A186421 -id:A186421 - cp's OEIS frontend

A186422 First differences of A186421.

1, 1, -1, 3, -1, 3, -3, 5, -3, 5, -5, 7, -5, 7, -7, 9, -7, 9, -9, 11, -9, 11, -11, 13, -11, 13, -13, 15, -13, 15, -15, 17, -15, 17, -17, 19, -17, 19, -19, 21, -19, 21, -21, 23, -21, 23, -23, 25, -23, 25, -25, 27, -25, 27, -27, 29, -27, 29, -29, 31, -29, 31, -31, 33, -31, 33, -33, 35, -33, 35, -35, 37, -35, 37, -37, 39, -37, 39, -39, 41, -39, 41, -41, 43

Offset: 0

Reinhard Zumkeller, Feb 21 2011

a(n) = A186421(n+1) - A186421(n);
a(2*n) = - A109613(n-1) for n>0; a(2*n+1) = A109613(n);
a(3*k) = A047270(floor((k+1)/2)) * (-1)^(k+1);
a(3*k+1) = A007310(floor((k+2)/2)) * (-1)^k;
a(3*k+2) = A047241(floor((k+3)/2)) * (-1)^(k+1).

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

Cf. A007310, A047241, A047270, A109613, A186421.

a186422 n = a186422_list !! n
a186422_list = zipWith (-) (tail a186421_list) a186421_list

/* By definition: */
A186421:=func;
[A186421(n+1)-A186421(n): n in [0..90]]; // Bruno Berselli, Mar 04 2013

Differences@ CoefficientList[Series[x (1 + 2 x + 2 x^3 + x^4)/((1 + x^2) (x - 1)^2 (1 + x)^2), {x, 0, 84}], x] (* Michael De Vlieger, Oct 02 2017 *)

makelist(-((2*n+1)*(-1)^n-2*%i^(n*(n+1))-3)/4,n,0,83); /* Bruno Berselli, Mar 04 2013 */

G.f.: -(x^4+2*x^3+2*x+1) / ((x-1)*(x+1)^2*(x^2+1)). - Colin Barker, Mar 04 2013
a(n) = -((2*n+1)*(-1)^n-2*i^(n*(n+1))-3)/4, where i=sqrt(-1). [Bruno Berselli, Mar 04 2013]
a(n) = cos((n-1)*Pi)*(2*n+1-2*cos(n*Pi/2)-3*cos(n*Pi)-2*sin(n*Pi/2))/4. - Wesley Ivan Hurt, Oct 02 2017
E.g.f.: (cos(x) + (1 + x)*cosh(x) - sin(x) - (x - 2)*sinh(x))/2. - Stefano Spezia, May 09 2021

A186423 Partial sums of A186421.

Original entry on oeis.org

0, 1, 3, 4, 8, 11, 17, 20, 28, 33, 43, 48, 60, 67, 81, 88, 104, 113, 131, 140, 160, 171, 193, 204, 228, 241, 267, 280, 308, 323, 353, 368, 400, 417, 451, 468, 504, 523, 561, 580, 620, 641, 683, 704, 748, 771, 817, 840, 888, 913, 963, 988, 1040, 1067, 1121, 1148, 1204, 1233, 1291, 1320

Offset: 0

Reinhard Zumkeller, Feb 21 2011

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

A062717 is the subsequence of even terms.

A186424 is the subsequence of odd terms.

List([0..65], n-> (6*n^2 +6*n +3 +(-1)^n*(2*n+1) -4*(-1)^Binomial(n+1, 2))/16); # G. C. Greubel, Oct 09 2019

a186423 n = a186423_list !! n
a186423_list = scanl1 (+) a186421_list

[(6*n^2 +6*n +3 +(-1)^n*(2*n+1) -4*(-1)^Binomial(n+1, 2))/16: n in [0..65]]; // G. C. Greubel, Oct 09 2019

A087960 := proc(n) op((n mod 4)+1,[1,-1,-1,1]) ; end proc:
A186423 := proc(n) 3*n*(n+1)/8 +3/16 +(-1)^n*(2*n+1)/16 -A087960(n)/4 ; end proc: # R. J. Mathar, Feb 28 2011

CoefficientList[Series[x(1+2x+2x^3+x^4)/((1-x)^3(1+x)^2(1+x^2)),{x, 0, 65}],x]  (* Harvey P. Dale, Mar 13 2011 *)
Table[(6*n^2 +6*n +3 +(-1)^n*(2*n+1) -4*(-1)^Binomial[n+1, 2])/16, {n, 0, 65}] (* G. C. Greubel, Oct 09 2019 *)

vector(66, n, my(m=n-1); (6*m^2 +6*m +3 +(-1)^m*(2*m+1) -4*(-1)^binomial(m+1, 2))/16) \\ G. C. Greubel, Oct 09 2019

def A186423(n): return (6*n*(n+1)+3+(-2*n-1 if n&1 else 2*n+1)+(4 if n+1&2 else -4))>>4 # Chai Wah Wu, Jan 31 2023

[(6*n^2 +6*n +3 +(-1)^n*(2*n+1) -4*(-1)^binomial(n+1, 2))/16 for n in (0..65)] # G. C. Greubel, Oct 09 2019

From R. J. Mathar, Feb 28 2011: (Start)
G.f.: x*(1 + 2*x + 2*x^3 + x^4)/( (1+x^2)*(1+x)^2*(1-x)^3 ).
a(n) = (6*n*(n+1) + 3 + (-1)^n*(2*n+1) - 4*A087960(n))/16. (End)
E.g.f.: ((2 + 5*x + 3*x^2)*cosh(x) + (1 + 7*x + 3*x^2)*sinh(x) + 2*sin(x) - 2*cos(x))/8. - G. C. Greubel, Oct 09 2019

More terms added by G. C. Greubel, Oct 09 2019

A109613 Odd numbers repeated.

Original entry on oeis.org

1, 1, 3, 3, 5, 5, 7, 7, 9, 9, 11, 11, 13, 13, 15, 15, 17, 17, 19, 19, 21, 21, 23, 23, 25, 25, 27, 27, 29, 29, 31, 31, 33, 33, 35, 35, 37, 37, 39, 39, 41, 41, 43, 43, 45, 45, 47, 47, 49, 49, 51, 51, 53, 53, 55, 55, 57, 57, 59, 59, 61, 61, 63, 63, 65, 65, 67, 67, 69, 69, 71, 71, 73

Offset: 0

Reinhard Zumkeller, Aug 01 2005

The number of rounds in a round-robin tournament with n competitors. - A. Timothy Royappa, Aug 13 2011
Diagonal sums of number triangle A113126. - Paul Barry, Oct 14 2005
When partitioning a convex n-gon by all the diagonals, the maximum number of sides in resulting polygons is 2*floor(n/2)+1 = a(n-1) (from Moscow Olympiad problem 1950). - Tanya Khovanova, Apr 06 2008
The inverse values of the coefficients in the series expansion of f(x) = (1/2)*(1+x)*log((1+x)/(1-x)) lead to this sequence; cf. A098557. - Johannes W. Meijer, Nov 12 2009
From Reinhard Zumkeller, Dec 05 2009: (Start)
First differences: A010673; partial sums: A000982;
A059329(n) = Sum_{k = 0..n} a(k)*a(n-k);
A167875(n) = Sum_{k = 0..n} a(k)*A005408(n-k);
A171218(n) = Sum_{k = 0..n} a(k)*A005843(n-k);
A008794(n+2) = Sum_{k = 0..n} a(k)*A059841(n-k). (End)
Dimension of the space of weight 2n+4 cusp forms for Gamma_0(5). - Michael Somos, May 29 2013
For n > 4: a(n) = A230584(n) - A230584(n-2). - Reinhard Zumkeller, Feb 10 2015
The arithmetic function v+-(n,2) as defined in A290988. - Robert Price, Aug 22 2017
For n > 0, also the chromatic number of the (n+1)-triangular (Johnson) graph. - Eric W. Weisstein, Nov 17 2017
a(n-1), for n >= 1, is also the upper bound a_{up}(b), where b = 2*n + 1, in the first (top) row of the complete coach system Sigma(b) of Hilton and Pedersen [H-P]. All odd numbers <= a_{up}(b) of the smallest positive restricted residue system of b appear once in the first rows of the c(2*n+1) = A135303(n) coaches. If b is an odd prime a_{up}(b) is the maximum. See a comment in the proof of the quasi-order theorem of H-P, on page 263 ["Furthermore, every possible a_i < b/2 ..."]. For an example see below. - Wolfdieter Lang, Feb 19 2020
Satisfies the nested recurrence a(n) = a(a(n-2)) + 2*a(n-a(n-1)) with a(0) = a(1) = 1. Cf. A004001. - Peter Bala, Aug 30 2022
The binomial transform is 1, 2, 6, 16, 40, 96, 224, 512, 1152, 2560,.. (see A057711). - R. J. Mathar, Feb 25 2023

			G.f. = 1 + x + 3*x^2 + 3*x^3 + 5*x^4 + 5*x^5 + 7*x^6 + 7*x^7 + 9*x^8 + 9*x^9 + ...
Complete coach system for (a composite) b = 2*n + 1 = 33: Sigma(33) ={[1; 5], [5, 7, 13; 2, 1, 2]} (the first two rows are here 1 and 5, 7, 13), a_{up}(33) = a(15) = 15. But 15 is not in the reduced residue system modulo 33, so the maximal (odd) a number is 13. For the prime b = 31, a_{up}(31) = a(14) = 15 appears as maximum of the first rows. - _Wolfdieter Lang_, Feb 19 2020

Peter Hilton and Jean Pedersen, A Mathematical Tapestry: Demonstrating the Beautiful Unity of Mathematics, Cambridge University Press, 2010, 3rd printing 2012, pp. (260-281).

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Chromatic Number
Eric Weisstein's World of Mathematics, Johnson Graph
Eric Weisstein's World of Mathematics, Triangular Graph
Wikipedia, Round-robin tournament
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A000217, A063196, A110660, A186421, A186422, A211955, A230584, A290988.
Complement of A052928 with respect to the universe A004526. - Guenther Schrack, Aug 21 2018
First differences of A000982, A061925, A074148, A105343, A116940, and A179207. - Guenther Schrack, Aug 21 2018

a109613 = (+ 1) . (* 2) . (`div` 2)
a109613_list = 1 : 1 : map (+ 2) a109613_list
-- Reinhard Zumkeller, Oct 27 2012, Feb 21 2011

A109613:=n->2*floor(n/2)+1; seq(A109613(k), k=0..100); # Wesley Ivan Hurt, Oct 22 2013

Flatten@ Array[{2# - 1, 2# - 1} &, 37] (* Robert G. Wilson v, Jul 07 2012 *)
(# - Boole[EvenQ[#]] &) /@ Range[80] (* Alonso del Arte, Sep 11 2019 *)
With[{c=2*Range[0,40]+1},Riffle[c,c]] (* Harvey P. Dale, Jan 02 2020 *)

A109613(n)=n>>1<<1+1 \\ Charles R Greathouse IV, Feb 24 2011

def a(n) : return( len( CuspForms( Gamma0( 5), 2*n + 4, prec=1). basis())); # Michael Somos, May 29 2013

((1 to 49) by 2) flatMap { List.fill(2)() } // _Alonso del Arte, Sep 11 2019

a(n) = 2*floor(n/2) + 1.
a(n) = A052928(n) + 1 = 2*A004526(n) + 1.
a(n) = A028242(n) + A110654(n).
a(n) = A052938(n-2) + A084964(n-2) for n > 1. - Reinhard Zumkeller, Aug 27 2005
G.f.: (1 + x + x^2 + x^3)/(1 - x^2)^2. - Paul Barry, Oct 14 2005
a(n) = 2*a(n-2) - a(n-4), a(0) = 1, a(1) = 1, a(2) = 3, a(3) = 3. - Philippe Deléham, Nov 03 2008
a(n) = A001477(n) + A059841(n). - Philippe Deléham, Mar 31 2009
a(n) = 2*n - a(n-1), with a(0) = 1. - Vincenzo Librandi, Nov 13 2010
a(n) = R(n, -2), where R(n, x) is the n-th row polynomial of A211955. a(n) = (-1)^n + 2*Sum_{k = 1..n} (-1)^(n - k - 2)*4^(k-1)*binomial(n+k, 2*k). Cf. A084159. - Peter Bala, May 01 2012
a(n) = A182579(n+1, n). - Reinhard Zumkeller, May 06 2012
G.f.: ( 1 + x^2 ) / ( (1 + x)*(x - 1)^2 ). - R. J. Mathar, Jul 12 2016
E.g.f.: x*exp(x) + cosh(x). - Ilya Gutkovskiy, Jul 12 2016
From Guenther Schrack, Sep 10 2018: (Start)
a(-n) = -a(n-1).
a(n) = A047270(n+1) - (2*n + 2).
a(n) = A005408(A004526(n)). (End)
a(n) = A000217(n) / A004526(n+1), n > 0. - Torlach Rush, Nov 10 2023

A109043 a(n) = lcm(n,2).

Original entry on oeis.org

0, 2, 2, 6, 4, 10, 6, 14, 8, 18, 10, 22, 12, 26, 14, 30, 16, 34, 18, 38, 20, 42, 22, 46, 24, 50, 26, 54, 28, 58, 30, 62, 32, 66, 34, 70, 36, 74, 38, 78, 40, 82, 42, 86, 44, 90, 46, 94, 48, 98, 50, 102, 52, 106, 54, 110, 56, 114, 58, 118, 60, 122, 62, 126, 64, 130, 66, 134

Offset: 0

Mitch Harris, Jun 18 2005

Exponent of the dihedral group D(2n) = . - Arkadiusz Wesolowski, Sep 10 2013
Second column of table A210530. - Boris Putievskiy, Jan 29 2013
For n > 1, the basic period of A000166(k) (mod n) (Miska, 2016). - Amiram Eldar, Mar 03 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Dorin Andrica, Sorin Rădulescu, and George Cătălin Ţurcaş, The Exponent of a Group: Properties, Computations and Applications, Disc. Math. and Applications, Springer, Cham (2020), 57-108.
Piotr Miska, Arithmetic properties of the sequence of derangements, Journal of Number Theory, Vol. 163 (2016), pp. 114-145; arXiv preprint, arXiv:1508.01987 [math.NT], 2015. See p. 124 (p. 14 in the preprint).
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).
Index entries for sequences related to lcm's.

Cf. A000166, A109042, A152749 (partial sums).
Cf. A066043 (essentially the same), A000034 (=a(n)/n), A026741 (=a(n)/2).
Cf. A186421, A210530.

a109043 = (lcm 2)
a109043_list = zipWith (*) [0..] a000034_list
-- Reinhard Zumkeller, Mar 31 2012

[0, 2, 2] cat [Exponent(DihedralGroup(n)) : n in [3..65]]; // Arkadiusz Wesolowski, Sep 10 2013

LCM[Range[0,70],2] (* Harvey P. Dale, Aug 19 2012 *)

a(n)=lcm(n,2) \\ Charles R Greathouse IV, Sep 24 2015

def A109043(n): return n<<1 if n&1 else n # Chai Wah Wu, Aug 05 2024

[lcm(n,2) for n in range(0, 68)] # Zerinvary Lajos, Jun 07 2009

For n > 0: a(n) = A186421(n) + A186421(n+2).
a(n) = n*2 / gcd(n, 2).
a(n) = -(n*((-1)^n-3))/2. - Stephen Crowley, Feb 11 2007
From R. J. Mathar, Aug 20 2008: (Start)
a(n) = A066043(n), n > 1.
a(n) = 2*A026741(n).
G.f.: 2*x(1+x+x^2)/((1-x)^2*(1+x)^2). (End)
a(n) = n*A000034(n). - Paul Curtz, Mar 25 2011
E.g.f.: x*(2*cosh(x) + sinh(x)). - Stefano Spezia, May 09 2021
Sum_{k=1..n} a(k) ~ (3/4) * n^2. - Amiram Eldar, Nov 26 2022

A186424 Odd terms in A186423.

Original entry on oeis.org

1, 3, 11, 17, 33, 43, 67, 81, 113, 131, 171, 193, 241, 267, 323, 353, 417, 451, 523, 561, 641, 683, 771, 817, 913, 963, 1067, 1121, 1233, 1291, 1411, 1473, 1601, 1667, 1803, 1873, 2017, 2091, 2243, 2321, 2481, 2563, 2731, 2817, 2993, 3083, 3267, 3361, 3553, 3651

Offset: 0

Reinhard Zumkeller, Feb 21 2011

Sum of odd square and half of even square. - Vladimir Joseph Stephan Orlovsky, May 20 2011

Numbers m such that 6*m-2 is a square. - Bruno Berselli, Apr 29 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A186421, A186423, A203568, A091999, A080859, A126587, A053253.

a186424 n = a186424_list !! n
a186424_list = filter odd a186423_list

Table[If[OddQ[n],n^2+((n+1)^2)/2,(n^2)/2+(n+1)^2],{n,0,100}] (* Vladimir Joseph Stephan Orlovsky, May 20 2011 *)

def A186424(n): return (n*(3*n + 2) + 1 if n&1 else n*(3*n + 4) + 2)>>1 # Chai Wah Wu, Jan 31 2023

From R. J. Mathar, Feb 28 2011: (Start)
G.f.: ( -1-2*x-6*x^2-2*x^3-x^4 ) / ( (1+x)^2*(x-1)^3 ).
a(n) = 3*(1+2*n+2*n^2)/4 + (-1)^n*(1+2*n)/4. (End)
a(n+2) = a(n) + A091999(n+2).
Union of A080859 and A126587: a(2*n) = A080859(n) and a(2*n+1) = A126587(n+1).
From Peter Bala, Feb 13 2021: (Start)
Appears to be the sequence of exponents in the following series expansion:
Sum_{n >= 0} (-1)^n * x^n/Product_{k = 1..n} 1 - x^(2*k-1) = 1 - x - x^3 + x^11 + x^17 - x^33 - x^43 + + - - .... Cf. A053253.
More generally, for nonnegative integer N, we appear to have the identity
Product_{j = 1..N} 1/(1 + x^(2*j-1))*( P(N,x) + Sum_{n >= 1} (-1)^n * x^((2*N+1)*n-N)/Product_{k = 1..n} 1 - x^(2*k-1) ) = 1 - x - x^3 + x^11 + x^17 - x^33 - x^43 + + - - ..., where P(N,x) is a polynomial in x of degree N^2 - 1, with the first few values given empirically by
P(0,x) = 0, P(1,x) = 1, P(2,x) = 1 - x^2 + x^3, P(3,x) = 1 - x^2 + x^5 - x^7 + x^8 and P(4,x) =  1 - x^2 - x^4 + x^5 + x^8 - x^9 + x^12 - x^14 + x^15. Cf. A203568. (End)
E.g.f.: ((2 + 5*x + 3*x^2)*cosh(x) + (1 + 7*x + 3*x^2)*sinh(x))/2. - Stefano Spezia, May 08 2021
Sum_{n>=0} 1/a(n) = sqrt(2)*Pi*sinh(sqrt(2)*Pi/3)/(1+2*cosh(sqrt(2)*Pi/3)). - Amiram Eldar, May 11 2025

A212831 a(4n) = 2n, a(2n+1) = 2n+1, a(4n+2) = 2n+2.

Original entry on oeis.org

0, 1, 2, 3, 2, 5, 4, 7, 4, 9, 6, 11, 6, 13, 8, 15, 8, 17, 10, 19, 10, 21, 12, 23, 12, 25, 14, 27, 14, 29, 16, 31, 16, 33, 18, 35, 18, 37, 20, 39, 20, 41, 22, 43, 22, 45, 24, 47, 24, 49, 26, 51, 26, 53, 28, 55, 28, 57, 30, 59, 30, 61, 32, 63, 32, 65, 34, 67, 34, 69, 36, 71, 36, 73, 38, 75

Offset: 0

Paul Curtz, Aug 14 2012

First differences: (1, 1, 1, -1, 3, -1, 3, -3, 5,...) = (1, A186422).
Second differences: (0, 0, -2, 4, -4, 4, -6, 8, ...)  = (-1)^(n+1) * A201629(n).
Interleave the terms with even indices of the companion A215495 and this one to get (A215495(0), A212831(0), A215495(2), A212831(2),...) = (1, 0, 1, 2, 3, 2, 3, 4, 5, 4,...) = A106249, up to the initial term = A083219 = A083220/2.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,1,0,1,0,-1).

Cf. A214282, A129756.

[(1/4)*((1 +(-1)^n)*(1 - (-1)^Floor(n/2)) + (3 -(-1)^n)*n): n in [0..50]]; // G. C. Greubel, Apr 25 2018

a[n_] := (1/4)*((-(1 + (-1)^n))*(-1 + (-1)^Floor[n/2]) - (-3 + (-1)^n)*n ); Table[a[n], {n, 0, 84}] (* Jean-François Alcover, Sep 18 2012 *)
LinearRecurrence[{0,1,0,1,0,-1},{0,1,2,3,2,5},80] (* Harvey P. Dale, May 29 2016 *)

A212831(n)=if(bittest(n,0), n, n\2+bittest(n,1)) \\ M. F. Hasler, Oct 21 2012

for(n=0,50, print1((1/4)*((1 +(-1)^n)*(1 - (-1)^floor(n/2)) + (3 -(-1)^n)*n), ", ")) \\ G. C. Greubel, Apr 25 2018

a(n) + A215495(n) = A043547(n).
a(n) = -A214283(n)/A000108([n/2]).
a(n+1) = (A186421(n)=0,1,2,1,4,...) + 1.
a(2*n) = A052928(n+1).
a(n+2) - a(n) = 2, 2, 0, 2. (period 4).
a(n) = a(n-2) +a(n-4) -a(n-6); also holds for A215495(n).
G.f.: x*(1+2*x+2*x^2+x^4) / ( (x^2+1)*(x-1)^2*(1+x)^2 ). - R. J. Mathar, Aug 21 2012
a(n) = (1/4)*((1 +(-1)^n)*(1 - (-1)^floor(n/2)) + (3 -(-1)^n)*n). - G. C. Greubel, Apr 25 2018

Corrected and edited by M. F. Hasler, Oct 21 2012

A257857 Sequentially filled binary triangle rotated 180 degrees and then superimposed and added to the original triangle.

Original entry on oeis.org

2, 1, 1, 0, 2, 0, 1, 1, 1, 1, 2, 0, 2, 0, 2, 1, 1, 1, 1, 1, 1, 0, 2, 0, 2, 0, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 2, 0, 2, 0, 2, 0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 2, 0, 2

Offset: 1

Craig Knecht, Jul 12 2015

The integers in the LINKS illustration hang like ornaments on a tree.

			Triangle T(n,k) begins:       Row sums
2;                                2
1,  1;                            2
0,  2,  0;                        2
1,  1,  1,  1;                    4
2,  0,  2,  0,  2;                6
1,  1,  1,  1,  1,  1;            6
0,  2,  0,  2,  0,  2,  0;        6
1,  1,  1,  1,  1,  1,  1,  1;    8

Craig Knecht, binary triangle rotated 180 degrees and superimposed on itself
Craig Knecht, Color coded contributions of the individual triangles

For row sums for the three other variations of this build process, see A186421, A201629, A240828.

A257857 := proc(n,k)
    if type(n,'even') then
        1 ;
    elif type((n+1)/2+k,'even') then
        2 ;
    else
        0;
    end if;
end proc:

T(n,k)=1 if n even, 1<=k<=n.
T(n,k)=2 if n odd and (n+1)/2+k even, 1<=k<=n.
T(n,k)=0 if n odd and (n+1)/2+k odd, 1<=k<=n.

cp's OEIS Frontend

A186422 First differences of A186421.

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A186423 Partial sums of A186421.

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A109613 Odd numbers repeated.

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A109043 a(n) = lcm(n,2).

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A186424 Odd terms in A186423.

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A212831 a(4*n) = 2*n, a(2*n+1) = 2*n+1, a(4*n+2) = 2*n+2.

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A212831 a(4n) = 2n, a(2n+1) = 2n+1, a(4n+2) = 2n+2.