A152746 -id:A152746 - cp's OEIS frontend

A164016 6 times centered hexagonal numbers: 18n(n+1) + 6.

6, 42, 114, 222, 366, 546, 762, 1014, 1302, 1626, 1986, 2382, 2814, 3282, 3786, 4326, 4902, 5514, 6162, 6846, 7566, 8322, 9114, 9942, 10806, 11706, 12642, 13614, 14622, 15666, 16746, 17862, 19014, 20202, 21426, 22686, 23982, 25314

Offset: 0

Omar E. Pol, Nov 07 2009

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

Cf. A003215, A152746, A164013, A108099, A164015.

Table[18n(n+1)+6,{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{6,42,114},40] (* Harvey P. Dale, Dec 16 2012 *)

a(n)=18*n*(n+1)+6 \\ Charles R Greathouse IV, Jul 17 2011

a(n) = A003215(n)*6.
a(n) = a(n-1) + 36*n (with a(0)=6). - Vincenzo Librandi, Nov 30 2010
a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) with a(0)=6, a(1)=42, a(2)=114. - Harvey P. Dale, Dec 16 2012
From G. C. Greubel, Sep 07 2017: (Start)
G.f.: 6*(1 + 4*x + x^2)/(1 - x)^3.
E.g.f.: 6*(1 + 6*x + 3*x^2)*exp(x). (End)

A331952 a(n) = (-7 + (-1)^(1+n) + 6*n^2) / 8.

Original entry on oeis.org

-1, 0, 2, 6, 11, 18, 26, 36, 47, 60, 74, 90, 107, 126, 146, 168, 191, 216, 242, 270, 299, 330, 362, 396, 431, 468, 506, 546, 587, 630, 674, 720, 767, 816, 866, 918, 971, 1026, 1082, 1140, 1199, 1260, 1322, 1386, 1451, 1518, 1586, 1656, 1727, 1800, 1874, 1950, 2027

Offset: 0

Paul Curtz, Feb 02 2020

a(n+1) is once in the hexagonal spiral in A330707. a(n+2) is twice in the same spiral.
a(n) has one odd followed by three evens.
Difference table:
  -1, 0, 2, 6, 11, 18, 26, 36, ... = a(n)
   1, 2, 4, 5,  7,  8, 10, 11, ... = A001651(n+1)
   1, 2, 1, 2,  1,  2,  1,  2, ... = A000034.

			G.f. = -1 + 2*x^2 + 6*x^3 + 11*x^4 + 18*x^5 + 26*x^6 + 36*x^7 + 47*x^8 + ... - _Michael Somos_, Sep 08 2023

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).
Index entries for two-way infinite sequences

Cf. A000034, A001651, A158463, 6*A014105, 2*A003154, A152746(n+1), A008585(1+n).

Equals 2 less than A084684, 1 less than A077043, and 1 more than A276382(n-1). - Greg Dresden, Feb 22 2020

a:=[-1,0,2,6]; [n le 4 select a[n] else 2*Self(n-1)-2*Self(n-3)+Self(n-4): n in [1..45]]; // Marius A. Burtea, Feb 02 2020

LinearRecurrence[{2, 0, -2, 1}, {-1, 0, 2, 6}, 100] (* Amiram Eldar, Feb 02 2020 *)
a[n_] := Floor[(n^2 - 1)*3/4]; (* Michael Somos, Sep 08 2023 *)

Vec(-(1 - 2*x - 2*x^2) / ((1 - x)^3*(1 + x)) + O(x^40)) \\ Colin Barker, Feb 03 2020

{a(n) = (n^2 - 1)*3\4}; /* Michael Somos, Sep 08 2023 */

a(-n) = a(n).
a(20+n) - a(n) = 30*(10+n).
a(2+n) = a(n) + 3*(1+n), a(0)=-1 and a(1)=0.
a(4*n) = 12*n^2 - 1, a(1+4*n) = 6*n*(1+2*n), a(2+4*n) = 2 + 12*n*(1+n), a(3+4*n) = 6*(1+n)*(1+2*n) for n>= 0.
From Colin Barker, Feb 02 2020: (Start)
G.f.: -(1 - 2*x - 2*x^2) / ((1 - x)^3*(1 + x)).
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n>3.
a(n) = (-7 + (-1)^(1+n) + 6*n^2) / 8.
(End)
E.g.f.: (1/8)*(exp(x)*(6*x^2 + 6*x - 7) - exp(-x)). - Stefano Spezia, Feb 02 2020 after Colin Barker
a(n) = floor((n^2 - 1)*3/4). - Michael Somos, Sep 09 2023

a(42)-a(52) from Stefano Spezia, Feb 02 2020

A386477 a(0) = 1; thereafter a(n) = 2(6n^2 - 3*n + 1).

Original entry on oeis.org

1, 8, 38, 92, 170, 272, 398, 548, 722, 920, 1142, 1388, 1658, 1952, 2270, 2612, 2978, 3368, 3782, 4220, 4682, 5168, 5678, 6212, 6770, 7352, 7958, 8588, 9242, 9920, 10622, 11348, 12098, 12872, 13670, 14492, 15338, 16208, 17102, 18020, 18962, 19928, 20918, 21932, 22970, 24032, 25118, 26228, 27362, 28520, 29702, 30908, 32138, 33392, 34670

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Jul 22 2025

Definition: A regular hexagram of radius R is formed by placing six equally-spaced points P_0 .. P_5 around the boundary of a circle of radius R, and drawing line segments P_0 - P_2 - P_4 - P_0 and P_1 - P_3 - P_5 - P_1.
Theorem 1: a(n) is the maximum number of regions that can be formed in the plane by drawing n regular hexagrams with the same radius and the same center.
Conjecture 2: a(n) is the maximum number of regions that can be formed in the plane by drawing n regular hexagrams with any radii and any centers.
The following construction works for any n >= 1. Take 6*n equally-spaced points P_i around a circle, and draw hexagrams starting at P_i for i = 0, ..., n-1.
The resulting planar graph decomposes into 6*n triangular cells each with 2*n-1 cells (see the red triangle in the "Three pentagons" illustration), plus the interior and exterior regions, for a total of 12*n^2 - 6*n + 2 regions. There are 12*n^2 vertices, for n>0.

Paolo Xausa, Table of n, a(n) for n = 0..10000
Scott R. Shannon, Illustration for a(1) = 8 [Note that the cell counts shown on these five figures do not include the black exterior region, so the totals are off by 1]
Scott R. Shannon, Illustration for a(2) = 38
Scott R. Shannon, Illustration for a(3) = 92
Scott R. Shannon, Illustration for a(8) = 722
Scott R. Shannon, Illustration for a(10) = 1142
N. J. A. Sloane, Sketch to illustrate a(2) = 38. The two hexagrams are colored red and black, respectively.
N. J. A. Sloane, Sketch to illustrate a(3) = 92. The three hexagrams are colored red, blue, and black, respectively.
N. J. A. Sloane, Analogous illustration for three pentagrams
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A094159, A152746.

See A077588, A069894, and A383466 for analogous sequences based on triangles, squares, and pentagrams.

A386477[n_] := If[n == 0, 1, 6*n*(2*n - 1) + 2]; Array[A386477, 50, 0] (* or *)
Join[{1}, 6*PolygonalNumber[6, Range[49]] + 2] (* or *)
LinearRecurrence[{3, -3, 1}, {1, 8, 38, 92}, 50] (* Paolo Xausa, Jul 24 2025 *)

From Stefano Spezia, Jul 23 2025: (Start)
G.f.: (1 + 5*x + 17*x^2 + x^3)/(1 - x)^3.
E.g.f.: 2*exp(x)*(1 + 3*x + 6*x^2) - 1. (End)
a(n) = A152746(n) + 2, for n >= 1. - Paolo Xausa, Jul 24 2025

A277980 a(n) = 12n^2 + 18n.

Original entry on oeis.org

0, 30, 84, 162, 264, 390, 540, 714, 912, 1134, 1380, 1650, 1944, 2262, 2604, 2970, 3360, 3774, 4212, 4674, 5160, 5670, 6204, 6762, 7344, 7950, 8580, 9234, 9912, 10614, 11340, 12090, 12864, 13662, 14484, 15330, 16200, 17094, 18012, 18954, 19920

Offset: 0

Emeric Deutsch, Nov 08 2016

For n>=3, a(n) is the second Zagreb index of the double-wheel graph DW[n]. The second Zagreb index of a simple connected graph g is the sum of the degree products d(i) d(j) over all edges ij of g.
The double-wheel graph DW[n] consists of two cycles C[n], whose vertices are connected to an additional vertex.
The M-polynomial of the double-wheel graph DW[n] is M(DW[n],x,y) = 2*n*x^3*y^3 + 2*n*x^3*y^{2*n}.

			a(3) = 162. Indeed, the double-wheel graph DW[3] has 6 edges with end-point degrees 3,3 and 6 edges with end-point degrees 3,6. Then the second Zagreb index is 6*9 + 6*18 = 162.

E. Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A014106, A152746, A154105, A277979.
First bisection of A277978.
After 0, subsequence of A255265.

[12*n^2+18*n: n in [0..40]]; // Vincenzo Librandi, Nov 09 2016

Maple
```
seq(12*n^2+18*n, n = 0 .. 50);
```

Table[12 n^2 + 18 n, {n, 0, 45}] (* Vincenzo Librandi, Nov 09 2016 *)

a(n)=12*n^2+18*n \\ Charles R Greathouse IV, Nov 09 2016

G.f.: 6*x*(5-x)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 6*A014106(n).
a(n) = A152746(n+1) - 6 = A154105(n) - 7. - Omar E. Pol, May 08 2018

cp's OEIS Frontend

A164016 6 times centered hexagonal numbers: 18*n*(n+1) + 6.

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A331952 a(n) = (-7 + (-1)^(1+n) + 6*n^2) / 8.

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A386477 a(0) = 1; thereafter a(n) = 2*(6*n^2 - 3*n + 1).

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A277980 a(n) = 12*n^2 + 18*n.

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Author

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Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A164016 6 times centered hexagonal numbers: 18n(n+1) + 6.

A386477 a(0) = 1; thereafter a(n) = 2(6n^2 - 3*n + 1).

A277980 a(n) = 12n^2 + 18n.