A081272 -id:A081272 - cp's OEIS frontend

A038764 a(n) = (9n^2 + 3n + 2)/2.

1, 7, 22, 46, 79, 121, 172, 232, 301, 379, 466, 562, 667, 781, 904, 1036, 1177, 1327, 1486, 1654, 1831, 2017, 2212, 2416, 2629, 2851, 3082, 3322, 3571, 3829, 4096, 4372, 4657, 4951, 5254, 5566, 5887, 6217, 6556, 6904, 7261, 7627, 8002, 8386, 8779, 9181

Offset: 0

N. J. A. Sloane, May 03 2000

Coefficients of x^2 of certain rook polynomials (for n>=1; see p. 18 of the Riordan paper). - Emeric Deutsch, Mar 08 2004

a(n) is also the least weight of self-conjugate partitions having n+1 different parts such that each part is congruent to 1 modulo 3. The first such self-conjugate partitions, corresponding to a(n) = 0, 1, 2, 3, are 1, 4+3, 7+4+4+4+3, 10+7+7+7+4+4+4+3. - Augustine O. Munagi, Dec 18 2008

J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23.

Colin Barker, Table of n, a(n) for n = 0..1000
S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, Unbranched catacondensed polygonal systems containing hexagons and tetragons, Croatica Chem. Acta, 69 (1996), 757-774.
A. O. Munagi, Pairing conjugate partitions by residue classes, Discrete Math., 308 (2008), 2492-2501.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Reflection of A060544 in A081272.
Second column of A024462. Also = A064641(n+1, 2).
Shallow diagonal of triangular spiral in A051682.
Cf. A027468, A080855, A283394.
Partial sums of A122709.

LinearRecurrence[{3, -3, 1}, {1, 7, 22}, 50] (* Paolo Xausa, Jul 03 2025 *)

a(n)=n*(9*n+3)/2+1 \\ Charles R Greathouse IV, Jun 17 2017

Vec((1 + 2*x)^2 / (1 - x)^3 + O(x^60)) \\ Colin Barker, Jan 22 2018

a = lambda n: hypergeometric([-n, -2], [1], 3)
print([simplify(a(n)) for n in range(46)]) # Peter Luschny, Nov 19 2014

a(n) = binomial(n,0) + 6*binomial(n,1) + 9*binomial(n,2).
From Paul Barry, Mar 15 2003: (Start)
G.f.: (1 + 2*x)^2/(1 - x)^3.
Binomial transform of (1, 6, 9, 0, 0, 0, ...). (End)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2. -  Colin Barker, Jan 22 2018
a(n) = a(n-1) + 3*(3*n-1) for n>0, a(0)=1. - Vincenzo Librandi, Nov 17 2010
a(n) = hypergeometric([-n, -2], [1], 3). - Peter Luschny, Nov 19 2014
E.g.f.: exp(x)*(2 + 12*x + 9*x^2)/2. - Stefano Spezia, Mar 07 2023

More terms from James Sellers, May 03 2000

Entry revised by N. J. A. Sloane, Jan 23 2018

A198392 a(n) = (6n(3n+7)+(2n+13)*(-1)^n+3)/16 + 1.

Original entry on oeis.org

2, 4, 12, 18, 31, 41, 59, 73, 96, 114, 142, 164, 197, 223, 261, 291, 334, 368, 416, 454, 507, 549, 607, 653, 716, 766, 834, 888, 961, 1019, 1097, 1159, 1242, 1308, 1396, 1466, 1559, 1633, 1731, 1809, 1912, 1994, 2102, 2188, 2301, 2391, 2509, 2603, 2726, 2824, 2952

Offset: 0

Bruno Berselli, Oct 25 2011

For an origin of this sequence, see the triangular spiral illustrated in the Links section.

First bisection gives A117625 (without the initial term).

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, Illustration of initial terms.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A152832 (by Superseeker).

Cf. sequences related to the triangular spiral: A022266, A022267, A027468, A038764, A045946, A051682, A062708, A062725, A062728, A062741, A064225, A064226, A081266-A081268, A081270-A081272, A081275 [incomplete list].

[(6*n*(3*n+7)+(2*n+13)*(-1)^n+3)/16+1: n in [0..50]];

LinearRecurrence[{1,2,-2,-1,1},{2,4,12,18,31},60] (* Harvey P. Dale, Jun 15 2022 *)

for(n=0, 50, print1((6*n*(3*n+7)+(2*n+13)*(-1)^n+3)/16+1", "));

G.f.: (2+2*x+4*x^2+2*x^3-x^4)/((1+x)^2*(1-x)^3).
a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5).
a(n)-a(-n-1) = A168329(n+1).
a(n)+a(n-1) = A102214(n).
a(2n)-a(2n-1) = A016885(n).
a(2n+1)-a(2n) = A016825(n).

A363080 Number of hexagonal lattice points within a hexagram centered at a lattice point and with outermost vertices at the six lattice points n steps outward from the central point.

Original entry on oeis.org

1, 7, 13, 25, 43, 61, 85, 115, 145, 181, 223, 265, 313, 367, 421, 481, 547, 613, 685, 763, 841, 925, 1015, 1105, 1201, 1303, 1405, 1513, 1627, 1741, 1861, 1987, 2113, 2245, 2383, 2521, 2665, 2815, 2965, 3121, 3283, 3445, 3613, 3787, 3961, 4141, 4327, 4513, 4705, 4903, 5101, 5305, 5515, 5725

Offset: 0

Aaron David Fairbanks, May 17 2023

In contrast, A003154 (the star numbers) counts the hexagonal lattice points within a hexagram centered at a lattice point and with the vertices of the central hexagon at the six lattice points a given number of steps outward from the central point.
Besides the first term, the first differences are given by six times A004396.
A005448 (the centered triangular numbers) counts just the lattice points within one of the two triangles that make up the hexagram.

			Illustration of initial terms:
.
.                                 o     o
.                   o   o          o o o
.         o o        o o          o o o o
.   o    o o o    o o o o o    o o o o o o o
.         o o        o o          o o o o
.                   o   o          o o o
.                                 o     o
.
.   1      7          13             25
.

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

Cf. A003154, A004396, A005448, A007980, A081272.

Table[6*Ceiling[n*(n + 1)/3] + 1, {n, 0, 60}] (* Amiram Eldar, Jul 28 2023 *)

a(n) = 6*ceil(n*(n+1)/3) + 1; \\ Michel Marcus, Jun 14 2024

a(n) = 6*ceiling(n*(n+1)/3) + 1.
a(n) = 6*A007980(n-1) + 1 for n >= 1.
a(n+1) - a(n) = 6*A004396(n+1).
a(3n) = A081272(n).
G.f.: (1 + 5*x + 5*x^3 + x^4)/((1 - x)^3*(1 + x + x^2)). - Stefano Spezia, Feb 06 2025

cp's OEIS Frontend

A038764 a(n) = (9n^2 + 3n + 2)/2.

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A198392 a(n) = (6n(3n+7)+(2n+13)*(-1)^n+3)/16 + 1.

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A363080 Number of hexagonal lattice points within a hexagram centered at a lattice point and with outermost vertices at the six lattice points n steps outward from the central point.

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cp's OEIS Frontend

A038764 a(n) = (9*n^2 + 3*n + 2)/2.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Sage

Formula

Extensions

A198392 a(n) = (6*n*(3*n+7)+(2*n+13)*(-1)^n+3)/16 + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A363080 Number of hexagonal lattice points within a hexagram centered at a lattice point and with outermost vertices at the six lattice points n steps outward from the central point.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A038764 a(n) = (9n^2 + 3n + 2)/2.

A198392 a(n) = (6n(3n+7)+(2n+13)*(-1)^n+3)/16 + 1.