A142717 -id:A142717 - cp's OEIS frontend

A144396 The odd numbers greater than 1.

3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 133

Offset: 1

Paul Curtz, Oct 03 2008

Last number of the n-th row of the triangle described in A142717.
If negated, these are also the values at local minima of the sequence A141620.
a(n) is the shortest leg of the n-th Pythagorean triple with consecutive longer leg and hypotenuse. The n-th such triple is given by (2n+1,2n^2+2n, 2n^2+2n+1), so that the longer legs are A046092(n) and the hypotenuses are A099776(n). - Ant King, Feb 10 2011
Numbers k such that the symmetric representation of sigma(k) has a pair of bars as its ends (cf. A237593). - Omar E. Pol, Sep 28 2018
Numbers k such that there is a prime knot with k crossings and braid index 2. (IS this true with "prime" removed?) - Charles R Greathouse IV, Feb 14 2023

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

Complement of A004275 and of A004277.
Cf. A000217, A002378, A005408, A007395, A046092, A099776, A237593, A254858.
Essentially the same as A140139, A130773, A062545, A020735, A005818.

List([3,5..200],n->n); # Muniru A Asiru, Sep 28 2018

a144396 = (+ 1) . (* 2)
a144396_list = [3, 5 ..] -- Reinhard Zumkeller, Feb 09 2015

seq(n,n=3..200,2); # Muniru A Asiru, Sep 28 2018

Range[3, 200, 2] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2012 *)

a(n)=2*n+1 \\ Charles R Greathouse IV, Feb 14 2023

a(n) = A005408(n+1) = A000290(n+1) - A000290(n).
G.f.: x*(3-x)/(1-x)^2. - Jaume Oliver Lafont, Aug 30 2009
a(n) = A254858(n-1,2). - Reinhard Zumkeller, Feb 09 2015

Edited by R. J. Mathar, May 21 2009

A214297 a(0)=-1, a(1)=0, a(2)=-3; thereafter a(n+2) - 2*a(n+1) + a(n) has period 4: repeat -4, 8, -4, 2.

Original entry on oeis.org

-1, 0, -3, 2, 3, 6, 5, 12, 15, 20, 21, 30, 35, 42, 45, 56, 63, 72, 77, 90, 99, 110, 117, 132, 143, 156, 165, 182, 195, 210, 221, 240, 255, 272, 285, 306, 323, 342, 357, 380, 399, 420, 437, 462, 483, 506, 525, 552, 575, 600, 621, 650, 675, 702, 725, 756, 783, 812, 837, 870, 899, 930, 957, 992, 1023, 1056, 1085, 1122, 1155, 1190

Offset: 0

Paul Curtz, Jul 11 2012

Let a(n)/A000290(n) = [-1/0, 0/1, -3/4, 2/9, 3/16, 6/25, 5/36, 12/49, 15/64, 20/81, 21/100, 30/121, ...] = a(n)/b(n) (say).
Then b(n)-4*a(n)=4, 1, 16, 1 (period of length 4).
Permutation from a(n) to A061037(n): 1, 3, 2, 7, 5, 11, 4, 15, 9, 19, 6, ... = shifted A145979 + 1.
A061037(n) - a(n) = 0, 3, -3, -3, 0, -15, 3, -33, 0 -57, 15, -87, 0, -123, ...
First 3 rows:
-1 0 -3 2  3  6  5 12 15 20 21 30 35
1 -3  5 1  3 -1  7  3  5  1  9  5  7
-4 8 -4 2 -4  8 -4  2 -4  8 -4  2 -4.
Note that the terms of a(n) increase from 12. Compare to increasing terms permutation of A061037(n): -3,-1,0,2,3,5,6,12,15, .... and A129647.
c(n) =  0, -1, 0, -1, 2, 1, 2, 1, 4, 3, 4, 3, 6, 5, 6, 5, ... (cf. A134967)
d(n) = -1,  1, 1,  3, 1, 3, 3, 5, 3, 5, 5, 7, 5, 7, 7, 9, ..., hence:
a(n) = c(n+1) * d(n+1).

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).

[(2*n^2-11-9*(-1)^n+6*((-1)^((2*n+1-(-1)^n)/4)+(-1)^((2*n-1+(-1)^n)/4)))/8: n in [0..100]]; // G. C. Greubel, Sep 19 2018

A214297 := proc(n)
    option remember;
    if n <=5 then
        op(n+1,[-1,0,-3,2,3,6]) ;
    else
        2*procname(n-1)-procname(n-2)+procname(n-4)-2*procname(n-5)+procname(n-6) ;
    end if;
end proc: # R. J. Mathar, Jun 28 2013

Table[(2 n^2 - 11 - 9 (-1)^n + 6 ((-1)^((2 n + 1 - (-1)^n)/4) + (-1)^((2 n - 1 + (-1)^n)/4)))/8, {n, 0, 69}] (* or *)
CoefficientList[Series[-(1 - 2 x + 4 x^2 - 8 x^3 + 3 x^4)/((1 - x)^2*(1 - x^4)), {x, 0, 69}], x] (* Michael De Vlieger, Mar 24 2017 *)

vector(100, n, n--; (2*n^2-11-9*(-1)^n+6*((-1)^((2*n+1-(-1)^n)/4)+(-1)^((2*n-1+(-1)^n)/4)))/8) \\ G. C. Greubel, Sep 19 2018

a(k+4)  - a(k)    =  2*k + 4.
a(k+2)  - a(k-2)  =  2*k.
a(k+6)  - a(k-6)  =  6*k.
a(k+10) - a(k-10) = 10*k.
a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12).
a(2*k)   = -1, -3, followed by 3, 5, 15, 21, 35, 45, ... (A142717);
a(2*k+1) = k*(k+1) (see A002378).
A198442(n) = -1,0,0,2,3,6,8,12,  minus 3 at A198442(4*n+2).
G.f. -( 1-2*x+4*x^2-8*x^3+3*x^4 )/( (1-x)^2*(1-x^4) ). - R. J. Mathar, Jul 17 2012; edited by N. J. A. Sloane, Jul 22 2012
From R. J. Mathar, Jun 28 2013: (Start)
a(4*k)   = A000466(k);
a(4*k+1) = A002943(k);
a(4*k+2) = A078371(k-1) for k>0;
a(4*k+3) = A002939(k+1). (End)
a(n) = (2*n^2-11-9*(-1)^n+6*((-1)^((2*n+1-(-1)^n)/4)+(-1)^((2*n-1+(-1)^n)/4)))/8. - Luce ETIENNE, Oct 27 2016

Edited by N. J. A. Sloane, Jul 22 2012

A142954 a(n) = 2n + 3 + 3(-1)^n.

Original entry on oeis.org

2, 10, 6, 14, 10, 18, 14, 22, 18, 26, 22, 30, 26, 34, 30, 38, 34, 42, 38, 46, 42, 50, 46, 54, 50, 58, 54, 62, 58, 66, 62, 70, 66, 74, 70, 78, 74, 82, 78, 86, 82, 90, 86, 94, 90, 98, 94, 102, 98, 106, 102, 110, 106, 114, 110

Offset: 1

Paul Curtz, Sep 29 2008

First differences of A142717.

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

First differences of A214345.

List([1..55],n->2*n+3+3*(-1)^n); # Muniru A Asiru, Nov 01 2018

I:=[2,10,6]; [n le 3 select I[n] else Self(n-1)+Self(n-2)-Self(n-3): n in [1..60]]; // Vincenzo Librandi, Apr 03 2013

seq(2*n+3+3*(-1)^n,n=1..55); # Muniru A Asiru, Nov 01 2018

Table[2 n + 3 + 3 (-1)^n, {n, 1, 60}] (* Vincenzo Librandi, Apr 03 2013 *)
LinearRecurrence[{1,1,-1},{2,10,6},60] (* Harvey P. Dale, Aug 20 2015 *)

for n in range(1,50): print(2*n+3+3*(-1)**n, end=', ') # Stefano Spezia, Nov 01 2018

a(2n+1) = A016825(n). a(2n) = A016825(n+1).
From R. J. Mathar, Oct 24 2008: (Start)
G.f.: 2*x*(1+4*x-3*x^2)/((1+x)*(1-x)^2). [corrected by Jason Yuen, Oct 01 2024]
a(n) = a(n-1)+a(n-2)-a(n-3) = 2*A097062(n+2). (End)

Edited by R. J. Mathar, Oct 24 2008

A247829 a(3k) = k(k+1), a(3k+1) = (2k-1)(2k+1), a(3k+2) = (2k-1)(2k+3).

Original entry on oeis.org

0, -1, -3, 2, 3, 5, 6, 15, 21, 12, 35, 45, 20, 63, 77, 30, 99, 117, 42, 143, 165, 56, 195, 221, 72, 255, 285, 90, 323, 357, 110, 399, 437, 132, 483, 525, 156, 575, 621, 182, 675, 725, 210, 783, 837, 240, 899, 957, 272, 1023, 1085, 306, 1155, 1221, 342, 1295

Offset: 0

Paul Curtz, Dec 01 2014

A permutation of A061037(n) = -1, -3, 0, 5, 3, 21, 2, 45, 15, 77, 6, ... and of A214297(n) = -1, 0, -3, 2, 3, 6, 5, 12, ... .
Among consequences: b(n) = 4*a(n) + (sequence of period 3:repeat 1, 4, 16) = 1, 0, 4, 9, 16, 36, 25, 64, 100, ... , is a permutation of the squares of the nonnegative integers A000290(n).
And a(n)/b(n) = 0/1, -1/0, -3/4, 2/9, 3/16, 5/36, ... is a permutation of the Balmer series A061037(n)/A061038(n) = -1/0, -3/4, 0/1, 5/36, 3/16, ... .
a(5n) is divisible by 5.

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

Cf. A142717 (2 terms out of 3), A000290, A000466, A002378, A061037/A061038, A078371, A214297.

m:=50; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(3*x^7-3*x^6-14*x^4-6*x^3-2*x^2+3*x+1)/((x-1)^3*(x^2 +x+1)^3))); // G. C. Greubel, Sep 20 2018

seq(op([k*(k+1),(2*k-1)*(2*k+1),(2*k-1)*(2*k+3)]), k=0..100); # Robert Israel, Dec 01 2014

Table[Sequence @@ {n*(n+1), (2*n-1)*(2*n+1), (2*n-1)*(2*n+3)}, {n, 0, 18}] (* Jean-François Alcover, Dec 16 2014 *)

concat(0, Vec(x*(3*x^7-3*x^6-14*x^4-6*x^3-2*x^2+3*x+1)/((x-1)^3*(x^2+x+1)^3) + O(x^100))) \\ Colin Barker, Dec 02 2014

a(n) = 3*a(n-3) - 3*a(n-6) + a(n-9).
a(3*k) + a(3*k+1) + a(3*k+2) = 9*k^2 + 5*k - 4.
G.f.: x*(3*x^7 - 3*x^6 - 14*x^4 - 6*x^3 - 2*x^2 + 3*x + 1)/((x-1)^3*(x^2 +x+1)^3). - Robert Israel, Dec 01 2014
a(n) = -(n^2 + n + floor(n/3)*(27*floor(n/3)^3 - 18*(n+1)*floor(n/3)^2 + (3*n^2 + 21*n - 14)*floor(n/3) - (5*n^2 - n + 5)))/2. - Luce ETIENNE, Mar 13 2017
From Amiram Eldar, Oct 08 2023: (Start)
Sum_{n>=1} 1/a(n) = 1/2.
Sum_{n>=1} (-1)^n/a(n) = Pi/4 + 1 - 2*log(2). (End)

cp's OEIS Frontend

A144396 The odd numbers greater than 1.

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A214297 a(0)=-1, a(1)=0, a(2)=-3; thereafter a(n+2) - 2*a(n+1) + a(n) has period 4: repeat -4, 8, -4, 2.

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A142954 a(n) = 2*n + 3 + 3*(-1)^n.

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A247829 a(3*k) = k*(k+1), a(3*k+1) = (2*k-1)*(2*k+1), a(3*k+2) = (2*k-1)*(2*k+3).

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Formula

A142954 a(n) = 2n + 3 + 3(-1)^n.

A247829 a(3k) = k(k+1), a(3k+1) = (2k-1)(2k+1), a(3k+2) = (2k-1)(2k+3).