Feb 19 2004

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

Cf. A001651, A017617, A017569, A069587, A092899, A145439, A157697.

Fourth row of A092260.

[n : n in [0..400] | n mod 12 in [4, 8]]; // Wesley Ivan Hurt, May 21 2016

A092259:=n->6*n-3-(-1)^n: seq(A092259(n), n=1..100); # Wesley Ivan Hurt, May 21 2016

Table[6n-3-(-1)^n, {n, 80}] (* Wesley Ivan Hurt, May 21 2016 *)
LinearRecurrence[{1,1,-1},{4,8,16},60] (* Harvey P. Dale, Oct 07 2021 *)

G.f.: 4*x*(1+x+x^2) / ( (1+x)*(x-1)^2 ).
a(n) = 4 * A001651(n).
Iff phi(n) = phi(3n/2), then n is in A069587. - Labos Elemer, Feb 25 2004
a(n) = 12*(n-1)-a(n-1) (with a(1)=4). - Vincenzo Librandi, Nov 16 2010
From Wesley Ivan Hurt, May 21 2016: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3) for n>3.
a(n) = 6n - 3 - (-1)^n.
a(2n) = A017617(n-1) for n>1, a(2n-1) = A017569(n-1) for n>1.
a(n) = -a(1-n), a(n) = A092899(n) + 1 for n>0. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi*sqrt(3)/36. - Amiram Eldar, Dec 30 2021
From Amiram Eldar, Nov 24 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = 1/sqrt(2) + 1/sqrt(6) (A145439).
Product_{n>=1} (1 + (-1)^n/a(n)) = sqrt(2/3) (A157697). (End)

Edited and extended by Ray Chandler, Feb 21 2004

A088520 Permutation of natural numbers generated by 3-rowed array shown below.

Original entry on oeis.org

1, 5, 2, 7, 4, 3, 11, 8, 6, 13, 10, 9, 17, 14, 12, 19, 16, 15, 23, 20, 18, 25, 22, 21, 29, 26, 24, 31, 28, 27, 35, 32, 30, 37, 34, 33, 41, 38, 36, 43, 40, 39, 47, 44, 42, 49, 46, 45, 53, 50, 48, 55, 52, 51, 59, 56, 54, 61, 58, 57, 65, 62, 60, 67, 64, 63, 71, 68, 66, 73, 70, 69

Offset: 1

Giovanni Teofilatto, Nov 14 2003

5 7 11 13 17 19 23 25 29 31 35 37 41 43...
4 8 10 14 16 20 22 26 28 32 34 38 40 44...
6 9 12 15 18 21 24 27 30 33 36 39 42 45...

M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Bologna 1988
Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta,UTET, CittaStudiEdizioni, Milano 1997

Cf. A090243, A090298, A092260, A113807.

Corrected and extended by Ray Chandler, Nov 16 2003

A090243 Permutation of natural numbers generated by 4-rowed array shown below.

Original entry on oeis.org

1, 7, 2, 11, 6, 3, 13, 10, 5, 4, 19, 14, 9, 8, 23, 18, 15, 12, 25, 22, 17, 16, 31, 26, 21, 20, 35, 30, 27, 24, 37, 34, 29, 28, 43, 38, 33, 32, 47, 42, 39, 36, 49, 46, 41, 40, 55, 50, 45, 44, 59, 54, 51, 48, 61, 58, 53, 52, 67, 62, 57, 56, 71, 66, 63, 60, 73, 70, 65, 64, 79, 74

Offset: 1

Giovanni Teofilatto, Jan 24 2004

7 11 13 19 23 25 31 35 37 43 47 ...
6 10 14 18 22 26 30 34 38 42 46 ...
5  9 15 17 21 27 29 33 39 41 45 ...
8 12 16 20 24 28 32 36 40 44 48 ...

M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Bologna 1988
Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta,UTET, CittaStudiEdizioni, Milano 1997

Cf. A088520, A090298, A092260, A113807.

Corrected and extended by Ray Chandler, Jan 30 2004

A092242 Numbers that are congruent to {5, 7} (mod 12).

Original entry on oeis.org

5, 7, 17, 19, 29, 31, 41, 43, 53, 55, 65, 67, 77, 79, 89, 91, 101, 103, 113, 115, 125, 127, 137, 139, 149, 151, 161, 163, 173, 175, 185, 187, 197, 199, 209, 211, 221, 223, 233, 235, 245, 247, 257, 259, 269, 271, 281, 283, 293, 295, 305, 307, 317, 319, 329, 331

Offset: 1

Giovanni Teofilatto, Feb 19 2004

L. B. W. Jolley, Summation of Series, Dover Publications, 1961, p. 64.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Fifth row of A092260.

Cf. A010527, A120683.

Select[Range[331], MemberQ[{5, 7}, Mod[#, 12]] &] (* Amiram Eldar, Dec 04 2021 *)

1/5^2 + 1/7^2 + 1/17^2 + 1/19^2 + 1/29^2 + 1/31^2 + ... = Pi^2*(2 - sqrt(3))/36 = 0.073459792... [Jolley] - Gary W. Adamson, Dec 20 2006
a(n) = 12*n - a(n-1) - 12 (with a(1)=5). - Vincenzo Librandi, Nov 16 2010
From R. J. Mathar, Oct 08 2011: (Start)
a(n) = 6*n - 3 - 2*(-1)^n.
G.f.: x*(5+2*x+5*x^2) / ( (1+x)*(x-1)^2 ). (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (2 - sqrt(3))*Pi/12. - Amiram Eldar, Dec 04 2021
From Amiram Eldar, Nov 24 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = sec(Pi/12) (A120683).
Product_{n>=1} (1 + (-1)^n/a(n)) = (sqrt(3)/2)*sec(Pi/12) (= A010527 * A120683). (End)

Edited and extended by Ray Chandler, Feb 21 2004

cp's OEIS Frontend

A016945 a(n) = 6*n+3.

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A203571 Period length 10: [0, 1, 2, 3, 4, 0, 4, 3, 2, 1] repeated.

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A091999 Numbers that are congruent to {2, 10} mod 12.

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A091998 Numbers that are congruent to {1, 11} mod 12.

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A113807 Permutation of natural numbers generated by 7-rowed array shown below.

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A090298 Permutation of natural numbers generated by 5-row array shown below.

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A092259 Numbers that are congruent to {4, 8} mod 12.

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