A033428 -id:A033428 - cp's OEIS frontend

A024606 Numbers of form x^2 + xy + y^2 with distinct x and y > 0.

7, 13, 19, 21, 28, 31, 37, 39, 43, 49, 52, 57, 61, 63, 67, 73, 76, 79, 84, 91, 93, 97, 103, 109, 111, 112, 117, 124, 127, 129, 133, 139, 147, 148, 151, 156, 157, 163, 169, 171, 172, 175, 181, 183, 189, 193, 196, 199, 201, 208, 211, 217, 219, 223, 228, 229, 237, 241, 244, 247

Offset: 1

Clark Kimberling

Alternatively, numbers expressible in more than one way as i^2 - ij + j^2, where 1 <= i <= j or 1 <= i < j. The following argument shows that the conditions i <= j or i < j are here equivalent. Note first that i^2 - ij + j^2 = (j - i)^2 - (j - i)*j + j^2, so the only non-duplicated values i^2 - ij + j^2 with 1 <= i < j are when j = 2i, whence i^2 - ij + j^2 = 3i^2. On the other hand, the values with i = j are j^2. There are no integer solutions to 3i^2 = j^2 with i >= 1. - Franklin T. Adams-Watters, May 03 2006
Numbers whose prime factorization contains at least one prime congruent to 1 mod 6 and any prime factor congruent to 2 mod 3 has even multiplicity. - Franklin T. Adams-Watters, May 03 2006
This is a subsequence of Loeschian numbers A003136, closed under multiplication. Its primitive elements are those with exactly one prime factor of form 6k + 1 with multiplicity one (A232436). - Jean-Christophe Hervé, Nov 22 2013
a(1)*a(2)*a(3) = 1729, the Hardy-Ramanujan taxicab number. 1729 is then in the sequence, by the argument of the preceding comment. - Jean-Christophe Hervé, Nov 24 2013
1729 is also the least term that can be written in 4 distinct ways in the given form. Sequence A024614 does not include the restriction x != y, it is the disjoint union of this sequence and A033428 (i.e., 3*x^2) (without 0). - M. F. Hasler, Mar 05 2018

			a(1) = 7 = 1^2 + 2 + 2^2.

G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
Michael Somos, A Multisection of q-Series
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for sequences related to A2 = hexagonal = triangular lattice

Cf. A003136, A004016, A024614, A074628, A088534, A118886, A232436.

Take[Union[Flatten[Table[x^2 + x*y + y^2, {x, 15}, {y, x - 1}]]], 60] (* Robert G. Wilson v, Nov 24 2013 *)

for(k=1,247,my(a088534=sum(x=0,sqrt(k\3),sum(y=max(x,sqrtint(k-x^2)\2),sqrtint(k-2*x^2),x^2+x*y+y^2==k)),a004016d6=sumdiv(k,d,(d%3==1)-(d%3==2)));if(a088534!=a004016d6,print1(k,", "))) \\ Hugo Pfoertner, Sep 22 2019

A004016(a(n)) >= 12. - Jean-Christophe Hervé, Nov 24 2013

Definition modified by Alonso del Arte and Jean-Christophe Hervé, Nov 25 2013

A140678 a(n) = n(3n + 10).

Original entry on oeis.org

0, 13, 32, 57, 88, 125, 168, 217, 272, 333, 400, 473, 552, 637, 728, 825, 928, 1037, 1152, 1273, 1400, 1533, 1672, 1817, 1968, 2125, 2288, 2457, 2632, 2813, 3000, 3193, 3392, 3597, 3808, 4025, 4248, 4477, 4712, 4953, 5200, 5453, 5712

Offset: 0

Omar E. Pol, May 22 2008

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. A033428, A045944, A140676, A067725, A140677, A067707, A140679, A140680, A140681, A140689.

Table[n (3 n + 10), {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 13, 32}, 50] (* Harvey P. Dale, Jun 05 2012 *)

a(n)=n*(3*n+10) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 3*n^2 + 10*n.
a(n) = 6*n + a(n-1) + 7, with a(0)=0. - Vincenzo Librandi, Aug 03 2010
G.f.: x*(13 - 7*x)/(1 - x)^3. - Arkadiusz Wesolowski, Dec 24 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), with a(0)=0, a(1)=13, a(2)=32. - Harvey P. Dale, Jun 05 2012
E.g.f.: (3*x^2 + 13*x)*exp(x). - G. C. Greubel, Jul 20 2017

A255211 a(n) = n(n+1)(7*n+2)/6.

Original entry on oeis.org

0, 3, 16, 46, 100, 185, 308, 476, 696, 975, 1320, 1738, 2236, 2821, 3500, 4280, 5168, 6171, 7296, 8550, 9940, 11473, 13156, 14996, 17000, 19175, 21528, 24066, 26796, 29725, 32860, 36208, 39776, 43571, 47600, 51870, 56388, 61161, 66196, 71500, 77080, 82943

Offset: 0

Luce ETIENNE, Feb 17 2015

a(n) is the number of triangles of all sizes in a polyiamond of trapezoid shape with 3 sides of length n and the base of length 2*n. The number of triangular cells in the trapezoid is 3*n^2. This is half of a regular hexagon with side lengths n.

The number of triangles oriented with their bases aligned with the base of the trapezoid is n*(n+1)*(2*n+1)/3 and the number oriented in the opposite direction is n^2*(n+1)/2. a(n) is the sum of these two.

			From the second comment: a(1)= 2+1, a(2)= 10+6, a(3)= 28+18, a(4)= 60+40.

Colin Barker, Table of n, a(n) for n = 0..1000
Luce ETIENNE, Illustration a(1), a(2), a(3), a(4) and a(5)
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Partial sums of A022264.

Cf. A000292, A000330, A006331, A002411, A033428, A212977.

[n*(n+1)*(7*n+2)/6 : n in [0..50]]; // Wesley Ivan Hurt, Apr 11 2021

Table[n (n + 1) (7 n + 2)/6, {n, 0, 50}] (* Bruno Berselli, Feb 17 2015 *)

concat(0, Vec(x*(4*x+3)/(x-1)^4 + O(x^100))) \\ Colin Barker, Feb 17 2015

vector(50, n, n--; n*(n+1)*(7*n+2)/6) \\ Bruno Berselli, Feb 17 2015

G.f.: x*(3 + 4*x) / (1 - x)^4. - Colin Barker, Feb 17 2015
a(n) = Sum_{j=0..n-1} (n-j)*(3*n-2*j) = Sum_{j=1..n} j*(n+2*j) for n>0.
a(n) = A000292(2*n) - A000292(n). - Bruno Berselli, Sep 22 2016
Sum_{n>=1} 1/a(n) = 21*HarmonicNumber(2/7)/5 - 6/5 = 0.44513027538601361333... . - Vaclav Kotesovec, Sep 22 2016
E.g.f.: exp(x)*x*(18 + 30*x + 7*x^2)/6. - Stefano Spezia, Mar 02 2025

Edited and extended by Bruno Berselli, Dec 01 2016

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

A016910 a(n) = (6*n)^2.

Original entry on oeis.org

0, 36, 144, 324, 576, 900, 1296, 1764, 2304, 2916, 3600, 4356, 5184, 6084, 7056, 8100, 9216, 10404, 11664, 12996, 14400, 15876, 17424, 19044, 20736, 22500, 24336, 26244, 28224, 30276, 32400, 34596, 36864, 39204, 41616, 44100, 46656, 49284, 51984, 54756, 57600, 60516, 63504, 66564, 69696, 72900

Offset: 0

N. J. A. Sloane

Areas A of two classes of triangles with integer sides (a,b,c) where a = 9k, b=10k and c = 17k, or a = 3k, b = 25k and c = 26k for k=0,1,2,... These areas are given by Heron's formula A = sqrt(s(s-a)(s-b)(s-c)) = (6k)^2, with the semiperimeter s = (a+b+c)/2. This sequence is a subsequence of A188158. - Michel Lagneau, Oct 11 2013

Sequence found by reading the line from 0, in the direction 0, 36, ..., in the square spiral whose vertices are the generalized 20-gonal numbers A218864. - Omar E. Pol, May 13 2018.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A089491, A188158, A243941.

Cf. similar sequences of the type k*n^2: A000290 (k=1), A001105 (k=2), A033428 (k=3), A016742 (k=4), A033429 (k=5), A033581 (k=6), A033582 (k=7), A139098 (k=8), A016766 (k=9), A033583 (k=10), A033584 (k=11), A135453 (k=12), A152742 (k=13), A144555 (k=14), A064761 (k=15), A016802 (k=16), A244630 (k=17), A195321 (k=18), A244631 (k=19), A195322 (k=20), A064762 (k=21), A195323 (k=22), A244632 (k=23), A195824 (k=24), A016850 (k=25), A244633 (k=26), A244634 (k=27), A064763 (k=28), A244635 (k=29), A244636 (k=30).

[(6*n)^2: n in [0..40]]; // Vincenzo Librandi, May 03 2011

(6*Range[0,40])^2 (* or *) LinearRecurrence[{3,-3,1},{0,36,144},40] (* Harvey P. Dale, Dec 25 2017 *)
Table[36 n^2, {n, 0, 45}] (* Omar E. Pol, Jun 07 2018 *)

a(n)=36*n^2 \\ Charles R Greathouse IV, Jun 10 2016

From Ilya Gutkovskiy, Jun 09 2016: (Start)
O.g.f.: 36*x*(1 + x)/(1 - x)^3.
E.g.f.: 36*x*(1 + x)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
Sum_{n>=1} 1/a(n) = Pi^2/216 = A086726. (End)
Product_{n>=1} a(n)/A136017(n) = Pi/3. - Fred Daniel Kline, Jun 09 2016
a(n) = t(9*n) - 9*t(n), where t(i) = i*(i+k)/2 for any k. Special case (k=1): a(n) = A000217(9*n) - 9*A000217(n). - Bruno Berselli, Aug 31 2017
a(n) = 36*A000290(n) = 18*A001105(n) = 12*A033428 = 9*A016742(n) = 6*A033581(n) = 4*A016766(n) = 3*A135453(n) = 2*A195321(n). - Omar E. Pol, Jun 07 2018
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/432. - Amiram Eldar, Jun 27 2020
From Amiram Eldar, Jan 25 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = sinh(Pi/6)/(Pi/6).
Product_{n>=1} (1 - 1/a(n)) = sin(Pi/6)/(Pi/6) = 3/Pi (A089491). (End)

A140679 a(n) = n(3n+14).

Original entry on oeis.org

0, 17, 40, 69, 104, 145, 192, 245, 304, 369, 440, 517, 600, 689, 784, 885, 992, 1105, 1224, 1349, 1480, 1617, 1760, 1909, 2064, 2225, 2392, 2565, 2744, 2929, 3120, 3317, 3520, 3729, 3944, 4165, 4392, 4625, 4864, 5109, 5360, 5617, 5880, 6149, 6424, 6705, 6992

Offset: 0

Omar E. Pol, May 22 2008

			a(1)=6*1+0+11=17; a(2)=6*2+17+11=40; a(3)=6*3+40+11=69. See 2nd formula.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A033428, A045944, A140676, A067725, A140677, A140678, A067707, A140680, A140681, A140689.

Table[n(3n+14),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,17,40},50] (* Harvey P. Dale, Apr 29 2011 *)

a(n)=n*(3*n+14) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 3*n^2 + 14*n.
a(n) = a(n-1) + 6*n + 11, with a(0)=0. - Vincenzo Librandi, Aug 03 2010
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), with a(1)=0, a(2)=17, a(3)=40. - Harvey P. Dale, Apr 29 2011
E.g.f.: (3*x^2 + 17*x)*exp(x). - G. C. Greubel, Jul 20 2017

A140680 a(n) = n(3n+16).

Original entry on oeis.org

0, 19, 44, 75, 112, 155, 204, 259, 320, 387, 460, 539, 624, 715, 812, 915, 1024, 1139, 1260, 1387, 1520, 1659, 1804, 1955, 2112, 2275, 2444, 2619, 2800, 2987, 3180, 3379, 3584, 3795, 4012, 4235, 4464, 4699, 4940, 5187, 5440, 5699

Offset: 0

Omar E. Pol, May 22 2008

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A033428, A045944, A140676, A067725, A140677, A140678, A067707, A140679, A140681, A140689.

a[n_]:=Sum[6*i+13, {i, 1, n}]; (* Vladimir Joseph Stephan Orlovsky, Dec 04 2008 *)
Table[n(3n+16),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,19,44},50] (* Harvey P. Dale, Aug 23 2020 *)

a(n)=n*(3*n+16) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 3*n^2 + 16*n.
a(n) = 6*n + a(n-1) + 13 (with a(0)=0). - Vincenzo Librandi, Aug 03 2010
E.g.f.: (3*x^2 + 19*x)*exp(x). - G. C. Greubel, Jul 20 2017

A140689 a(n) = n(3n + 20).

Original entry on oeis.org

0, 23, 52, 87, 128, 175, 228, 287, 352, 423, 500, 583, 672, 767, 868, 975, 1088, 1207, 1332, 1463, 1600, 1743, 1892, 2047, 2208, 2375, 2548, 2727, 2912, 3103, 3300, 3503, 3712, 3927, 4148, 4375, 4608, 4847, 5092, 5343, 5600, 5863

Offset: 0

Omar E. Pol, May 22 2008

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. A033428, A045944, A140676, A067725, A140677, A140678, A067707, A140679, A140680, A140681.

a[n_]:=Sum[6*i+17, {i, 1, n}]; (* Vladimir Joseph Stephan Orlovsky, Dec 04 2008 *)
Table[n(3n+20),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,23,52},50] (* Harvey P. Dale, Apr 29 2016 *)

a(n)=n*(3*n+20) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 3*n^2 + 20*n.
a(n) = a(n-1) + 6*n + 17 (with a(0)=0). - Vincenzo Librandi, Dec 15 2010
a(n) = 3*a(n-1)-3*a(n-2)+a(n-3), with a(0)=0, a(1)=23, a(2)=52. - Harvey P. Dale, Apr 29 2016
From G. C. Greubel, Jul 21 2017: (Start)
G.f.: x*(23 - 17*x)/(1 - x)^3.
E.g.f.: x*(3*x + 23)*exp(x). (End)

A171108 a(n) is the Severi degree for curves of degree n and cogenus 2.

Original entry on oeis.org

0, 0, 21, 225, 882, 2370, 5175, 9891, 17220, 27972, 43065, 63525, 90486, 125190, 168987, 223335, 289800, 370056, 465885, 579177, 711930, 866250, 1044351, 1248555, 1481292, 1745100, 2042625, 2376621, 2749950, 3165582, 3626595, 4136175, 4697616, 5314320

Offset: 1

N. J. A. Sloane, Sep 27 2010

Severi degree N(n, delta) is the number of degree n plane curves which have delta nodes and pass through a generic configuration of n*(n+3)/2-delta points on the plane. delta is called the cogenus of these curves. See Fomin and Mikhalkin (2010), Section 1.2 "Combinatorial rules for Gromov-Witten invariants and Severi degrees" and 5 "Node polynomials". - Andrey Zabolotskiy, Jan 18 2021

Colin Barker, Table of n, a(n) for n = 1..1000
Florian Block, Computing node polynomials for plane curves, arXiv:1006.0218 [math.AG], 2010-2011; Math. Res. Lett. 18, (2011), no. 4, 621-643.
Florian Block, Susan Jane Colley, and Gary Kennedy, Computing Severi degrees with long-edge graphs, Bulletin of the Brazilian Mathematical Society, New Series 45.4 (2014): 625-647. Also arXiv:1303.5308 [math.AG], 2013 (see first page).
Sergey Fomin and Grigory Mikhalkin, Labeled floor diagrams for plane curves, Journal of the European Mathematical Society 012.6 (2010): 1453-1496; arXiv:0906.3828 [math.AG], 2009-2010.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Severi degrees N(n, delta) for other values of delta: A033428(n-1) (1), A171113 (3), A328551 (4), A328552 (5), A171116 (6).

Table[3(n-1)(n-2)(3n^2-3n-11)/2,{n,40}] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{0,0,21,225,882},40] (* Harvey P. Dale, Feb 01 2013 *)

concat([0,0], Vec(3*x^3*(7 + 40*x - 11*x^2) / (1 - x)^5 + O(x^40))) \\ Colin Barker, Nov 01 2019

a(n) = 3*(n-1)*(n-2)*(3*n^2-3*n-11)/2.
a(1)=0, a(2)=0, a(3)=21, a(4)=225, a(5)=882, a(n) = 5*a(n-1)-10*a(n-2)+ 10*a(n-3)-5*a(n-4)+a(n-5). - Harvey P. Dale, Feb 01 2013
G.f.: 3*x^2*(-7-40*x+11*x^2) / (x-1)^5 . - R. J. Mathar, Dec 19 2013

New name from Andrey Zabolotskiy, Jan 18 2021

A214295 a(n) = 1 if n is a square, -1 if n is three times a square, 0 otherwise.

Original entry on oeis.org

1, 0, -1, 1, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0

Offset: 1

Michael Somos, Jul 10 2012

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

a(A092206(n)) = 0; a(A000290(n)) = 1; a(A033428(n)) = -1.

			G.f. = q - q^3 + q^4 + q^9 - q^12 + q^16 + q^25 - q^27 + q^36 - q^48 + q^49 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Shaun Cooper and Michael Hirschhorn, On some infinite product identities, Rocky Mountain J. Math., 31 (2001) 131-139. see p. 133 Theorem 3.
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A000700, A010052, A214293, A244612, A247133, A245485.

a214295 n = a010052 n - a010052 (3*n)  -- Reinhard Zumkeller, Jul 12 2012

Basis( ModularForms( Gamma1(12), 1/2), 50) [2] ; /* Michael Somos, Jun 10 2014 */

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] - EllipticTheta[ 3, 0, q^3]) / 2, {q, 0, n}];
a[ n_] := Boole[ IntegerQ[ Sqrt[ n]]] - Boole[ IntegerQ[ Sqrt[ 3 n]]]; (* Michael Somos, Jun 10 2014 *)
Table[Which[IntegerQ[Sqrt[n]],1,IntegerQ[Sqrt[n/3]],-1,True,0],{n,120}] (* Harvey P. Dale, Apr 08 2013 *)

PARI
```
{a(n) = issquare(n) - issquare(3*n)};
```

{a(n) = if( n<1, 0, direuler( p=2, n, if( p==3, 1 - X, 1) / (1 - X^2 ))[n])};

Expansion of q * psi(q^3) * f(-q^2, -q^10) / f(-q^5, -q^7) in powers of q where psi(), f() are Ramanujan theta functions.
Multiplicative with a(3^e) = (-1)^e, a(p^e) = 1 if e even, 0 otherwise.
G.f.: (theta_3(q) - theta_3(q^3)) / 2 = Sum_{k>0} x^(k^2) - x^(3*k^2).
Dirichlet g.f.: zeta(2*s) * (1 - 3^(-s)). [corrected by Amiram Eldar, Oct 24 2023]
a(3*n) = - a(n). - Reinhard Zumkeller, Jul 12 2012
Expansion of (phi(q) - phi(q^3)) / 2 = q * chi(q) * f(-q, -q^11) in powers fof q where phi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Jan 10 2015
Euler transform of period 12 sequence [ 0, -1, 1, 0, 1, -1, 1, 0, 1, -1, 0, -1, ...]. - Michael Somos, Jan 10 2015
Convolution product of A000700 and A247133. - Michael Somos, Jan 10 2015
Sum_{k=1..n} a(k) ~ c*sqrt(n), where c = 1 - 1/sqrt(3) = 0.42264973... . - Amiram Eldar, Oct 24 2023

cp's OEIS Frontend

A024606 Numbers of form x^2 + xy + y^2 with distinct x and y > 0.

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A140678 a(n) = n*(3*n + 10).

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A255211 a(n) = n*(n+1)*(7*n+2)/6.

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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.

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A016910 a(n) = (6*n)^2.

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A140679 a(n) = n*(3*n+14).

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A140680 a(n) = n*(3*n+16).

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A140678 a(n) = n(3n + 10).

A255211 a(n) = n(n+1)(7*n+2)/6.

A140679 a(n) = n(3n+14).

A140680 a(n) = n(3n+16).

A140689 a(n) = n(3n + 20).