A059029 -id:A059029 - cp's OEIS frontend

A022998 If n is odd then n, otherwise 2n.

0, 1, 4, 3, 8, 5, 12, 7, 16, 9, 20, 11, 24, 13, 28, 15, 32, 17, 36, 19, 40, 21, 44, 23, 48, 25, 52, 27, 56, 29, 60, 31, 64, 33, 68, 35, 72, 37, 76, 39, 80, 41, 84, 43, 88, 45, 92, 47, 96, 49, 100, 51, 104, 53, 108, 55, 112, 57, 116, 59, 120, 61, 124, 63, 128, 65, 132, 67

Offset: 0

N. J. A. Sloane

Also for n > 0: numerator of Sum_{i=1..n} 2/(i*(i+1)), denominator=A026741. - Reinhard Zumkeller, Jul 25 2002
For n > 2: a(n) = gcd(A143051((n-1)^2), A143051(1+(n-1)^2)) = A050873(A000290(n-1), A002522(n-1)). - Reinhard Zumkeller, Jul 20 2008
Partial sums give the generalized octagonal numbers A001082. - Omar E. Pol, Sep 10 2011
Multiples of 4 and odd numbers interleaved. - Omar E. Pol, Sep 25 2011
The Pisano period lengths modulo m appear to be A066043(m). - R. J. Mathar, Oct 08 2011
The partial sums a(n)/A026741(n+1) given by R. Zumkeller in a comment above are 2*n/(n+1) (telescopic sum), and thus converge to 2. - Wolfdieter Lang, Apr 09 2013
a(n) = numerator(H(n,1)), where H(n,1) = 2*n/(n+1) is the harmonic mean of 1 and n. a(n) = 2*n/gcd(2n, n+1) = 2*n/gcd(n+1,2). a(n) = A227041(n,1), n>=1. - Wolfdieter Lang, Jul 04 2013
a(n) = numerator of the mean (2n/(n+1), after reduction), of the compositions of n; denominator is given by A001792(n-1). - Clark Kimberling, Mar 11 2014
A strong divisibility sequence, that is, gcd(a(n), a(m)) = a(gcd(n,m)) for all natural numbers n and m. The sequence of convergents of the 2-periodic continued fraction [0; 1, -4, 1, -4, ...] = 1/(1 - 1/(4 - 1/(1 - 1/(4 - ...)))) begins [0/1, 1/1, 4/3, 3/2, 8/5, 5/3, 12/7, ...]. The present sequence is the sequence of numerators. The sequence of denominators of the continued fraction convergents [1, 1, 3, 2, 5, 3, 7, ...] is A026741, also a strong divisibility sequence. Cf. A203976. - Peter Bala, May 19 2014
a(n) is also the length of the n-th line segment of a rectangular spiral on the infinite square grid. The vertices of the spiral are the generalized octagonal numbers. - Omar E. Pol, Jul 27 2018
a(n) is the number of petals of the Rhodonea curve r = a*cos(n*theta) or r = a*sin(n*theta). - Matt Westwood, Nov 19 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Rose
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A026741, A059026, A203976, A345082.
Column 4 of A195151. - Omar E. Pol, Sep 25 2011
Cf. A000034, A001082 (partial sums).
Cf. A227041 (first column). - Wolfdieter Lang, Jul 04 2013
Row 2 of A349593. A385555, A385556, A385557, A385558, A385559, and A385560 are respectively rows 3, 4, 5-6, 7, 8, and 9-10.

a022998 n = a000034 (n + 1) * n
a022998_list = zipWith (*) [0..] $ tail a000034_list
-- Reinhard Zumkeller, Mar 31 2012

[((-1)^n+3)*n/2: n in [0..70]]; // Vincenzo Librandi, Sep 17 2011

A022998 := proc(n) if type(n,'odd') then n ; else 2*n; end if; end proc: # R. J. Mathar, Mar 10 2011

Table[n (3 + (-1)^n)/2, {n, 0, 100}] (* Wesley Ivan Hurt, Dec 13 2013 *)
Table[If[OddQ[n],n,2n],{n,0,150}] (* or *) Riffle[ 2*Range[ 0,150,2], Range[ 1,150,2]] (* Harvey P. Dale, Feb 06 2017 *)

PARI
```
a(n)=if(n%2,n,2*n)
```

def A022998(n): return n if n&1 else n<<1 # Chai Wah Wu, Mar 05 2024

[n*(1+((n+1)%2)) for n in (0..80)] # G. C. Greubel, Jul 31 2022

Denominator of (n+1)*(n-1)*(2*n+1)/(2*n) (for n > 0).
a(n+1) = lcm(n, n+2)/n + lcm(n, n+2)/(n+2) for all n >= 1. - Asher Auel, Dec 15 2000
Multiplicative with a(2^e) = 2^(e+1), a(p^e) = p^e, p > 2.
G.f. x*(1 + 4*x + x^2)/(1-x^2)^2. - Ralf Stephan, Jun 10 2003
a(n) = 3*n/2 + n*(-1)^n/2 = n*(3 + (-1)^n)/2. - Paul Barry, Sep 04 2003
a(n) = A059029(n-1) + 1 = A043547(n+2) - 2.
a(n)*a(n+3) = -4 + a(n+1)*a(n+2).
a(n) = n*(((n+1) mod 2) + 1) = n^2 + 2*n - 2*n*floor((n+1)/2). - William A. Tedeschi, Feb 29 2008
a(n) = denominator((n+1)/(2*n)) for n >= 1; A026741(n+1) = numerator((n+1)/(2*n)) for n >= 1. - Johannes W. Meijer, Jun 18 2009
a(n) = 2*a(n-2) - a(n-4).
Dirichlet g.f. zeta(s-1)*(1+2^(1-s)). - R. J. Mathar, Mar 10 2011
a(n) = n * (2 - n mod 2) = n * A000034(n+1). - Reinhard Zumkeller, Mar 31 2012
a(n) = floor(2*n/(1 + (n mod 2))). - Wesley Ivan Hurt, Dec 13 2013
From Ilya Gutkovskiy, Mar 16 2017: (Start)
E.g.f.: x*(2*sinh(x) + cosh(x)).
It appears that a(n) is the period of the sequence k*(k + 1)/2 mod n. (End) [This is correct; see A349593. - Jianing Song, Jul 03 2025]
a(n) = Sum_{d | n} A345082(d). - Peter Bala, Jan 13 2024

More terms from Michael Somos, Aug 07 2000

A043547 Odd numbers interspersed with double the previous odd number.

Original entry on oeis.org

1, 2, 3, 6, 5, 10, 7, 14, 9, 18, 11, 22, 13, 26, 15, 30, 17, 34, 19, 38, 21, 42, 23, 46, 25, 50, 27, 54, 29, 58, 31, 62, 33, 66, 35, 70, 37, 74, 39, 78, 41, 82, 43, 86, 45, 90, 47, 94, 49, 98, 51, 102, 53, 106, 55, 110, 57, 114, 59, 118, 61, 122, 63, 126, 65, 130, 67, 134

Offset: 1

Jim Cook (jcook(AT)halcyon.com), Mar 01 2000

As pointed out by E. Angelini on the SeqFan list (cf. link), this is the lexicographically earliest sequence of positive integers without repetitions such that the sum of four consecutive terms is always a multiple of 4. - M. F. Hasler, Mar 22 2013

			a(1)=1 because n is odd. a(2)=2 because a(1)*2=2.

E. Angelini, k-chunks sum and division by k, post to the SeqFan list, Mar 22 2013
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

For n>3, a(n) = A059029(n-3)+3. For n>1, a(n) = A022998(n-2)+2.

[n*(-1)^n/2-(-1)^n+3*n/2-1 : n in [1..50]]; // Wesley Ivan Hurt, Nov 22 2015

A043547:=n->n*(-1)^n/2-(-1)^n+3*n/2-1: seq(A043547(n), n=1..100); # Wesley Ivan Hurt, Nov 22 2015

Flatten[Table[Accumulate[{2 n - 1, 2 n - 1}], {n, 40}]] (* Wesley Ivan Hurt, Nov 22 2015 *)
With[{o=Range[1,71,2]},Riffle[o,2o]] (* Harvey P. Dale, Sep 17 2019 *)

A043547(n)=n+!bittest(n,0)*(n-2) \\ M. F. Hasler, Mar 22 2013

Vec(x*(1+2*x)*(1+x^2)/(1-x^2)^2 + O(x^100)) \\ Altug Alkan, Nov 22 2015

a(n) = (2 - n) * (n - floor(n/2) * 2) + 2 * (n - 1).
G.f.: x*(1+2*x)*(1+x^2)/(1-x^2)^2. - Ralf Stephan, Jun 10 2003
a(2n-1) = 2n-1, a(2n) = 4n-2. - M. F. Hasler, Mar 22 2013
From Wesley Ivan Hurt, Nov 22 2015: (Start)
a(n) = 2*a(n-2) - a(n-4) for n>4.
a(n) = n*(-1)^n/2 - (-1)^n + 3*n/2 - 1. (End)

More terms from James Sellers, Mar 01 2000

A059026 Table B(n,m) read by rows: B(n,m) = LCM(n,m)/n + LCM(n,m)/m - 1 for all 1<=m<=n.

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 4, 2, 6, 1, 5, 6, 7, 8, 1, 6, 3, 2, 4, 10, 1, 7, 8, 9, 10, 11, 12, 1, 8, 4, 10, 2, 12, 6, 14, 1, 9, 10, 3, 12, 13, 4, 15, 16, 1, 10, 5, 12, 6, 2, 7, 16, 8, 18, 1, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 1, 12, 6, 4, 3, 16, 2, 18, 4, 6, 10, 22, 1, 13, 14, 15, 16

Offset: 1

Asher Auel, Dec 15 2000

In an n X m box, a ball makes B(n,m) "bounces" starting at one corner until it reaches another corner, only allowed to travel on diagonal grid lines. B(n+2,n) = A022998(n+1) for all n >= 1. B(2n-1,n) = A016777(n) = 3n + 1 for all n >= 1 (central vertical).

Cf. A022998, A059028, A059029, A059030, A059031.

B := (n,m) -> lcm(n,m)/n + lcm(n,m)/m - 1: seq(seq(B(n,m), n=1..m),m=1..15);

A032438 a(n) = n^2 - floor((n+1)/2)^2.

Original entry on oeis.org

0, 0, 3, 5, 12, 16, 27, 33, 48, 56, 75, 85, 108, 120, 147, 161, 192, 208, 243, 261, 300, 320, 363, 385, 432, 456, 507, 533, 588, 616, 675, 705, 768, 800, 867, 901, 972, 1008, 1083, 1121, 1200, 1240, 1323, 1365, 1452, 1496, 1587, 1633, 1728, 1776, 1875, 1925

Offset: 0

N. J. A. Sloane

The answer to a question from Mike and Laurie Crain (2crains(AT)concentric.net): how many even numbers are there in an n X n multiplication table starting at 1 X 1?
a(n+1) is the number of pairs (x,y) with x and y in {0,...,n}, x and y of the same parity, and x+y >= n. - Clark Kimberling, Jul 02 2012
From J. M. Bergot, Aug 08 2013: (Start)
Define a triangle to have T(1,1)=0 and T(n,c) = n^2 - c^2. The difference of the sum of the terms in antidiagonal(n+1) and those in antidiagonal(n)=a(n).
Column 0 is vertical and T(n,n)=0. The first few rows are 0; 3,0; 8,5,0; 15,12,7,0; 24,21,16,9,0; 35,32,27,20,11,0; the first few antidiagonals are 0; 3; 8,0; 15,5; 24,12,0; 35,21,7; 48,32,16,0; the first few sum of terms in the antidiagonals are 0, 3, 8, 20, 36, 63, 96, 144, 200, 275, 360, 468, 588, 735, 896, 1088, 1296, 1539. (End)
Sum of the largest parts in the partitions of 2n into two distinct odd parts. For example, a(5) = 16; the partitions of 2(5) = 10 into two distinct odd parts are (9,1) and (7,3). The sum of the largest parts is then 9+7 = 16. - Wesley Ivan Hurt, Nov 27 2017

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

First differences are in A059029, partial sums in A143785.

Cf. A008794, A014255.

[n^2-Floor( (n+1)/2 )^2 : n in [0..60]]; // Vincenzo Librandi, Sep 27 2011

A032438:=n->n^2-floor((n+1)/2)^2; seq(A032438(n), n=0..100) # Wesley Ivan Hurt, Nov 25 2013

Table[n^2-Floor[((n+1)/2)]^2,{n,0,50}] (* or *) LinearRecurrence[ {1,2,-2,-1,1},{0,0,3,5,12},51]

a(n)=n^2 - ((n+1)\2)^2 \\ Charles R Greathouse IV, Feb 19 2017

a(n) = n^2 - A008794(n+1).
G.f.: x^2*(x^2 + 2*x + 3)/(1-x^2)^2/(1-x). - Ralf Stephan, Jun 10 2003
a(n) = (1/8)*(2*n*(3*n-1)+(2*n+1)*(-1)^n-1). a(-n-1) = A014255(n). - Bruno Berselli, Sep 27 2011
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5), n > 4. - Harvey P. Dale, Nov 24 2011
E.g.f.: (x*(1 + 3*x)*cosh(x) + (3*x^2 + 3*x - 1)*sinh(x))/4. - Stefano Spezia, Aug 01 2022

A059030 Fourth main diagonal of A059026: a(n) = B(n+3,n) = lcm(n+3,n)/(n+3) + lcm(n+3,n)/n - 1 for all n >= 1.

Original entry on oeis.org

4, 6, 2, 10, 12, 4, 16, 18, 6, 22, 24, 8, 28, 30, 10, 34, 36, 12, 40, 42, 14, 46, 48, 16, 52, 54, 18, 58, 60, 20, 64, 66, 22, 70, 72, 24, 76, 78, 26, 82, 84, 28, 88, 90, 30, 94, 96, 32, 100, 102, 34, 106, 108, 36, 112, 114, 38, 118, 120, 40, 124, 126, 42, 130

Offset: 1

Asher Auel, Dec 15 2000

Cf. A059026, A059029, A059031.

B := (n,m) -> lcm(n,m)/n + lcm(n,m)/m - 1: seq(B(m+3,m),m=1..90);

G.f.: x(2x^3+2x^2+6x+4)/(1-x^3)^2.

A059031 Fifth main diagonal of A059026: a(n) = B(n+4,n) = lcm(n+4,n)/(n+4) + lcm(n+4,n)/n - 1 for all n >= 1.

Original entry on oeis.org

5, 3, 9, 2, 13, 7, 17, 4, 21, 11, 25, 6, 29, 15, 33, 8, 37, 19, 41, 10, 45, 23, 49, 12, 53, 27, 57, 14, 61, 31, 65, 16, 69, 35, 73, 18, 77, 39, 81, 20, 85, 43, 89, 22, 93, 47, 97, 24, 101, 51, 105, 26, 109, 55, 113, 28, 117, 59, 121, 30, 125, 63, 129, 32, 133, 67

Offset: 1

Asher Auel, Dec 15 2000

Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,-1).

Cf. A059026, A059029, A059030.

B := (n,m) -> lcm(n,m)/n + lcm(n,m)/m - 1: seq(B(m+4,m),m=1..90);

LinearRecurrence[{0,0,0,2,0,0,0,-1},{5,3,9,2,13,7,17,4},70] (* Harvey P. Dale, Aug 19 2019 *)

a(2n+1) = 4n+5, a(4n+2) = 4n+3, a(4n+4) = 4n+2. - Ralf Stephan, Jun 10 2005

G.f.: -x*(-5-3*x-9*x^2-2*x^3-3*x^4-x^5+x^6) / ( (x-1)^2*(1+x)^2*(x^2+1)^2 ). - R. J. Mathar, Sep 20 2011

A133146 Antidiagonal sums of the triangle A133128.

Original entry on oeis.org

2, 5, 7, 14, 18, 29, 35, 50, 58, 77, 87, 110, 122, 149, 163, 194, 210, 245, 263, 302, 322, 365, 387, 434, 458, 509, 535, 590, 618, 677, 707, 770, 802, 869, 903, 974, 1010, 1085, 1123, 1202, 1242, 1325, 1367, 1454, 1498, 1589, 1635, 1730, 1778, 1877, 1927, 2030

Offset: 0

Paul Curtz, Aug 27 2008

			a(2) = A133128(2,0) + A133128(1,1) = 10 - 3 = 7.
a(3) = A133128(3,0) + A133128(2,1) = 17 - 3 = 14.

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

[19/8 +5*n/4 +3*n^2/4 -(-1)^n*(n/4+3/8): n in [0..60]]; // Vincenzo Librandi, Aug 10 2011

CoefficientList[Series[(1+2x)(2-x+x^3)/((1-x)^3(1+x)^2),{x,0,60}],x] (* or *) LinearRecurrence[{1,2,-2,-1,1},{2,5,7,14,18},60] (* Harvey P. Dale, Aug 26 2013 *)

First differences: a(n+1) - a(n) = A059029(n+1).
Bisections: a(2n+1) = A005918(n+1). a(2n) = A141631(n+1).
G.f.: (1+2*x)(2 - x + x^3)/((1-x)^3*(1+x)^2). - R. J. Mathar, Oct 15 2008
a(n) = 19/8 + 5*n/4 + 3*n^2/4 - (-1)^n*(n/4 + 3/8). - R. J. Mathar, Oct 15 2008
From Harvey P. Dale, Aug 26 2013: (Start)
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5); a(0)=2, a(1)=5, a(2)=7, a(3)=14, a(4)=18. (End)

Edited and extended by R. J. Mathar, Oct 15 2008

A115379 Number of positive integers k < n such that n XOR k < n and gcd(n,k) is odd.

Original entry on oeis.org

0, 1, 0, 3, 0, 3, 2, 7, 0, 3, 2, 7, 4, 11, 6, 15, 0, 3, 2, 7, 4, 11, 6, 15, 8, 19, 10, 23, 12, 27, 14, 31, 0, 3, 2, 7, 4, 11, 6, 15, 8, 19, 10, 23, 12, 27, 14, 31, 16, 35, 18, 39, 20, 43, 22, 47, 24, 51, 26, 55, 28, 59, 30, 63, 0, 3, 2, 7, 4, 11, 6, 15, 8, 19, 10, 23, 12, 27, 14, 31, 16, 35

Offset: 0

Paul D. Hanna, Jan 21 2006

nonn

A059029 equals the limiting sequence of 2^k consecutive terms of this sequence starting at position 2^k as k increases, where A059029(n) = n if n is even, 2n+1 if n is odd.

Indranil Ghosh, Table of n, a(n) for n = 0..1000
Index entries for sequences related to the Josephus Problem

Cf. A059029, A006257 (Josephus problem).

Table[Sum[If[BitXor[n, k]< n && OddQ[GCD[n, k]], 1, 0], {k, 0, n}], {n, 0, 81}] (* Indranil Ghosh, Mar 16 2017 *)

PARI
```
a(n)=sum(k=0,n,if(bitxor(n,k)
				
```

a(2^n) = 0, a(2^n-1) = 2^n-1, for n >= 0. a(2^n+1)=3 (n>0), a(2^n+2)=2 (n>1), a(2^n+3)=7 (n>1), a(2^n+4)=4 (n>2), a(2^n+5)=11 (n>2), etc.

cp's OEIS Frontend

A022998 If n is odd then n, otherwise 2n.

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A043547 Odd numbers interspersed with double the previous odd number.

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A059026 Table B(n,m) read by rows: B(n,m) = LCM(n,m)/n + LCM(n,m)/m - 1 for all 1<=m<=n.

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A032438 a(n) = n^2 - floor((n+1)/2)^2.

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A059030 Fourth main diagonal of A059026: a(n) = B(n+3,n) = lcm(n+3,n)/(n+3) + lcm(n+3,n)/n - 1 for all n >= 1.

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A059031 Fifth main diagonal of A059026: a(n) = B(n+4,n) = lcm(n+4,n)/(n+4) + lcm(n+4,n)/n - 1 for all n >= 1.

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A133146 Antidiagonal sums of the triangle A133128.

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