A035019 -id:A035019 - cp's OEIS frontend

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

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

A055666 Number of inequivalent Eisenstein-Jacobi primes of successive norms (indexed by A055664).

Original entry on oeis.org

1, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

N. J. A. Sloane, Jun 09 2000

These are the primes in the ring of integers a+b*omega, a and b rational integers, omega = (1+sqrt(-3))/2.

Two primes are considered equivalent if they differ by multiplication by a unit (+-1, +-omega, +-omega^2).

			There are 6 Eisenstein-Jacobi primes of norm 3, omega-omega^2 times one of the 6 units [ +-1, +-omega, +-omega^2 ] but only one up to equivalence.

R. K. Guy, Unsolved Problems in Number Theory, A16.
L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910, see Chap. VI.

Jean-François Alcover, Table of n, a(n) for n = 1..1000

Cf. A055664-A055668, A055025-A055029. See A004016 and A035019 for theta series of Eisenstein (or hexagonal) lattice.

norms = Join[{3}, Select[Range[2000], (PrimeQ[#] && Mod[#, 6] == 1) || (PrimeQ[Sqrt[#]] && Mod[Sqrt[#], 3] == 2) &]]; r[n_] := Length[Reduce[n == a^2 - a*b + b^2, {a, b}, Integers]]/6; A055666 = r /@ norms (* Jean-François Alcover, Oct 24 2013 *)

More terms from Franklin T. Adams-Watters, May 05 2006

A055665 Number of Eisenstein-Jacobi primes of successive norms (indexed by A055664).

Original entry on oeis.org

6, 6, 12, 12, 12, 6, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 6, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 6, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 6, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12

Offset: 1

N. J. A. Sloane, Jun 09 2000

These are the primes in the ring of integers a+b*omega, a and b rational integers, omega = (1+sqrt(-3))/2.

			There are 6 Eisenstein-Jacobi primes of norm 3, omega-omega^2 times one of the 6 units [ +-1, +-omega, +-omega^2 ] but only one up to equivalence.

R. K. Guy, Unsolved Problems in Number Theory, A16.
L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910, see Chap. VI.

Antti Karttunen, Table of n, a(n) for n = 1..1000 (computed from the b-file of A055666 with the formula of _Franklin T. Adams-Watters_)

Cf. A055664-A055668, A055025-A055029. See A004016 and A035019 for theta series of Eisenstein (or hexagonal) lattice.

norms = Join[{3}, Select[Range[1000], (PrimeQ[#] && Mod[#, 6] == 1) || (PrimeQ[Sqrt[#]] && Mod[Sqrt[#], 3] == 2) &]]; r[n_] := Reduce[n == a^2 - a*b + b^2, {a, b}, Integers] // Length; A055665 = r /@ norms (* Jean-François Alcover, Oct 24 2013 *)

a(n) = 6 * A055666(n) - Franklin T. Adams-Watters, May 05 2006

More terms from Franklin T. Adams-Watters, May 05 2006

A038161 (A038590-1)/6.

Original entry on oeis.org

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.

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.

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}

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

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A055666 Number of inequivalent Eisenstein-Jacobi primes of successive norms (indexed by A055664).

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A055665 Number of Eisenstein-Jacobi primes of successive norms (indexed by A055664).

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A038161 (A038590-1)/6.

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A038162 Numbers missing from A038161.

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

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