A038590 -id:A038590 - cp's OEIS frontend

A038161 (A038590-1)/6.

0, 1, 2, 3, 5, 6, 7, 9, 10, 12, 14, 15, 16, 18, 20, 21, 23, 25, 27, 28, 31, 33, 35, 37, 39, 40, 42, 44, 45, 47, 49, 50, 52, 56, 58, 60, 61, 63, 64, 66, 68, 70, 72, 73, 75, 77, 79, 83, 85, 86, 89, 91, 93, 95, 97, 99, 102, 104, 106, 108, 110, 112, 114, 115

Offset: 0

N. J. A. Sloane

nonn

Hugo Pfoertner, Table of n, a(n) for n = 0..10000

Cf. A035019, A038162, A038590.

A038160 Intersection of A003136 and A038590.

Original entry on oeis.org

1, 7, 13, 19, 31, 37, 43, 61, 73, 91, 97, 109, 121, 127, 139, 151, 163, 169, 199, 211, 223, 241, 271, 283, 301, 313, 337, 349, 361, 367, 379, 397, 409, 421, 433, 439, 463, 475, 499, 511, 547, 559, 571, 613, 625, 637, 661, 673, 691, 703, 721

Offset: 1

N. J. A. Sloane

nonn

Cf. A003136, A038590.

A035019 Sizes of successive shells in hexagonal (or A_2) lattice.

Original entry on oeis.org

1, 6, 6, 6, 12, 6, 6, 12, 6, 12, 12, 6, 6, 12, 12, 6, 12, 12, 12, 6, 18, 12, 12, 12, 12, 6, 12, 12, 6, 12, 12, 6, 12, 24, 12, 12, 6, 12, 6, 12, 12, 12, 12, 6, 12, 12, 12, 24, 12, 6, 18, 12, 12, 12, 12, 12, 18, 12, 12, 12, 12, 12, 12, 6, 12, 18, 12, 12, 12, 12

Offset: 0

N. J. A. Sloane

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 111.

T. D. Noe, Table of n, a(n) for n = 0..10000
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2

Cf. A003136, A004016, A038590 (partial sums), A357112.

S:=series(JacobiTheta2(0,q)*JacobiTheta2(0,q^3)+JacobiTheta3(0,q)*JacobiTheta3(0,q^3),q,1001):
subs(0=NULL,[seq(coeff(S,q,j),j=0..1000)]); # Robert Israel, Jul 29 2016

s = EllipticTheta[2, 0, q]*EllipticTheta[2, 0, q^3] + EllipticTheta[3, 0, q]* EllipticTheta[3, 0, q^3] + O[q]^1000; CoefficientList[s, q] /. 0 -> Nothing (* Jean-François Alcover, Sep 19 2016, after Robert Israel *)

Nonzero coefficients in expansion of theta_3(q)*theta_3(q^3) + theta_2(q)*theta_2(q^3).

The corresponding powers of q are A003136. - Robert Israel, Jul 29 2016

A307014 List coordinates (x,y) of the points in an hexagonal grid, sorted first by radial coordinate r and in case of ties, by polar angle 0 <= phi < 2*Pi in a polar coordinate system. Sequence gives the first coordinate in a barycentric coordinate system.

Original entry on oeis.org

0, 1, 0, -1, -1, 0, 1, 1, -1, -2, -1, 1, 2, 2, 0, -2, -2, 0, 2, 2, 1, -1, -2, -3, -3, -2, -1, 1, 2, 3, 3, 3, 0, -3, -3, 0, 3, 2, -2, -4, -2, 2, 4, 3, 1, -1, -3, -4, -4, -3, -1, 1, 3, 4, 4, 4, 0, -4, -4, 0, 4, 3, 2, -2, -3, -5, -5, -3, -2, 2, 3, 5, 5, 4, 1

Offset: 0

Hugo Pfoertner, Mar 21 2019

Cartesian coordinates (x,y) of the grid points are converted to barycentric coordinates (i,j,k) by i = x - y/sqrt(3), j = 2*y/sqrt(3), k = x + y/sqrt(3). The sequence gives i. j is given in A307016, k is given in A307017.

The sorting by polar angle affects the grid points in the shells of size A035019, starting at indices given by A038590.

Hugo Pfoertner, Table of n, a(n) for n = 0..9060, covering range r <= 50.
Hugo Pfoertner, Illustration of initial terms of A307014 and A307016.
Hugo Pfoertner, Illustration of A307016 vs A307014, grid points.
Hugo Pfoertner, Illustration of A307016 vs A307014, spiral.
Hugo Pfoertner, Illustration of A307017 vs A307014, spiral.
Hugo Pfoertner, PARI program for A307014 and A307016.

Cf. A004016, A035019, A038590, A305575, A305576, A307011, A307012, A307013, A307016, A307017.

\\ See Link
\\ To create the data of this sequence load program from file and call
a307014_16(5, 4) \\ Hugo Pfoertner, Nov 07 2023

A307016 List coordinates (x,y) of the points in an hexagonal grid, sorted first by radial coordinate r and in case of ties, by polar angle 0 <= phi < 2*Pi in a polar coordinate system. Sequence gives the second coordinate in a barycentric coordinate system.

Original entry on oeis.org

0, 0, 1, 1, 0, -1, -1, 1, 2, 1, -1, -2, -1, 0, 2, 2, 0, -2, -2, 1, 2, 3, 3, 2, 1, -1, -2, -3, -3, -2, -1, 0, 3, 3, 0, -3, -3, 2, 4, 2, -2, -4, -2, 1, 3, 4, 4, 3, 1, -1, -3, -4, -4, -3, -1, 0, 4, 4, 0, -4, -4, 2, 3, 5, 5, 3, 2, -2, -3, -5, -5, -3, -2, 1

Offset: 0

Hugo Pfoertner, Mar 21 2019

Cartesian coordinates (x,y) of the grid points are converted to barycentric coordinates (i,j,k) by i = x - y/sqrt(3), j = 2*y/sqrt(3), k = x + y/sqrt(3). The sequence gives j. i is given in A307014, k is given in A307017.

Hugo Pfoertner, Table of n, a(n) for n = 0..9060, covering range r <= 50.
Hugo Pfoertner, PARI program for A307014 and A307016.

Cf. A004016, A035019, A038590, A307014, A307017.

\\ See Link
\\ To create the data of this sequence load program from file and call
a307014_16(5, 6) \\ Hugo Pfoertner, Nov 07 2023

A307017 List coordinates (x,y) of the points in an hexagonal grid, sorted first by radial coordinate r and in case of ties, by polar angle 0 <= phi < 2*Pi in a polar coordinate system. Sequence gives the third coordinate in a barycentric coordinate system.

Original entry on oeis.org

0, 1, 1, 0, -1, -1, 0, 2, 1, -1, -2, -1, 1, 2, 2, 0, -2, -2, 0, 3, 3, 2, 1, -1, -2, -3, -3, -2, -1, 1, 2, 3, 3, 0, -3, -3, 0, 4, 2, -2, -4, -2, 2, 4, 4, 3, 1, -1, -3, -4, -4, -3, -1, 1, 3, 4, 4, 0, -4, -4, 0, 5, 5, 3, 2, -2, -3, -5, -5, -3, -2, 2, 3, 5, 5, 4, 1

Offset: 0

Hugo Pfoertner, Mar 21 2019

Cartesian coordinates (x,y) of the grid points are converted to barycentric coordinates (i,j,k) by i = x - y/sqrt(3), j = 2*y/sqrt(3), k = x + y/sqrt(3). The sequence gives k. i is given in A307014, j is given in A307016.

Hugo Pfoertner, Table of n, a(n) for n = 0..9060, covering range r <= 50.

Cf. A004016, A035019, A038590, A307014, A307016.

A038589 Sizes of successive clusters in hexagonal lattice A_2 centered at lattice point.

Original entry on oeis.org

1, 7, 7, 13, 19, 19, 19, 31, 31, 37, 37, 37, 43, 55, 55, 55, 61, 61, 61, 73, 73, 85, 85, 85, 85, 91, 91, 97, 109, 109, 109, 121, 121, 121, 121, 121, 127, 139, 139, 151, 151, 151, 151, 163, 163, 163, 163, 163, 169, 187, 187, 187, 199, 199, 199

Offset: 0

N. J. A. Sloane

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.

			1 + 7*x + 7*x^2 + 13*x^3 + 19*x^4 + 19*x^5 + 19*x^6 + 31*x^7 + 31*x^8 + 37*x^9 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Benoit Cloitre, On the circle and divisor problems.
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2

Cf. A004016, A014201, A038589, A038590.
Cf. A035019.
Cf. A057655 (for square lattice).

a[n_] := 1 + Sum[ Length[ {ToRules[ Reduce[ x^2 + x*y + y^2 == k, {x, y}, Integers] ]}], {k, 1, n}]; Table[a[n], {n, 0, 54}] (* Jean-François Alcover, Feb 23 2012, after Neven Juric *)

a(n)=1+6*sum(k=0,n\3,(n\(3*k+1))-(n\(3*k+2)))

Partial sums of A004016.
Expansion of a(x) / (1 - x) in powers of x where a() is a cubic AGM theta function (cf. A004016). - Michael Somos, Aug 21 2012
Equals 1 + A014201(n). - Neven Juric, May 10 2010
a(n) = 1 + 6*Sum_{k=1..n/3} floor(n/(3k+1)) - floor(n/(3k+2)). a(n) is asymptotic to 2*(Pi/sqrt(3))*n. Conjecture: a(n) = 2*(Pi/sqrt(3))*n + O(n^(1/4 + epsilon)) as for the Gauss circle or Dirichlet divisor problems. - Benoit Cloitre, Oct 27 2012
a(n) = A014201(n) + 1. - Hugo Pfoertner, Nov 09 2023

A038162 Numbers missing from A038161.

Original entry on oeis.org

4, 8, 11, 13, 17, 19, 22, 24, 26, 29, 30, 32, 34, 36, 38, 41, 43, 46, 48, 51, 53, 54, 55, 57, 59, 62, 65, 67, 69, 71, 74, 76, 78, 80, 81, 82, 84, 87, 88, 90, 92, 94, 96, 98, 100, 101, 103, 105, 107, 109, 111, 113, 116, 118, 119, 121, 123, 125, 127

Offset: 1

N. J. A. Sloane

nonn

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A035019, A038161, A038590.

A038591 Sizes of clusters in hexagonal lattice A_2 with 3-fold symmetry.

Original entry on oeis.org

1, 3, 6, 7, 12, 13, 18, 19, 21, 27, 30, 31, 36, 37, 42, 43, 48, 54, 55, 61, 63, 69, 73, 75, 78, 84, 85, 90, 91, 96, 97, 102, 109, 114, 120, 121, 123, 127, 129, 135, 139, 141, 144, 150, 151, 156, 163, 168, 169, 174, 180

Offset: 0

N. J. A. Sloane

nonn

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.

G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2

Union of A038588 and A038590.

A218146 The number of centered circles that can form hexagonal symmetry. Contains all hexagonal numbers.

Original entry on oeis.org

1, 7, 13, 19, 31, 37, 43, 55, 61, 91, 97, 109, 121, 127, 139, 151, 163, 169, 217, 229, 241, 253, 265, 271, 283, 295, 307, 319, 331, 343, 355, 367, 379, 391, 397

Offset: 1

Jason Betts, Oct 21 2012

nonn

B(n) contain all of the hexagonal numbers such that: 1. Bn(m) = H(n), each subsequence always starts with a hexagonal number; 2. Bn(m) < H(n+1), all values of Bn(m) are less than the next hexagonal number; 3. If n=odd, the number of terms of Bn(m), m = { (n-1)/2 + 1}; 4. If n=even, the number of terms of Bn(m), m = { (n/2) + 1}.
All items describe a radial symmetrical geometric shape around a central circle.
All items are symmetrically reflective and are never non-trigional in geometry.
All hexagonal numbers are contained within this set of Radial Numbers, B(n).
The first term is 1, which is also the first Pythagorean Square and Triangular number. However, as the set B(n) must contain a central circle, this precludes most Pythagorean Triangular Numbers as they do not have a central circle, with 91 being the next one after 1.
Start with n=1, where n is the number of circles from the radius to the circumference of the figure and is also the number of circles along the side of the complete hexagonal figure, having the total hexagonal number of circles.
If n=odd, then the number of terms from the last hexagonal number to the next hexagonal number are {((n-1)/2) +1}, by adding 2 circles to the center of each side, symmetrically from middle outwards, for each term (12 circles), except for the last term which will always be 1 circle that fills in the corners to make the complete hexagonal geometry (6 circles). Thus { (m-1)x12 + 6 }
If n=even, then the number of terms from the last hexagonal number to the next hexagonal number are {(n/2) +1}, by adding 1 circle to the middle of each side as the first term (6 circles), then add 2 circles to each side for each consequent term (12 circles), except for the last term which will always be 1 circle that fills in the corners to make the complete hexagonal geometry (6 circles). Thus {6 + (m-2)x12 + 6} = { (m-1) x 12 }, for all m>2.
The definition is not clear to me, but it sounds like this should be the same as A038590 (the partial sums of A035019). See also A004016. - N. J. A. Sloane, Dec 08 2012

			First terms for Bn, where n denotes the number of circles of the radius, including the center:
B1(1) = H(1) = 1.
B1(2) = B(1) + 6 = 7, as n=odd, m=1 term, the only term is 6; H(2)=7.
B2(3) = B(2) + 6 = 13, as n=even, number of terms m=2, the first term is 6.
B2(4) = B(3) + 6 = 19, as n=even, number of terms m=2, last term is 6; H(3)=19.
B3(5) = B(4) + 12 = 31, as n=odd, number of terms m=2, first term is 12.
B3(6) = B(5) + 6 = 37, as n=odd, number of terms m=2, last term is 6; H(4)=37.
B4(7) = B(6) + 6 = 43, as n=even, number of terms m=3, first term is 6.
B4(8) = B(7) + 12 = 55, as n=even, number of terms m=3, second term is 12.
B4(9) = B(8) + 6 = 61, as n=even, number of terms m=3, third term is 6; H(5)=61.
B5(10) = B(9) + 12 = 73, as n=odd, number of terms m=3, first term is 12.
B5(11) = B(10) + 12 = 85, as n=odd, number of terms m=3, second term is 12.
B5(12) = B(11) + 6 = 91, as n=odd, number of terms m=3, third term is 6; H(6)=91.

Jason Betts, Maths Experiments, Software Publications, 2005, 36-40.

Jason Betts, Illustration of initial terms

A003215 is a subsequence.

Cf. A004016, A035019, A038589, A038590. - N. J. A. Sloane, Dec 08 2012

For all hexagonal numbers H(n), there exists a sequence B(n) such that H(n) < B(n) < .. < B(n+m) < H(n+1), where m = {(n/2)+1} if n=even and m={(n-1)/2 +1} if n=odd.
Bn(m) = H(n), each subsequence always starts with a hexagonal number.
Bn(m) < H(n+1), all values of Bn(m) are less than the next hexagonal number.
If n=odd, the number of terms of Bn(m), m = { (n-1)/2 + 1}
If n=even, the number of terms of Bn(m), m = { (n/2) + 1}

cp's OEIS Frontend

A038161 (A038590-1)/6.

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A038160 Intersection of A003136 and A038590.

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A035019 Sizes of successive shells in hexagonal (or A_2) lattice.

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A307014 List coordinates (x,y) of the points in an hexagonal grid, sorted first by radial coordinate r and in case of ties, by polar angle 0 <= phi < 2*Pi in a polar coordinate system. Sequence gives the first coordinate in a barycentric coordinate system.

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PARI

A307016 List coordinates (x,y) of the points in an hexagonal grid, sorted first by radial coordinate r and in case of ties, by polar angle 0 <= phi < 2*Pi in a polar coordinate system. Sequence gives the second coordinate in a barycentric coordinate system.

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Programs

PARI

A307017 List coordinates (x,y) of the points in an hexagonal grid, sorted first by radial coordinate r and in case of ties, by polar angle 0 <= phi < 2*Pi in a polar coordinate system. Sequence gives the third coordinate in a barycentric coordinate system.

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A038589 Sizes of successive clusters in hexagonal lattice A_2 centered at lattice point.

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Mathematica

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A038162 Numbers missing from A038161.

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A038591 Sizes of clusters in hexagonal lattice A_2 with 3-fold symmetry.

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A218146 The number of centered circles that can form hexagonal symmetry. Contains all hexagonal numbers.

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