A049451 -id:A049451 - cp's OEIS frontend

A033580 Four times second pentagonal numbers: a(n) = 2n(3*n+1).

0, 8, 28, 60, 104, 160, 228, 308, 400, 504, 620, 748, 888, 1040, 1204, 1380, 1568, 1768, 1980, 2204, 2440, 2688, 2948, 3220, 3504, 3800, 4108, 4428, 4760, 5104, 5460, 5828, 6208, 6600, 7004, 7420, 7848, 8288, 8740, 9204, 9680, 10168, 10668, 11180, 11704, 12240

Offset: 0

N. J. A. Sloane

Subsequence of A062717: A010052(6*a(n)+1) = 1. - Reinhard Zumkeller, Feb 21 2011
Sequence found by reading the line from 0, in the direction 0, 8,..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. Opposite numbers to the members of A139267 in the same spiral - Omar E. Pol, Sep 09 2011
a(n) is the number of edges of the octagonal network O(n,n); O(m,n) is defined by Fig. 1 of the Siddiqui et al. reference. - Emeric Deutsch May 13 2018
The partial sums of this sequence give A035006. - Leo Tavares, Oct 03 2021

Ivan Panchenko, Table of n, a(n) for n = 0..1000
M. K. Siddiqui, M. Naeem, N. A. Rahman, and M. Imran, Computing topological indices of certain networks, J. of Optoelectronics and Advanced Materials, 18, No. 9-10 (2016), pp. 884-892.
Leo Tavares, Illustration: Crossed Stars
Leo Tavares, Illustration: Four Quarter Star Crosses
Leo Tavares, Illustration: Triangulated Star Crosses
Leo Tavares, Illustration: Oblong Star Crosses
Leo Tavares, Illustration: Crossed Diamond Stars
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A033579, A049451, A045945, A186423.
Cf. sequences listed in A254963.
Cf. A003154, A016813.
Cf. A035006.
Cf. A005449, A046092, A033996, A002378, A016742, A000290, A016754, A001105.

List([0..50], n-> 2*n*(3*n+1)); # G. C. Greubel, Oct 09 2019

[2*n*(3*n+1): n in [0..50]]; // G. C. Greubel, Oct 09 2019

seq(2*n*(3*n+1), n=0..50); # G. C. Greubel, Oct 09 2019

4*Binomial[3*Range[50]-2, 2]/3 (* G. C. Greubel, Oct 09 2019 *)

a(n)=2*n*(3*n+1) \\ Charles R Greathouse IV, Sep 28 2015

[2*n*(3*n+1) for n in (0..50)] # G. C. Greubel, Oct 09 2019

a(n) = a(n-1) +12*n -4 (with a(0)=0). - Vincenzo Librandi, Aug 05 2010
G.f.: 4*x*(2+x)/(1-x)^3. - Colin Barker, Feb 13 2012
a(-n) = A033579(n). - Michael Somos, Jun 09 2014
E.g.f.: 2*x*(4 + 3*x)*exp(x). - G. C. Greubel, Oct 09 2019
From Amiram Eldar, Jan 14 2021: (Start)
Sum_{n>=1} 1/a(n) = 3/2 - Pi/(4*sqrt(3)) - 3*log(3)/4.
Sum_{n>=1} (-1)^(n+1)/a(n) =  -3/2 + Pi/(2*sqrt(3)) + log(2). (End)
From Leo Tavares, Oct 12 2021: (Start)
a(n) = A003154(n+1) - A016813(n). See Crossed Stars illustration.
a(n) = 4*A005449(n). See Four Quarter Star Crosses illustration.
a(n) = 2*A049451(n).
a(n) = A046092(n-1) + A033996(n). See Triangulated Star Crosses illustration.
a(n) = 4*A000217(n-1) + 8*A000217(n).
a(n) = 4*A000217(n-1) + 4*A002378. See Oblong Star Crosses illustration.
a(n) = A016754(n) + 4*A000217(n). See Crossed Diamond Stars illustration.
a(n) = 2*A001105(n) + 4*A000217(n).
a(n) = A016742(n) + A046092(n).
a(n) = 4*A000290(n) + 4*A000217(n). (End)

A098301 Member r=16 of the family of Chebyshev sequences S_r(n) defined in A092184.

Original entry on oeis.org

0, 1, 16, 225, 3136, 43681, 608400, 8473921, 118026496, 1643897025, 22896531856, 318907548961, 4441809153600, 61866420601441, 861688079266576, 12001766689130625, 167163045568562176, 2328280871270739841, 32428769152221795600, 451674487259834398561

Offset: 0

Wolfdieter Lang, Oct 18 2004

Also m such that (3*m^2 + m)/4 = m*(3*m + 1)/4 is a perfect square. - Ctibor O. Zizka, Oct 15 2010

Consequently A049451(k) is a square if and only if k = a(n). - Bruno Berselli, Oct 14 2011

Colin Barker, Table of n, a(n) for n = 0..874
S. Barbero, U. Cerruti, and N. Murru, On polynomial solutions of the Diophantine equation (x + y - 1)^2 = wxy, Rendiconti Sem. Mat. Univ. Pol. Torino (2020) Vol. 78, No. 1, 5-12.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (15,-15,1).

Cf. A001075, A001353, A049451, A092184.

LinearRecurrence[{# - 1, -# + 1, 1}, {0, 1, #}, 20] &[16] (* Michael De Vlieger, Feb 23 2021 *)

concat(0, Vec(x*(1+x)/((1-x)*(1-14*x+x^2)) + O(x^50))) \\ Colin Barker, Jun 15 2015

a(n) = (T(n, 7)-1)/6 with Chebyshev's polynomials of the first kind evaluated at x=7: T(n, 7) = A011943(n) = ((7 + 4*sqrt(3))^n + (7 - 4*sqrt(3))^n)/2; therefore: a(n) = ((7 + 4*sqrt(3))^n + (7 - 4*sqrt(3))^n - 2)/12.
a(n) = A001353(n)^2 = S(n-1, 4)^2 with Chebyshev's polynomials of the second kind evaluated at x=4, S(n, 4):=U(n, 2).
a(n) = 14*a(n-1) - a(n-2) + 2, n >= 2, a(0)=0, a(1)=1.
a(n) = 15*a(n-1) - 15*a(n-2) + a(n-3), n >= 3.
G.f.: x*(1+x)/((1-x)*(1 - 14*x + x^2)) = x*(1+x)/(1 - 15*x + 15*x^2 - x^3) (from the Stephan link, see A092184).
Conjecture: 4*A007655(n+1) + A046184(n) = A055793(n+2) + a(n+1). - Creighton Dement, Nov 01 2004
a(n) = (A001075(n)^2-1)/3. - Parker Grootenhuis, Nov 28 2017

A144390 a(n) = 3*n^2 - n - 1.

Original entry on oeis.org

1, 9, 23, 43, 69, 101, 139, 183, 233, 289, 351, 419, 493, 573, 659, 751, 849, 953, 1063, 1179, 1301, 1429, 1563, 1703, 1849, 2001, 2159, 2323, 2493, 2669, 2851, 3039, 3233, 3433, 3639, 3851, 4069, 4293, 4523, 4759, 5001, 5249, 5503, 5763, 6029, 6301, 6579

Offset: 1

Paul Curtz, Oct 02 2008

Sequence's original Name was "First bisection of A135370."

The partial sums of this sequence give A081437. - Leo Tavares, Dec 26 2021

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
John Elias, Illustration: Belted Hexagrams
Leo Tavares, Illustration: Bounded Hexagons
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A016933, A049450, A144391.
Cf. A003215, A005408, A016754, A002378.
Cf. A081437 (partial sums).

[3*n^2-n-1: n in [1..60]]; // Vincenzo Librandi, Jun 14 2011

A144390:=n->3*n^2-n-1; seq(A144390(n), n=1..50); # Wesley Ivan Hurt, Mar 26 2014

Table[3*n^2 -n -1 , {n,0,50}] (* G. C. Greubel, Jul 19 2017 *)

a(n)=3*n^2-n-1 \\ Charles R Greathouse IV, Oct 07 2015

a(n+1) = a(n) + 6*n + 2; see A016933.
G.f.: x*(1+6*x-x^2)/(1-x)^3. a(n) = A049450(n)-1. - R. J. Mathar, Oct 24 2008
a(-n) = A144391(n). - Michael Somos, Mar 27 2014
E.g.f.: (3*x^2 + 2*x -1)*exp(x) + 1. - G. C. Greubel, Jul 19 2017
From Leo Tavares, Dec 26 2021: (Start)
a(n) = A003215(n) - 2*A005408(n). See Bounded Hexagons illustration.
a(n) = A016754(n-1) - A002378(n-2). (End)
a(n) = A003154(n) - A049451(n-1). - John Elias, Dec 22 2022

Edited by R. J. Mathar, Oct 24 2008

More terms from Vladimir Joseph Stephan Orlovsky, Oct 25 2008

A316466 a(n) = 2n(7*n - 3).

Original entry on oeis.org

0, 8, 44, 108, 200, 320, 468, 644, 848, 1080, 1340, 1628, 1944, 2288, 2660, 3060, 3488, 3944, 4428, 4940, 5480, 6048, 6644, 7268, 7920, 8600, 9308, 10044, 10808, 11600, 12420, 13268, 14144, 15048, 15980, 16940, 17928, 18944, 19988, 21060, 22160, 23288, 24444, 25628, 26840

Offset: 0

Bruno Berselli, Jul 04 2018

This is the case k = 9 of Sum_{i = 2..k} P(i,n) = (k - 1)*n*((k - 2)*n - (k - 6))/4, where P(k,n) = n*((k - 2)*n - (k - 4))/2 (see Crossrefs for similar sequences and "Square array in A139600" in Links section).

14*x + 9 is a square for x = a(n) or x = a(-n).

Colin Barker, Table of n, a(n) for n = 0..1000
Bruno Berselli, Square array in A139600.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A139600, A218471.

Similar sequences (see the first comment): A000096 (k = 3), A045943 (k = 4), A049451 (k = 5), A033429 (k = 6), A167469 (k = 7), A152744 (k = 8), this sequence (k = 9), A152994 (k = 10).

GAP
```
List([0..50], n -> 2*n*(7*n-3));
```
Julia
```
[2*n*(7*n-3) for n in 0:50] |> println
```
Magma
```
[2*n*(7*n-3): n in [0..50]];
```

Table[2 n (7 n - 3), {n, 0, 50}]
LinearRecurrence[{3,-3,1},{0,8,44},50] (* Harvey P. Dale, Jan 24 2021 *)

Maxima
```
makelist(2*n*(7*n-3), n, 0, 50);
```
PARI
```
vector(50, n, n--; 2*n*(7*n-3))
```

concat(0, Vec(4*x*(2 + 5*x)/(1 - x)^3 + O(x^40))) \\ Colin Barker, Jul 05 2018

Python
```
[2*n*(7*n-3) for n in range(50)]
```
Sage
```
[2*n*(7*n-3) for n in (0..50)]
```

O.g.f.: 4*x*(2 + 5*x)/(1 - x)^3.
E.g.f.: 2*x*(4 + 7*x)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 4*A218471(n).

A144391 a(n) = 3*n^2 + n - 1.

Original entry on oeis.org

3, 13, 29, 51, 79, 113, 153, 199, 251, 309, 373, 443, 519, 601, 689, 783, 883, 989, 1101, 1219, 1343, 1473, 1609, 1751, 1899, 2053, 2213, 2379, 2551, 2729, 2913, 3103, 3299, 3501, 3709, 3923, 4143, 4369, 4601, 4839, 5083, 5333, 5589, 5851, 6119, 6393, 6673

Offset: 1

Paul Curtz, Oct 02 2008

Vincenzo Librandi, Table of n, a(n) for n = 1..3000
Leo Tavares, Illustration: Cropped Hexagons
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A135370, A144390.

[3*n^2+n-1: n in [1..50]]; // Vincenzo Librandi, May 06 2011

Table[3 n^2 + n - 1, {n, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {3, 13, 29}, 50] (* Harvey P. Dale, Sep 18 2016 *)

a(n)=3*n^2+n-1 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = A135370(2*n).
First differences: a(n+1) - a(n) = A016957(n).
a(n) - A144390(n) = 6*n + 4 = A005843(n).
From R. J. Mathar, Oct 24 2008: (Start)
G.f.: x*(3 + 4*x - x^2)/(1 - x)^3.
a(n) = A049451(n) - 1. (End)
E.g.f.: (3*x^2 + 4*x - 1)*exp(x) + 1. - G. C. Greubel, Jul 19 2017
a(n) = 1 + Sum_{i = n-1..2*n-1} 2*i. - Bruno Berselli, Feb 16 2018
a(n) = A003215(n) - (n+1)*2. - Leo Tavares, Jul 04 2021

Edited by R. J. Mathar, Oct 24 2008

More terms from Vladimir Joseph Stephan Orlovsky, Mar 01 2009

A045946 Star of David matchstick numbers: a(n) = 6n(3*n+1).

Original entry on oeis.org

0, 24, 84, 180, 312, 480, 684, 924, 1200, 1512, 1860, 2244, 2664, 3120, 3612, 4140, 4704, 5304, 5940, 6612, 7320, 8064, 8844, 9660, 10512, 11400, 12324, 13284, 14280, 15312, 16380, 17484, 18624, 19800, 21012, 22260, 23544, 24864, 26220, 27612, 29040, 30504, 32004

Offset: 0

R. K. Guy

Vertical spoke of triangular spiral in A051682. - Paul Barry, Mar 15 2003

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

Cf. A005449, A033580, A049451, A045945, A051682, A081266.

Table[6n(3n+1),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{0,24,84},40] (* Harvey P. Dale, Nov 23 2012 *)

a(n)=18*n^2+6*n \\ Charles R Greathouse IV, Feb 19 2017

a(n) = 24*C(n,1) + 36*C(n,2); binomial transform of (0, 24, 36, 0, 0, 0, ...). - Paul Barry, Mar 15 2003
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=0, a(1)=24, a(2)=84. - Harvey P. Dale, Nov 23 2012
G.f.: 12*x*(2+x)/(1-x)^3. - Ivan Panchenko, Nov 13 2013
a(n) = 2*A045945(n). - Michel Marcus, Nov 13 2013
a(n) = 12*A005449(n). - R. J. Mathar, Feb 08 2016
From Amiram Eldar, Jan 14 2021: (Start)
Sum_{n>=1} 1/a(n) = 1/2 - Pi/(12*sqrt(3)) - log(3)/4.
Sum_{n>=1} (-1)^(n+1)/a(n) = -1/2 + Pi/(6*sqrt(3)) + log(2)/3. (End)
From Elmo R. Oliveira, Dec 12 2024: (Start)
E.g.f.: 6*exp(x)*x*(4 + 3*x).
a(n) = 6*A049451(n) = 4*A081266(n) = 3*A033580(n). (End)

A100104 a(n) = n^3 - n^2 + 1.

Original entry on oeis.org

1, 1, 5, 19, 49, 101, 181, 295, 449, 649, 901, 1211, 1585, 2029, 2549, 3151, 3841, 4625, 5509, 6499, 7601, 8821, 10165, 11639, 13249, 15001, 16901, 18955, 21169, 23549, 26101, 28831, 31745, 34849, 38149, 41651, 45361, 49285, 53429, 57799, 62401, 67241, 72325

Offset: 0

N. J. A. Sloane, Jan 12 2005

Appears to be the number of possible distinct sums of a set of n distinct integers between 1 and n^2. Checked up to n=6. - Dylan Hamilton, Sep 21 2010

a(n) = A100104(n+1) - A100104(n). - Reinhard Zumkeller, Jul 07 2012

T. A. Gulliver, Sequences from Cubes of Integers, Int. Math. Journal, 4 (2003), 439-445.

Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

Cf. A162611. - Vincenzo Librandi, May 27 2010

Cf. A049451 (first differences).

a049451 n = n * (3 * n + 1)  -- Reinhard Zumkeller, Jul 07 2012

f[n_]:=n^3-n^2+1;Table[f[n],{n,5!}] (* Vladimir Joseph Stephan Orlovsky, Feb 07 2010 *)
Array[#^3-#^2+1&,50,0] (* or *) LinearRecurrence[{4,-6,4,-1},{1,1,5,19},50] (* Harvey P. Dale, Sep 11 2011 *)

a(n)=n^3-n^2+1 \\ Charles R Greathouse IV, Oct 07 2015

From Harvey P. Dale, Sep 11 2011: (Start)
a(0)=1, a(1)=1, a(2)=5, a(3)=19, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: (x^3+7*x^2-3*x+1)/(x-1)^4. (End)

A187377 T(n,k)=Number of n-step S, NW and NE-moving king's tours on a kXk board summed over all starting positions.

Original entry on oeis.org

1, 4, 0, 9, 4, 0, 16, 14, 4, 0, 25, 30, 25, 4, 0, 36, 52, 64, 40, 0, 0, 49, 80, 121, 132, 40, 0, 0, 64, 114, 196, 278, 188, 24, 0, 0, 81, 154, 289, 478, 487, 264, 18, 0, 0, 100, 200, 400, 732, 924, 832, 324, 0, 0, 0, 121, 252, 529, 1040, 1499, 1810, 1418, 404, 0, 0, 0, 144, 310, 676

Offset: 1

R. H. Hardin Mar 09 2011

Table starts
.1.4..9..16...25....36....49....64.....81....100....121....144....169....196
.0.4.14..30...52....80...114...154....200....252....310....374....444....520
.0.4.25..64..121...196...289...400....529....676....841...1024...1225...1444
.0.4.40.132..278...478...732..1040...1402...1818...2288...2812...3390...4022
.0.0.40.188..487...924..1499..2212...3063...4052...5179...6444...7847...9388
.0.0.24.264..832..1810..3154..4864...6940...9382..12190..15364..18904..22810
.0.0.18.324.1418..3448..6581.10688..15769..21824..28853..36856..45833..55784
.0.0..0.404.2140..6380.13220.22996..35366..50330..67888..88040.110786.136126
.0.0..0.340.3060.10320.24892.46412..75567.111492.154187.203652.259887.322892
.0.0..0.280.3792.17052.44464.92628.159328.245946.350386.472648.612732.770638

			Some n=4 solutions for 4X4
..0..4..0..0....0..0..0..0....1..0..0..0....0..0..0..0....0..0..0..0
..3..0..0..0....0..0..0..0....2..0..0..0....0..0..0..0....0..0..1..0
..0..2..0..0....0..0..3..1....3..0..0..0....4..2..0..0....0..0..2..4
..0..0..1..0....0..0..4..2....4..0..0..0....0..3..1..0....0..0..3..0

R. H. Hardin, Table of n, a(n) for n = 1..287

Row 2 is A049451(n-1)

Row 3 is A016790(n-2)

Empirical: T(1,k) = k^2
Empirical: T(2,k) = 3*k^2 - 5*k + 2
Empirical: T(3,k) = 9*k^2 - 24*k + 16 for k>1
Empirical: T(4,k) = 27*k^2 - 97*k + 88 for k>2
Empirical: T(5,k) = 69*k^2 - 322*k + 372 for k>3
Empirical: T(6,k) = 183*k^2 - 1035*k + 1432 for k>4
Empirical: T(7,k) = 487*k^2 - 3198*k + 5104 for k>5
Empirical: T(8,k) = 1297*k^2 - 9679*k + 17420 for k>6
Empirical: T(9,k) = 3385*k^2 - 28390*k + 56892 for k>7
Empirical: T(10,k) = 8911*k^2 - 82691*k + 181756 for k>8

A307011 First coordinate in a redundant hexagonal coordinate system of the points of a counterclockwise spiral on an hexagonal grid. Second and third coordinates are given in A307012 and A345978.

Original entry on oeis.org

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

Offset: 0

Hugo Pfoertner, Mar 19 2019

From Peter Munn, Jul 22 2021: (Start)
The points of the spiral are equally the points of a hexagonal lattice, the points of an isometric (triangular) grid and the center points of the cells of a honeycomb (regular hexagonal tiling or grid). The coordinate system can be described using 3 axes that pass through spiral point 0 and one of points 1, 2 or 3. Along each axis, one of the coordinates is 0.
a(n) is the signed distance from spiral point n to the axis that passes through point 2. The distance is measured along either of the lines through point n that are parallel to one of the other 2 axes and the sign is such that point 1 has positive distance.
This coordinate can be paired with either of the other coordinates to form oblique coordinates as described in A307012. Alternatively, all 3 coordinates can be used together, symmetrically, as described in A345978.
There is a negated variant of the 3rd coordinate, which is the conventional sense of this coordinate for specifying (with the 2nd coordinate) the Eisenstein integers that can be the points of the spiral when it is embedded in the complex plane. See A307013.
(End)

Hugo Pfoertner, Table of n, a(n) for n = 0..10034
Margherita Barile, Oblique Coordinates, entry in Eric Weisstein's World of Mathematics.
HandWiki, Hexagonal Lattice.
Peter Munn, Illustration of signed distance of spiral points.
Hugo Pfoertner, Illustration of A307012 vs A307011, spiral.
Hugo Pfoertner, Illustration of A345978 vs A307011, spiral.
Wikipedia, Signed distance function.

Cf. A307012, A307013, A307014, A307016, A307017, A328818, A345978.
Numbers on the spokes of the spiral: A000567, A028896, A033428, A045944, A049450, A049451.
Positions on the spiral that correspond to Eisenstein primes: A345435.

r=-1;d=-1;print1(m=0,", ");for(k=0,8,for(j=1,r,print1(s,", "));if(k%2,,m++;r++);for(j=-m,m+1,if(d*j>=-m,print1(s=d*j,", ")));d=-d)

Name revised by Peter Munn, Jul 08 2021

A242669 a(n) = n*floor(n/3).

Original entry on oeis.org

0, 0, 0, 3, 4, 5, 12, 14, 16, 27, 30, 33, 48, 52, 56, 75, 80, 85, 108, 114, 120, 147, 154, 161, 192, 200, 208, 243, 252, 261, 300, 310, 320, 363, 374, 385, 432, 444, 456, 507, 520, 533, 588, 602, 616, 675, 690, 705, 768, 784, 800, 867, 884, 901, 972, 990

Offset: 0

Bruno Berselli, Jul 01 2014

For n = 0, 1, 2, 4, 8, 49, 98, 676, 1352, 9409, 18818, 131044, 262088, 1825201, 3650402, ... a(n) is a square.

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

Cf. A000290 (n^2), A010762 (floor(n/2)*floor(n/3)), A093353 (n*floor(n/2)), A213033 (n*floor(n/2)*floor(n/3)), A233035 (n*floor(n/4)).
Cf. A033428, A045944, A049451.
Cf. A002264 (floor(n/3)).

Magma
```
[n*Floor(n/3): n in [0..60]];
```
Mathematica
```
Table[n Floor[n/3], {n, 0, 60}]
```

a(n)=n\3*n \\ Charles R Greathouse IV, Oct 07 2015

Sage
```
[n*floor(n/3) for n in (0..60)];
```

G.f.: x^3*(3 + x + x^2 + x^3)/((1 - x)^3*(1 + x + x^2)^2).
a(3m) = A033428(m), a(3m+1) = A049451(m), a(3m+2) = A045944(m).
Sum_{n>=3} (-1)^(n+1)/a(n) = 9/4 + Pi^2/36 - Pi/(2*sqrt(3)) - 2*log(2). - Amiram Eldar, Mar 30 2023

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A033580 Four times second pentagonal numbers: a(n) = 2*n*(3*n+1).

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A098301 Member r=16 of the family of Chebyshev sequences S_r(n) defined in A092184.

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A144390 a(n) = 3*n^2 - n - 1.

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A316466 a(n) = 2*n*(7*n - 3).

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A144391 a(n) = 3*n^2 + n - 1.

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A045946 Star of David matchstick numbers: a(n) = 6*n*(3*n+1).

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A033580 Four times second pentagonal numbers: a(n) = 2n(3*n+1).

A316466 a(n) = 2n(7*n - 3).

A045946 Star of David matchstick numbers: a(n) = 6n(3*n+1).