A109044 -id:A109044 - cp's OEIS frontend

A051176 If n mod 3 = 0 then n/3 else n.

0, 1, 2, 1, 4, 5, 2, 7, 8, 3, 10, 11, 4, 13, 14, 5, 16, 17, 6, 19, 20, 7, 22, 23, 8, 25, 26, 9, 28, 29, 10, 31, 32, 11, 34, 35, 12, 37, 38, 13, 40, 41, 14, 43, 44, 15, 46, 47, 16, 49, 50, 17, 52, 53, 18, 55, 56, 19, 58, 59, 20, 61, 62, 21, 64, 65, 22, 67

Offset: 0

N. J. A. Sloane

Numerator of n/3. - Wesley Ivan Hurt, Jul 18 2014

			G.f. = x + 2*x^2 + x^3 + 4*x^4 + 5*x^5 + 2*x^6 + 7*x^7 + 8*x^8 + 3*x^9 + ...

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

Cf. A026741, A051176, A060819, A060791, A060789 for n / GCD(n,k) for k=2..6. See also A106608 thru A106612 (k = 7 thru 11), A051724 (k = 12), A106614 thru A106621 (k = 13 thru 20).

Cf. A109044, A011655, A144437, A167192.

a051176 n = if m == 0 then n' else n  where (n',m) = divMod n 3
-- Reinhard Zumkeller, Aug 27 2012

[Numerator(n/3): n in [0..70]]; // G. C. Greubel, Feb 19 2019

A051176:=n->numer(n/3); seq(A051176(n), n=0..100); # Wesley Ivan Hurt, Jul 18 2014

If[Divisible[#,3],#/3,#]&/@Range[0,70] (* Harvey P. Dale, Feb 07 2011 *)
a[n_] := Numerator[n/3]; Array[a, 100, 0] (* Wesley Ivan Hurt, Jul 18 2014 *)
LinearRecurrence[{0,0,2,0,0,-1},{0,1,2,1,4,5},70] (* Harvey P. Dale, Jul 12 2025 *)

a(n) = if (n % 3, n, n/3); \\ Michel Marcus, Feb 02 2016

[numerator(n/3) for n in range(70)] # G. C. Greubel, Feb 19 2019

a(n) = n / gcd(n,3).
G.f.: x*(1+2*x+x^2+2*x^3+x^4)/(1-x^3)^2 = x*(1+2*x+x^2+2*x^3+x^4) / ( (x-1)^2*(1+x+x^2)^2 ). - Len Smiley, Apr 30 2001
Multiplicative with a(3^e) = 3^(e-1), a(p^e) = p^e otherwise. - Mitch Harris, Jun 09 2005
a(n) = A167192(n+3, 3). - Reinhard Zumkeller, Oct 30 2009
From R. J. Mathar, Apr 18 2011: (Start)
a(n) = A109044(n)/3.
Dirichlet g.f.: zeta(s-1)*(1-2/3^s). (End)
a(n) = n/3 * (1 + 2*A011655(n)) = n*A144437(n)/3. - Timothy Hopper, Feb 23 2017
G.f.: x /(1 - x)^2 - 2 * x^3/(1 - x^3)^2. - Michael Somos, Mar 05 2017
a(n) = a(-n) for all n in Z. - Michael Somos, Mar 05 2017
a(n) = n*(7 - 4*cos((2*Pi*n)/3)) / 9. - Colin Barker, Mar 05 2017
Sum_{k=1..n} a(k) ~ (7/18) * n^2. - Amiram Eldar, Nov 25 2022
Sum_{n>=1} (-1)^(n+1)/a(n) = 5*log(2)/3. - Amiram Eldar, Sep 08 2023

A109042 Square array read by antidiagonals: A(n, k) = lcm(n, k) for n >= 0, k >= 0.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 2, 3, 0, 0, 4, 6, 6, 4, 0, 0, 5, 4, 3, 4, 5, 0, 0, 6, 10, 12, 12, 10, 6, 0, 0, 7, 6, 15, 4, 15, 6, 7, 0, 0, 8, 14, 6, 20, 20, 6, 14, 8, 0, 0, 9, 8, 21, 12, 5, 12, 21, 8, 9, 0, 0, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 0, 0, 11, 10, 9, 8, 35, 6, 35, 8, 9, 10, 11, 0

Offset: 0

Mitch Harris, Jun 18 2005

			Seen as an array:
  [0] 0, 0,  0,  0,  0,  0,  0,  0,  0,  0, ...
  [1] 0, 1,  2,  3,  4,  5,  6,  7,  8,  9, ...
  [2] 0, 2,  2,  6,  4, 10,  6, 14,  8, 18, ...
  [3] 0, 3,  6,  3, 12, 15,  6, 21, 24,  9, ...
  [4] 0, 4,  4, 12,  4, 20, 12, 28,  8, 36, ...
  [5] 0, 5, 10, 15, 20,  5, 30, 35, 40, 45, ...
  [6] 0, 6,  6,  6, 12, 30,  6, 42, 24, 18, ...
  [7] 0, 7, 14, 21, 28, 35, 42,  7, 56, 63, ...
  [8] 0, 8,  8, 24,  8, 40, 24, 56,  8, 72, ...
  [9] 0, 9, 18,  9, 36, 45, 18, 63, 72,  9, ...
.
Seen as a triangle T(n, k) = lcm(n - k, k).
  [0] 0;
  [1] 0, 0;
  [2] 0, 1,  0;
  [3] 0, 2,  2,  0;
  [4] 0, 3,  2,  3,  0;
  [5] 0, 4,  6,  6,  4,  0;
  [6] 0, 5,  4,  3,  4,  5, 0;
  [7] 0, 6, 10, 12, 12, 10, 6,  0;
  [8] 0, 7,  6, 15,  4, 15, 6,  7, 0;
  [9] 0, 8, 14,  6, 20, 20, 6, 14, 8, 0;

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Rows A000027, A109043, A109044, A109045, A109046, A109047, A109048, A109049, A109050, A109051, A109052, A109053, A006580 (row sums of triangle), A001477 (main diagonal, central terms).

Variants: A003990 is (1, 1) based, A051173 (T(n,k) = lcm(n,k)).

T := (n, k) -> ilcm(n - k, k):
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;  # Peter Luschny, Mar 24 2025

lcm(n, k) = n*k / gcd(n, k) for (n, k) != (0, 0).

A130724 a(n) = lcm(n,3) / gcd(n,3).

Original entry on oeis.org

0, 3, 6, 1, 12, 15, 2, 21, 24, 3, 30, 33, 4, 39, 42, 5, 48, 51, 6, 57, 60, 7, 66, 69, 8, 75, 78, 9, 84, 87, 10, 93, 96, 11, 102, 105, 12, 111, 114, 13, 120, 123, 14, 129, 132, 15, 138, 141, 16, 147, 150, 17, 156, 159, 18, 165, 168, 19, 174, 177, 20, 183, 186

Offset: 0

W. Neville Holmes, Jul 04 2007

			a(7) = 21 because lcm(3,7) = 21, gcd(3,7) = 1 and 21/1 = 21.

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

Cf. A109007, A109044.

[Lcm(n,3)/Gcd(n,3) : n in [0..100]]; // Wesley Ivan Hurt, Jul 24 2016

A130724:=n->lcm(n,3)/gcd(n,3): seq(A130724(n), n=0..100); # Wesley Ivan Hurt, Jul 24 2016

LCM[3,#]/GCD[3,#]&/@Range[0,70] (* Harvey P. Dale, May 16 2013 *)

a(n) = A109044(n) / A109007(n).
From Wesley Ivan Hurt, Jul 24 2016: (Start)
G.f.: x*(3 + 6*x + x^2 + 6*x^3 + 3*x^4)/(x^3 - 1)^2.
a(n) = 2*a(n-3) - a(n-6) for n>5.
a(n) = 27*n/(5 + 4*cos(2*n*Pi/3))^2.
If n mod 3 = 0, then n/3, else 3*n.
a(n) = lcm(numerator(n/3), denominator(n/3)). (End)
Sum_{k=1..n} a(k) ~ (19/18)*n^2. - Amiram Eldar, Oct 07 2023

Corrected and extended by Harvey P. Dale, May 16 2013

A066393 Coordination sequence for (9^3, 3.9^2) net with respect to a vertex of type 9^3.

Original entry on oeis.org

1, 3, 6, 6, 12, 15, 12, 21, 24, 18, 30, 33, 24, 39, 42, 30, 48, 51, 36, 57, 60, 42, 66, 69, 48, 75, 78, 54, 84, 87, 60, 93, 96, 66, 102, 105, 72, 111, 114, 78, 120, 123, 84, 129, 132, 90, 138, 141, 96, 147, 150, 102, 156, 159, 108, 165, 168, 114

Offset: 0

N. J. A. Sloane, Dec 24 2001

nonn

This net may be regarded as a tiling of the plane by 9-gons and triangles. There are two kinds of vertices: (a) 9^3 vertices, where three 9-gons meet, and (b) 3.9^2 vertices, where a triangle and two 9-gons meet. The present sequence is the coordination sequence with respect to a vertex of type 9^3. See also A319980.

Muniru A Asiru, Table of n, a(n) for n = 0..300
Jean-Guillaume Eon, Geometrical relationships between nets mapped on isomorphic quotient graphs: examples, Journal of Solid State Chemistry 138.1 (1998): 55-65. See Fig. 1.
Jean-Guillaume Eon, Algebraic determination of generating functions for coordination sequences in crystal structures, Acta Cryst. A58 (2002), 47-53. See Section 8.
N. J. A. Sloane, A portion of the (9^3, 3.9^2) net

Cf. A319980, A109044, A186101.

seq(coeftayl((1+3*x+6*x^2+4*x^3+6*x^4+3*x^5+x^6)/(1-x^3)^2, x = 0, k), k=0..60); # Muniru A Asiru, Feb 13 2018

G.f.: (1+3*x+6*x^2+4*x^3+6*x^4+3*x^5+x^6)/(1-x^3)^2.
a(n) = (3*n + lcm(n,3))/2, for n>=1. - Ridouane Oudra, Jan 22 2021
a(n) = 3*A186101(n), for n>=1. - Ridouane Oudra, Jun 11 2025

A130723 Least common multiple of 3 and n^2+n+1.

Original entry on oeis.org

3, 3, 21, 39, 21, 93, 129, 57, 219, 273, 111, 399, 471, 183, 633, 723, 273, 921, 1029, 381, 1263, 1389, 507, 1659, 1803, 651, 2109, 2271, 813, 2613, 2793, 993, 3171, 3369, 1191, 3783, 3999, 1407, 4449, 4683, 1641, 5169, 5421, 1893, 5943, 6213, 2163, 6771

Offset: 0

W. Neville Holmes, Jul 04 2007

Harvey P. Dale, 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. A109044.

Table[LCM[3,n^2+n+1],{n,0,60}] (* Harvey P. Dale, Mar 03 2017 *)

a(n) = lcm(3, n^2+n+1) \\ Michel Marcus, Jul 11 2013

Vec(3*(1 + x + 7*x^2 + 10*x^3 + 4*x^4 + 10*x^5 + 7*x^6 + x^7 + x^8) / ((1 - x)^3*(1 + x + x^2)^3) + O(x^100)) \\ Colin Barker, Mar 08 2017

From Colin Barker, Mar 08 2017: (Start)
G.f.: 3*(1 + x + 7*x^2 + 10*x^3 + 4*x^4 + 10*x^5 + 7*x^6 + x^7 + x^8) / ((1 - x)^3*(1 + x + x^2)^3).
a(n) = 3*a(n-3) - 3*a(n-6) + a(n-9) for n>8.
(End)

Corrected by Harvey P. Dale, Mar 03 2017

A187601 n/2 times period 6 sequence [1, 2, 3, 4, 3, 2, ...].

Original entry on oeis.org

0, 1, 3, 6, 6, 5, 3, 7, 12, 18, 15, 11, 6, 13, 21, 30, 24, 17, 9, 19, 30, 42, 33, 23, 12, 25, 39, 54, 42, 29, 15, 31, 48, 66, 51, 35, 18, 37, 57, 78, 60, 41, 21, 43, 66, 90, 69, 47, 24, 49, 75, 102, 78, 53, 27, 55, 84, 114, 87, 59, 30, 61, 93, 126, 96, 65, 33, 67, 102

Offset: 0

Bruno Berselli, Mar 11 2011

A007310 is a subsequence.

Bruno Berselli, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (2,-1,-2,4,-2,-1,2,-1).

Cf. A186813.

Cf. A109044, A088439 (by Superseeker).

[(n/2)*[1, 2, 3, 4, 3, 2][n mod 6 + 1]: n in [0..68]]; /* Other: */ [n*(5-2*(-1)^Floor((n+1)/3)-(-1)^n)/4: n in [0..68]];

CoefficientList[Series[x (1 + x + x^2 - x^3 + x^4 + x^5 + x^6) / ((1 - x)^2 (1 + x)^2 (1 - x + x^2)^2), {x, 0, 70}], x] (* Vincenzo Librandi, Aug 19 2013 *)
LinearRecurrence[{2,-1,-2,4,-2,-1,2,-1},{0,1,3,6,6,5,3,7},90] (* Harvey P. Dale, Aug 20 2017 *)

a(n) = (n/2)*A028356(n).
G.f.: x*(1+x+x^2-x^3+x^4+x^5+x^6)/((1-x)^2*(1+x)^2*(1-x+x^2)^2).
a(-n) = -a(n).  a(n) =  2*a(n-1)-a(n-2)-2*a(n-3)+4*a(n-4)-2*a(n-5)-a(n-6)+2*a(n-7)-a(n-8) for n>7.
a(n) = n*(5-2*(-1)^floor((n+1)/3)-(-1)^n)/4.

cp's OEIS Frontend

A051176 If n mod 3 = 0 then n/3 else n.

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A109042 Square array read by antidiagonals: A(n, k) = lcm(n, k) for n >= 0, k >= 0.

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A130724 a(n) = lcm(n,3) / gcd(n,3).

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A066393 Coordination sequence for (9^3, 3.9^2) net with respect to a vertex of type 9^3.

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A130723 Least common multiple of 3 and n^2+n+1.

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A187601 n/2 times period 6 sequence [1, 2, 3, 4, 3, 2, ...].

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