A033581 -id:A033581 - cp's OEIS frontend

A016743 Even cubes: a(n) = (2*n)^3.

0, 8, 64, 216, 512, 1000, 1728, 2744, 4096, 5832, 8000, 10648, 13824, 17576, 21952, 27000, 32768, 39304, 46656, 54872, 64000, 74088, 85184, 97336, 110592, 125000, 140608, 157464, 175616, 195112, 216000, 238328, 262144, 287496, 314432

Offset: 0

N. J. A. Sloane

a(n) is also the number of non-degenerate triangles that can be drawn with vertices on a cross with n points on each branch. - James P. B. Hall, Nov 22 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Hilko Koning, 216 neodymium magnets for n=3.
Ana Rechtman, Mars 2022, 1er défi, Images des Mathématiques, CNRS, 2022 (in French).
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Even bisection of A000578, cf. A016755.

Cf. A016803 (even bisection), A016827 (odd bisection), A033581, A276712.

[(2*n)^3: n in [0..50]]; // Vincenzo Librandi, Sep 05 2011

A016743:=n->(2*n)^3: seq(A016743(n), n=0..50); # Wesley Ivan Hurt, Sep 15 2018

Range[0, 78, 2]^3 (* Alonso del Arte, Apr 06 2013 *)

a(n) = 8*n^3; \\ Joerg Arndt, Apr 07 2013

a(n) = (2*n)^3 = 8*n^3.
G.f.: x*(8+32*x+8*x^2)/(1-4*x+6*x^2-4*x^3+x^4). - Colin Barker, Jan 02 2012
E.g.f.: 8*x*(1 +3*x +x^2)*exp(x). - G. C. Greubel, Sep 15 2018
From Amiram Eldar, Oct 10 2020: (Start)
Sum_{n>=1} 1/a(n) = zeta(3)/8 (A276712).
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*zeta(3)/32. (End)

A045945 Hexagonal matchstick numbers: a(n) = 3n(3*n+1).

Original entry on oeis.org

0, 12, 42, 90, 156, 240, 342, 462, 600, 756, 930, 1122, 1332, 1560, 1806, 2070, 2352, 2652, 2970, 3306, 3660, 4032, 4422, 4830, 5256, 5700, 6162, 6642, 7140, 7656, 8190, 8742, 9312, 9900, 10506, 11130, 11772, 12432, 13110, 13806, 14520, 15252, 16002, 16770, 17556

Offset: 0

R. K. Guy

This may also be construed as the number of line segments illustrating the isometric projection of a cube of side length n. Moreover, a(n) equals the number of rods making a cube of side length n+1 minus the number of rods making a cube of side length n. See the illustration in the links and formula below.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Peter M. Chema, Illustration of initial terms as the first difference of number of rods required to make a 3-D cube.
Craig Knecht, Number of positions a frame shifted H1 hexagon can occupy in a hexagon of order n.
Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A000290, A005449, A033580, A059986.

The hexagon matchstick sequences are: Number of matchsticks: this sequence; size=1 triangles: A033581; larger triangles: A307253; total triangles: A045949. Analog for triangles: A045943; analog for stars: A045946. - John King, Apr 05 2019

a:= n-> 3*n*(3*n+1): seq(a(n), n=0..42); # Zerinvary Lajos, May 03 2007

f[n_]:=3*n*(3*n+1);f[Range[0,60]] (* Vladimir Joseph Stephan Orlovsky, Feb 05 2011 *)

a(n) = 3*n*(3*n+1) \\ Charles R Greathouse IV, Feb 27 2017

def a(n): return 3*n*(3*n+1) # Indranil Ghosh, Mar 26 2017

a(n) = a(n-1) + 6*(3*n-1) (with a(0)=0). - Vincenzo Librandi, Nov 18 2010
G.f.: 6*x*(2+x)/(1-x)^3. - Colin Barker, Feb 12 2012
a(n) = 6*A005449(n). - R. J. Mathar, Feb 13 2016
a(n) = A059986(n) - A059986(n-1). - Peter M. Chema, Feb 26 2017
a(n) = 6*(A000217(n) + A000290(n)). - Peter M. Chema, Mar 26 2017
From Amiram Eldar, Jan 14 2021: (Start)
Sum_{n>=1} 1/a(n) = 1 - Pi/(6*sqrt(3)) - log(3)/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = -1 + Pi/(3*sqrt(3)) + 2*log(2)/3. (End)
From Elmo R. Oliveira, Dec 12 2024: (Start)
E.g.f.: 3*exp(x)*x*(4 + 3*x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3. (End)

A061167 a(n) = n^5 - n.

Original entry on oeis.org

0, 0, 30, 240, 1020, 3120, 7770, 16800, 32760, 59040, 99990, 161040, 248820, 371280, 537810, 759360, 1048560, 1419840, 1889550, 2476080, 3199980, 4084080, 5153610, 6436320, 7962600, 9765600, 11881350, 14348880, 17210340, 20511120, 24299970, 28629120

Offset: 0

Henry Bottomley, Apr 18 2001

(b^2+c^2)/(bc+1) is an integer if {b,c} are of the form {0,n}, {n,n^3}, {n^3,n^5-n}, {n^5-n,n^7-2n^3}, {n^7-2n^3,n^9-3n^5+n}, etc. for some n, in which case the division results in n^2. Cf. A052530.

Convolution of A033429 by A033581. - R. J. Mathar, Aug 19 2008

			a(2) = 32 - 2 = 30.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
D. Zagier, Problems posed at the St Andrews Colloquium, 1996
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Subsequence of A249674.

Cf. A000584, A024002, A033429, A033455, A033581, A052530.

[n^5-n: n in [0..40]]; // Vincenzo Librandi, May 02 2011

Table[n^5 - n, {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2012 *)

a(n) = 30*A033455(n-1). [Corrected by Bernard Schott, Mar 16 2021]
a(n) = -n*A024002(n).
a(n) = A000584(n) - n.
O.g.f.: 30x^2(1+x)^2/(1-x)^6. - R. J. Mathar, Aug 19 2008
a(n) = n * (n-1) * (n+1) * (n^2+1). - Bernard Schott, Mar 16 2021
E.g.f.: exp(x)*x^2*(15 + 25*x + 10*x^2 + x^3). - Stefano Spezia, Dec 27 2021

A195321 a(n) = 18*n^2.

Original entry on oeis.org

0, 18, 72, 162, 288, 450, 648, 882, 1152, 1458, 1800, 2178, 2592, 3042, 3528, 4050, 4608, 5202, 5832, 6498, 7200, 7938, 8712, 9522, 10368, 11250, 12168, 13122, 14112, 15138, 16200, 17298, 18432, 19602, 20808, 22050, 23328, 24642, 25992, 27378, 28800, 30258, 31752

Offset: 0

Omar E. Pol, Sep 16 2011

Sequence found by reading the line from 0, in the direction 0, 18, ..., in the square spiral whose vertices are the generalized hendecagonal numbers A195160. Semi-axis opposite to A195316 in the same spiral.
Area of a square with diagonal 6n. - Wesley Ivan Hurt, Jun 19 2014
Number of identical tessellation tiles that are composed of 48 equilateral edge joined triangles that can be formed into a order n hexagon. The example tiles shown in the link below are tessellated with eight sphinx tiles. See A291582. - Craig Knecht, Sep 02 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Craig Knecht, Hexagon tessellation.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Bisection of A195147.
Cf. A016802, A033581, A033583, A135453, A139098, A144555, A195322, A195323.
Cf. A000290, A001105, A008600, A016766, A033428, A195160, A195316, A291582.

[18*n^2:n in [0..40]]; // Vincenzo Librandi, Sep 20 2011

A195321:=n->18*n^2; seq(A195321(n), n=0..50); # Wesley Ivan Hurt, Jun 19 2014

18 Range[0, 50]^2 (* or *) CoefficientList[Series[18 x*(1 + x)/(1 - x)^3, {x, 0, 30}], x] (* Wesley Ivan Hurt, Jun 20 2014 *)
LinearRecurrence[{3,-3,1},{0,18,72},50] (* Harvey P. Dale, Mar 26 2023 *)

a(n)=18*n^2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 18*A000290(n) = 9*A001105(n) = 6*A033428(n) = 3*A033581(n) = 2*A016766(n).
G.f.: 18*x*(1+x)/(1-x)^3. - Wesley Ivan Hurt, Jun 20 2014
From Elmo R. Oliveira, Dec 01 2024: (Start)
E.g.f.: 18*x*(1 + x)*exp(x).
a(n) = n*A008600(n) = A195147(2*n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A195322 a(n) = 20*n^2.

Original entry on oeis.org

0, 20, 80, 180, 320, 500, 720, 980, 1280, 1620, 2000, 2420, 2880, 3380, 3920, 4500, 5120, 5780, 6480, 7220, 8000, 8820, 9680, 10580, 11520, 12500, 13520, 14580, 15680, 16820, 18000, 19220, 20480, 21780, 23120, 24500, 25920, 27380, 28880, 30420, 32000, 33620, 35280

Offset: 0

Omar E. Pol, Sep 16 2011

Sequence found by reading the line from 0, in the direction 0, 20, ..., in the square spiral whose vertices are the generalized dodecagonal numbers A195162. Semiaxis opposite to A195317 in the same spiral.
a(n) is the sum of all the integers less than 10*n which are not multiple of 2 or 5. a(2) = (1 + 3 + 7 + 9) + (11 + 13 + 17 + 19) = 20 + 60 = 80 = 20 * 2^2. (Link Crux Mathematicorum). - Bernard Schott, May 15 2017
Number of terms less than 10^k (k=0, 1, 2, ...): 1, 1, 3, 8, 23, 71, 224, 708, 2237, 7072, 22361, 70711, ... - Muniru A Asiru, Feb 01 2018

			From _Muniru A Asiru_, Feb 01 2018: (Start)
n=0, a(0) = 20*0^2 = 0.
n=1, a(1) = 20*1^2 = 20.
n=1, a(2) = 20*2^2 = 80.
n=1, a(3) = 20*3^2 = 180.
n=1, a(4) = 20*4^2 = 320.
...
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Léo Sauvé, Problem 53, Crux Mathematicorum, Vol. 1, Nov. 1975, page 88.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Bisection of A195148.
Cf. A016802, A033581, A033583, A135453, A139098, A144555, A195321, A195323.
Cf. A000290, A001105, A005899, A010014, A016742, A033429, A195162, A195317.
Cf. A008602.

List([0..10^3],n->20*n^2); # Muniru A Asiru, Feb 01 2018

[20*n^2: n in [0..40]]; // Vincenzo Librandi, Sep 20 2011

a := n -> 20*n^2; seq(a(n), n=0..10^3); # Muniru A Asiru, Feb 01 2018

20 Range[0, 40]^2 (* or *) LinearRecurrence[{3, -3, 1}, {0, 20, 80}, 50] (* Harvey P. Dale, Jan 18 2013 *)

a(n) = 20*n^2 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 20*A000290(n) = 10*A001105(n) = 5*A016742(n) = 4*A033429(n) = 2*A033583(n).
a(0)=0, a(1)=20, a(2)=80; for n > 2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Jan 18 2013
a(n) = A010014(n) - A005899(n) for n > 0. - R. J. Cano, Sep 29 2015
From Elmo R. Oliveira, Nov 30 2024: (Start)
G.f.: 20*x*(1 + x)/(1-x)^3.
E.g.f.: 20*x*(1 + x)*exp(x).
a(n) = n*A008602(n) = A195148(2*n). (End)

A227776 a(n) = 6*n^2 + 1.

Original entry on oeis.org

1, 7, 25, 55, 97, 151, 217, 295, 385, 487, 601, 727, 865, 1015, 1177, 1351, 1537, 1735, 1945, 2167, 2401, 2647, 2905, 3175, 3457, 3751, 4057, 4375, 4705, 5047, 5401, 5767, 6145, 6535, 6937, 7351, 7777, 8215, 8665, 9127, 9601, 10087, 10585, 11095, 11617, 12151

Offset: 0

Clark Kimberling, Jul 30 2013

Least splitter is defined for x < y at A227631 as the least positive integer d such that x <= c/d < y for some integer c; the number c/d is called the least splitting rational of x and y. Conjecture: a(n) is the least splitter of s(n) and s(n+1), where s(n) = n*sin(1/n).

			The first eight least splitting rationals for {n*sin(1/n), n >=1 } are these fractions: 6/7, 24/25, 54/55, 96/97, 150/151, 216/217, 294/295, 384/385.

Clark Kimberling, Table of n, a(n) for n = 0..1000
Leo Tavares, Illustration: Hexagonal Star Rays.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A227631, A287326.

Cf. A003154, A008458, A003215, A000217, A028896, A054554, A046092, A080855, A045943, A172043, A002378, A033581.

z = 40; r[x_, y_] := Module[{c, d}, d = NestWhile[#1 + 1 &, 1, ! (c = Ceiling[#1 x - 1]) < Ceiling[#1 y] - 1 &]; (c + 1)/d]; s[n_] := s[n] = n*Sin[1/n]; t = Table[r[s[n], s[n + 1]], {n, 1, z}] (* least splitting rationals *); fd = Denominator[t] (* Peter J. C. Moses, Jul 15 2013 *)
Array[6 #^2 + 1 &, 45] (* Michael De Vlieger, Nov 08 2017 *)
LinearRecurrence[{3,-3,1},{7,25,55},50] (* Harvey P. Dale, Dec 16 2017 *)

a(n)=6*n^2+1 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: (1 + 4*x + 7*x^2)/(1 - x)^3.
a(n) = A287326(2n, n). - Kolosov Petro, Nov 06 2017
From Amiram Eldar, Jul 15 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + (Pi/sqrt(6))*coth(Pi/sqrt(6)))/2.
Sum_{n>=0} (-1)^n/a(n) = (1 + (Pi/sqrt(6))*csch(Pi/sqrt(6)))/2. (End)
From Amiram Eldar, Feb 05 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi/sqrt(6))*sinh(Pi/sqrt(3)).
Product_{n>=1} (1 - 1/a(n)) = (Pi/sqrt(6))*csch(Pi/sqrt(6)).(End)
From Leo Tavares, Nov 20 2021: (Start)
a(n) = A003154(n+1) - A008458(n). See Hexagonal Star Rays illustration.
a(n) = A003215(n) + A028896(n-1).
a(n) = A054554(n+1) + A046092(n).
a(n) = A080855(n) + A045943(n).
a(n) = A172043(n) + A002378(n).
a(n) = A033581(n) + 1. (End)
E.g.f.: exp(x)*(1 + 6*x + 6*x^2). - Stefano Spezia, Sep 14 2024

a(0) = 1 prepended by Robert P. P. McKone, Oct 09 2023

A277713 Positions of 3's in A264977; positions of 6's in A277330.

Original entry on oeis.org

3, 39, 75, 147, 207, 291, 411, 579, 819, 927, 1155, 1635, 1851, 2307, 2487, 2583, 2919, 3267, 3699, 3903, 4611, 4971, 5163, 5835, 6531, 7395, 7803, 9219, 9939, 10323, 10839, 11667, 13059, 14787, 15603, 15999, 17895, 18435, 19875, 20295, 20643, 21675, 23331, 26115, 29571, 31203, 31995, 33327, 34383, 35787, 36867, 39747, 40587, 41283, 43347

Offset: 1

Antti Karttunen, Oct 28 2016

nonn

Positions in A260443 of terms that are six times a perfect square (terms in A033581, although not all of them are present in A260443).

All terms are multiples of three.

Antti Karttunen, Table of n, a(n) for n = 1..555

Row 3 of A277710.
Cf. A033581, A260443, A264977, A277701, A277330.
Cf. also A277701, A277712, A277714.

A277714(n) = a(n)/3.

A064761 a(n) = 15*n^2.

Original entry on oeis.org

0, 15, 60, 135, 240, 375, 540, 735, 960, 1215, 1500, 1815, 2160, 2535, 2940, 3375, 3840, 4335, 4860, 5415, 6000, 6615, 7260, 7935, 8640, 9375, 10140, 10935, 11760, 12615, 13500, 14415, 15360, 16335, 17340, 18375, 19440, 20535, 21660, 22815

Offset: 0

Roberto E. Martinez II, Oct 18 2001

Number of edges in a complete 6-partite graph of order 6n, K_n,n,n,n,n,n.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A000290, A008585, A008587, A022272, A033428, A033429, A033581, A033583.

Cf. A008597, A195046.

Table[15*n^2, {n, 0, 45}] (* Amiram Eldar, Feb 03 2021 *)

a(n)=15*n^2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 15*A000290(n) = 5*A033428(n) = 3*A033429(n). - Omar E. Pol, Dec 13 2008
a(n) = A008587(n)*A008585(n). - Reinhard Zumkeller, Apr 12 2010
a(n) = a(n-1) + 30*n - 15 for n > 0, a(0)=0. - Vincenzo Librandi, Dec 15 2010
a(n) = A022272(n) + A022272(-n). - Bruno Berselli, Mar 31 2015
a(n) = t(6*n) - 6*t(n), where t(i) = i*(i+k)/2 for any k. Special case (k=1): a(n) = A000217(6*n) - 6*A000217(n). - Bruno Berselli, Aug 31 2017
From Amiram Eldar, Feb 03 2021: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/90.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/180.
Product_{n>=1} (1 + 1/a(n)) = sqrt(15)*sinh(Pi/sqrt(15))/Pi.
Product_{n>=1} (1 - 1/a(n)) = sqrt(15)*sin(Pi/sqrt(15))/Pi. (End)
From Elmo R. Oliveira, Nov 29 2024: (Start)
G.f.: 15*x*(1 + x)/(1 - x)^3.
E.g.f.: 15*x*(1 + x)*exp(x).
a(n) = n*A008597(n) = A195046(2*n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A070169 Rounded total surface area of a regular tetrahedron with edge length n.

Original entry on oeis.org

0, 2, 7, 16, 28, 43, 62, 85, 111, 140, 173, 210, 249, 293, 339, 390, 443, 501, 561, 625, 693, 764, 838, 916, 998, 1083, 1171, 1263, 1358, 1457, 1559, 1665, 1774, 1886, 2002, 2122, 2245, 2371, 2501, 2634, 2771, 2912, 3055, 3203, 3353, 3507, 3665, 3826, 3991

Offset: 0

Rick L. Shepherd, Apr 24 2002

a(n) is the integer k that minimizes |k/n^2 - sqrt(3)|. - Clark Kimberling, Oct 11 2017

			a(3)=16 because round(3^2*sqrt(3)) = round(9*1.73205...) = round(15.5884...) = 16.

S. Selby, editor, CRC Basic Mathematical Tables, CRC Press, 1970, pp. 10-11.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Tetrahedron
Eric Weisstein's World of Mathematics, Platonic Solid

Cf. A033581 (cube), A071396 (octahedron), A071397 (dodecahedron), A071398 (icosahedron), A071399 (volume of tetrahedron).

[Round(n^2 * Sqrt(3)): n in [0..50]]; // Vincenzo Librandi, May 21 2011

Round[Sqrt[3]#]&/@(Range[0,50]^2) (* Harvey P. Dale, Sep 24 2012 *)

for(n=0,100,print1(round(n^2*sqrt(3)),","))

from math import isqrt
def A070169(n): return (m:=isqrt(k:=3*n**4))+(k-m*(m+1)>=1) # Chai Wah Wu, Jun 19 2024

a(n) = round(n^2 * sqrt(3)).

a(n) = A000194(3*n^4). - Christian Krause, Aug 04 2021; corrected by Chai Wah Wu, Jun 19 2024

A195323 a(n) = 22*n^2.

Original entry on oeis.org

0, 22, 88, 198, 352, 550, 792, 1078, 1408, 1782, 2200, 2662, 3168, 3718, 4312, 4950, 5632, 6358, 7128, 7942, 8800, 9702, 10648, 11638, 12672, 13750, 14872, 16038, 17248, 18502, 19800, 21142, 22528, 23958, 25432, 26950, 28512, 30118, 31768, 33462, 35200, 36982, 38808

Offset: 0

Omar E. Pol, Sep 16 2011

Sequence found by reading the line from 0, in the direction 0, 22, ..., in the square spiral whose vertices are the generalized tridecagonal numbers A195313. Semi-axis opposite to A195318 in the same spiral.

Surface area of a rectangular prism with dimensions n, 2n and 3n. - Wesley Ivan Hurt, Apr 10 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Bisection of A195149.
Cf. A016802, A033581, A033583, A135453, A139098, A144555, A195321, A195322.
Cf. A000290, A001105, A008604, A033584, A195313, A195318.

[22*n^2 : n in [0..40]]; // Vincenzo Librandi, Sep 20 2011

A195323:=n->22*n^2: seq(A195323(n), n=0..80); # Wesley Ivan Hurt, Apr 10 2015

22Range[0,50]^2 (* or *) LinearRecurrence[{3,-3,1},{0,22,88},50] (* Harvey P. Dale, Sep 19 2011 *)

vector(50,n,22*(n-1)^2) \\ Derek Orr, Apr 10 2015

a(n) = 22*A000290(n) = 11*A001105(n) = 2*A033584(n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Sep 19 2011
G.f.: 22*x*(1+x)/(1-x)^3. - Wesley Ivan Hurt, Apr 10 2015
From Elmo R. Oliveira, Dec 01 2024: (Start)
E.g.f.: 22*x*(1 + x)*exp(x).
a(n) = n*A008604(n) = A195149(2*n). (End)

cp's OEIS Frontend

A016743 Even cubes: a(n) = (2*n)^3.

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A045945 Hexagonal matchstick numbers: a(n) = 3*n*(3*n+1).

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A061167 a(n) = n^5 - n.

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A195321 a(n) = 18*n^2.

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A195322 a(n) = 20*n^2.

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A227776 a(n) = 6*n^2 + 1.

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A045945 Hexagonal matchstick numbers: a(n) = 3n(3*n+1).