A001082 -id:A001082 - cp's OEIS frontend

A132209 a(0) = 0 and a(n) = 2n^2 + 2n - 1, for n>=1.

0, 3, 11, 23, 39, 59, 83, 111, 143, 179, 219, 263, 311, 363, 419, 479, 543, 611, 683, 759, 839, 923, 1011, 1103, 1199, 1299, 1403, 1511, 1623, 1739, 1859, 1983, 2111, 2243, 2379, 2519, 2663, 2811, 2963, 3119, 3279, 3443, 3611, 3783, 3959, 4139, 4323, 4511

Offset: 0

Mohamed Bouhamida, Nov 06 2007

nonn

Previous name was: Sequence gives X values that satisfy the integer equation 2*X^3 + 3*X^2 = Y^2.

To find Y values: b(n) = (2*n^2 + 2*n - 1)*(2*n - 1).

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. A005563, A046092, A001082, A002378, A036666, A062717, A028347, A087475, A000217.

[0] cat [2*n^2+2*n-1: n in [1..50]]; // Vincenzo Librandi, Sep 22 2015

Join[{0}, LinearRecurrence[{3, -3, 1}, {3, 11, 23}, 40]] (* Vincenzo Librandi, Sep 22 2015 *)

for(n=0,50, print1(if(n==0, 0, 2*n^2 + 2*n -1), ", ")) \\ G. C. Greubel, Jul 13 2017

a(n) = 2*n^2 + 2*n - 1 for n>=1.
G.f.: x*(1+x)*(3-x)/(1-x)^3. - R. J. Mathar, Nov 14 2007
E.g.f.: 1 + (2*x^2 + 4*x -1)*exp(x). - G. C. Greubel, Jul 13 2017
From Amiram Eldar, Mar 07 2021: (Start)
Sum_{n>=1} 1/a(n) = 1 + sqrt(3)*Pi*tan(sqrt(3)*Pi/2)/6.
Product_{n>=1} (1 + 1/a(n)) = -Pi*sec(sqrt(3)*Pi/2)/2.
Product_{n>=1} (1 - 1/a(n)) = cos(sqrt(5)*Pi/2)*sec(sqrt(3)*Pi/2)/2. (End)

Edited by the Associate Editors of the OEIS, Nov 15 2009
More terms from Vincenzo Librandi, Sep 22 2015
Shorter name (using formula given) from Joerg Arndt, Sep 27 2015

A089802 Expansion of q^(-1/3) * (theta_4(q^3) - theta_4(q^(1/3))) / 2 in powers of q.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Nov 12 2003

sign

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

Number 10 of the 14 primitive eta-products which are holomorphic modular forms of weight 1/2 listed by D. Zagier on page 30 of "The 1-2-3 of Modular Forms". - Michael Somos, May 04 2016

			G.f. = 1 - x - x^5 + x^8 + x^16 - x^21 - x^33 + x^40 + x^56 - x^65 - x^85 + ...
G.f. = q - q^4 - q^16 + q^25 + q^49 - q^64 - q^100 + q^121 + q^169 - q^196 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Jacobi Theta Functions
I. J. Zucker, Further Relations Amongst Infinite Series and Products. II. The Evaluation of Three-Dimensional Lattice Sums, J. Phys. A: Math. Gen. 23, 117-132, 1990.

Cf. A001082, A002448, A089801, A089812.

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, x^3] - EllipticTheta[ 4, 0, x^(1/3)]) / (2 x^(1/3)), {x, 0, n}]; (* Michael Somos, Jun 30 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] EllipticTheta[ 2, 0, x^(3/2)] / (2 x^(3/8)), {x, 0, n}]; (* Michael Somos, Jun 30 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] QPochhammer[ -x^3, x^3] QPochhammer[ x^6], {x, 0, n}]; (* Michael Somos, Jun 30 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^6] QPochhammer[ x^5, x^6] QPochhammer[ x^6], {x, 0, n}]; (* Michael Somos, Jun 30 2015 *)
a[ n_] := (-1)^n Sign @ SquaresR[ 1, 3 n + 1]; (* Michael Somos, Jun 30 2015 *)

{a(n) = (-1)^n * issquare(3*n + 1)}; /* Michael Somos, Apr 12 2005 */

Expansion of q^(-1/3) * (eta(q) * eta(q^6)^2) / (eta(q^2) * eta(q^3)) in powers of q. - Michael Somos, Apr 12 2005
Expansion of chi(-x) * psi(x^3) in powers of x where psi(), chi() are Ramanujan theta functions. - Michael Somos, Dec 23 2011
Expansion of f(-x, -x^5) in powers of x, where f(, ) is Ramanujan's general theta function.
a(n) = b(3*n + 1) where b() is multiplicative with b(3^e) = 0^e, b(2^e) = - (1 + (-1)^e) / 2 if e>0, b(p^e) = (1 + (-1)^e) / 2 if p>3.
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 8^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A089812. - Michael Somos, Dec 23 2011
Euler transform of period 6 sequence [-1, 0, 0, 0, -1, -1, ...]. - Michael Somos, Apr 12 2005
abs(a(n)) is the characteristic function of A001082. - Michael Somos, Oct 31 2005
G.f.: Sum_{k in Z} (-1)^k * x^((3*k^2 - 2*k)) = Product_{k>0} (1 - x^(6*k)) * (1 - x^(6*k - 1)) * (1 - x^(6*k - 5)). - Michael Somos, Oct 31 2005
A002448(3*n + 1) = -2 * a(n). - Michael Somos, Jul 07 2006
a(n) = (-1)^n * A089801(n).
a(n) = -(1/n)*Sum_{k=1..n} A284362(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017

Corrected by N. J. A. Sloane, Nov 05 2005

A195839 Triangle read by rows which arises from A195829, in the same way as A175003 arises from A195310. Column k starts at row A118277(k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 2, -1, 5, 3, -1, 7, 4, -1, 10, 4, -2, 12, 4, -3, 13, 4, -4, 13, 5, -4, 14, 7, -4, -1, 16, 10, -4, -1, 21, 12, -5, -1, 27, 13, -7, -1, 32, 13, -10, -1, 34, 14, -12, -1, 36, 16, -13, -2, 1

Offset: 1

Omar E. Pol, Sep 24 2011

The sum of terms of row n is equal to the leftmost term of row n+1. This sequence is related to the generalized enneagonal numbers A118277, A195829 and A195849 in the same way as A175003 is related to the generalized pentagonal numbers A001318, A195310 and A000041. See comments in A195825.

			Written as a triangle:
.  1;
.  1;
.  1;
.  1;
.  1;
.  1,  1;
.  2,  1;
.  3,  1;
.  4,  1, -1;
.  4,  1, -1;
.  4,  1, -1;
.  4,  2, -1;
.  5,  3, -1;
.  7,  4, -1;
. 10,  4, -2;
. 12,  4, -3;
. 13,  4, -4;
. 13,  5, -4;
. 14,  7, -4, -1;
. 16, 10, -4, -1;
. 21, 12, -5, -1;

Row sums give A195849.

Cf. A001082, A175003, A195828, A195836, A195837, A195838.

A195840 Triangle read by rows which arises from A195830, in the same way as A175003 arises from A195310. Column k starts at row A074377(k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 2, -1, 5, 3, -1, 7, 4, -1, 10, 4, -2, 12, 4, -3, 13, 4, -4, 13, 4, -4, 13, 5, -4, 14, 7, -4, -1, 16, 10, -4, -1, 21, 12, -5, -1, 27, 13, -7, -1, 32, 13, -10, -1, 34

Offset: 1

Omar E. Pol, Sep 24 2011

The sum of terms of row n is equal to the leftmost term of row n+1. This sequence is related to the generalized decagonal numbers A074377, A195830 and A195850 in the same way as A175003 is related to the generalized pentagonal numbers A001318, A195310 and A000041. See comments in A195825.

			Written as a triangle:
.  1;
.  1;
.  1;
.  1;
.  1;
.  1;
.  1,  1;
.  2,  1;
.  3,  1;
.  4,  1,  -1;
.  4,  1,  -1;
.  4,  1,  -1;
.  4,  1,  -1;
.  4,  2,  -1;
.  5,  3,  -1;
.  7,  4,  -1;
. 10,  4,  -2;
. 12,  4,  -3;
. 13,  4,  -4;
. 13,  4,  -4;
. 13,  5,  -4;
. 14,  7,  -4,  -1;
. 16, 10,  -4,  -1;
. 21, 12,  -5,  -1;
. 27, 13,  -7,  -1;
. 32, 13, -10,  -1;
. 34, 13, -12,  -1,  1;

Row sums give A195850.

Cf. A001082, A175003, A195828, A195836, A195837, A195838, A195839.

A195841 Triangle read by rows which arises from A195831, in the same way as A175003 arises from A195310. Column k starts at row A195160(k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 2, -1, 5, 3, -1, 7, 4, -1, 10, 4, -2, 12, 4, -3, 13, 4, -4, 13, 4, -4, 13, 4, -4, 13, 5, -4, 14, 7, -4, -1, 16, 10, -4, -1, 21, 12, -5, -1, 27, 13, -7, -1

Offset: 1

Omar E. Pol, Sep 24 2011

The sum of terms of row n is equal to the leftmost term of row n+1. This sequence is related to the generalized hendecagonal numbers A195160, A195831 and A195851 in the same way as A175003 is related to the generalized pentagonal numbers A001318, A195310 and A000041. See comments in A195825.

			Written as a triangle:
1;
1;
1;
1;
1;
1;
1;
1, 1;
2, 1;
3, 1;
4, 1, -1;
4, 1, -1;
4, 1, -1;
4, 1, -1;
4, 1, -1;
4, 2, -1;
5, 3, -1;
7, 4, -1;

Row sums give A195851.

Cf. A001082, A175003, A195828, A195836, A195837, A195838, A195839, A195840.

A195842 Triangle read by rows which arises from A195832, in the same way as A175003 arises from A195310. Column k starts at row A195162(k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 2, -1, 5, 3, -1, 7, 4, -1, 10, 4, -2, 12, 4, -3, 13, 4, -4, 13, 4, -4, 13, 4, -4, 13, 4, -4, 13, 5, -4, 14, 7, -4, -1, 16, 10, -4, -1, 21, 12, -5, -1

Offset: 1

Omar E. Pol, Sep 24 2011

The sum of terms of row n is equal to the leftmost term of row n+1. This sequence is related to the generalized dodecagonal numbers A195162, A195832 and A195852 in the same way as A175003 is related to the generalized pentagonal numbers A001318, A195310 and A000041. See comments in A195825.

			Written as a triangle:
1;
1;
1;
1;
1;
1;
1;
1;
1, 1;
2, 1;
3, 1;
4, 1, -1;
4, 1, -1;
4, 1, -1;
4, 1, -1;
4, 1, -1;
4, 1, -1;
4, 2, -1;
5, 3, -1;
7, 4, -1;

Row sums give A195852.

Cf. A001082, A175003, A195828, A195836, A195837, A195838, A195839, A195840, A195841.

A158463 a(n) = 12*n^2 - 1.

Original entry on oeis.org

-1, 11, 47, 107, 191, 299, 431, 587, 767, 971, 1199, 1451, 1727, 2027, 2351, 2699, 3071, 3467, 3887, 4331, 4799, 5291, 5807, 6347, 6911, 7499, 8111, 8747, 9407, 10091, 10799, 11531, 12287, 13067, 13871, 14699, 15551, 16427, 17327, 18251, 19199, 20171, 21167, 22187

Offset: 0

Vincenzo Librandi, Mar 19 2009

Related to Legendre polynomials - see Mma line. - N. J. A. Sloane, Nov 17 2009
One notices that this sequence produces an inordinate number of semiprimes, perhaps better than mere chance for large values of n. - J. M. Bergot, Jun 30 2011
Sequence found by reading the line from -1, in the direction -1, 11, ..., in the square spiral whose vertices are -1 together with the generalized octagonal numbers A001082. - Omar E. Pol, Jul 18 2012

			G.f. = -1 + 11*x + 47*x^2 + 107*x^3 + 191*x^4 + 299*x^5 + 431*x^6 + 587*x^7 + 767*x^8 + ...

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

Cf. A001082, A080663, A135453.

[12*n^2 - 1: n in [0..100]]; // G. C. Greubel, Sep 25 2018

Table[Numerator[LegendreP[2,2n]],{n,0,50}] (* N. J. A. Sloane, Nov 17 2009 *)
a[ n_] := 2 LegendreP[ 2, 2 n] (* Michael Somos, Jul 01 2011 *)
12*Range[0,40]^2-1 (* or *) LinearRecurrence[{3,-3,1},{-1,11,47},50] (* Harvey P. Dale, Jun 22 2019 *)

{a(n) = 12 * n^2 - 1} /* Michael Somos, Nov 12 2011 */

a(-n) = a(n).
G.f.: (1-14*x-11*x^2)/(x-1)^3. - R. J. Mathar, Aug 27 2011
a(n) = A135453(n) - 1. - Omar E. Pol, Jul 18 2012
From Amiram Eldar, Feb 04 2021: (Start)
Sum_{n>=1} 1/a(n) = (1 - (Pi/sqrt(12))*cot(Pi/sqrt(12)))/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = ((Pi/sqrt(12))*csc(Pi/sqrt(12)) - 1)/2.
Product_{n>=1} (1 + 1/a(n)) = (Pi/sqrt(12))*csc(Pi/sqrt(12)).
Product_{n>=1} (1 - 1/a(n)) = csc(Pi/sqrt(12))*sin(Pi/sqrt(6))/sqrt(2). (End)
From Gerry Martens, Apr 06 2024: (Start)
a(n) = Re((2*n*i-1)^3).
a(n) = -8*(1/4+n^2)^(3/2)*cos(3*arctan(2*n)). (End)
From Elmo R. Oliveira, Jan 16 2025: (Start)
E.g.f.: exp(x)*(12*x^2 + 12*x - 1).
a(n) = A080663(2*n) for n > 0. (End)

Minor edits by N. J. A. Sloane, Nov 17 2009

Comment rewritten by Bruno Berselli, Aug 27 2011

A195843 Triangle read by rows which arises from A195833, in the same way as A175003 arises from A195310. Column k starts at row A195313(k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 1, -1, 4, 2, -1, 5, 3, -1, 7, 4, -1, 10, 4, -2, 12, 4, -3, 13, 4, -4, 13, 4, -4, 13, 4, -4, 13, 4, -4, 13, 4, -4, 13, 5, -4, 14, 7, -4, -1

Offset: 1

Omar E. Pol, Sep 24 2011

The sum of terms of row n is equal to the leftmost term of row n+1. This sequence is related to the generalized tridecagonal numbers A195313, A195833 and A196933 in the same way as A175003 is related to the generalized pentagonal numbers A001318, A195310 and A000041. See comments in A195825.

			Written as a triangle:
1;
1;
1;
1;
1;
1;
1;
1;
1;
1, 1;
2, 1;
3, 1;
4, 1, -1;
4, 1, -1;
4, 1, -1;
4, 1, -1;
4, 1, -1;
4, 1, -1;
4, 1, -1;
4, 2, -1;
5, 3, -1;
7, 4, -1;

Row sums give A196933.

Cf. A001082, A175003, A195828, A195836, A195837, A195838, A195839, A195840, A195841, A195842.

A284362 a(n) = Sum_{d|n, d = 0, 1, or 5 mod 6} d.

Original entry on oeis.org

1, 1, 1, 1, 6, 7, 8, 1, 1, 6, 12, 19, 14, 8, 6, 1, 18, 25, 20, 6, 8, 12, 24, 43, 31, 14, 1, 8, 30, 42, 32, 1, 12, 18, 48, 73, 38, 20, 14, 6, 42, 56, 44, 12, 6, 24, 48, 91, 57, 31, 18, 14, 54, 79, 72, 8, 20, 30, 60, 114, 62, 32, 8, 1, 84, 84, 68, 18, 24, 48, 72

Offset: 1

Seiichi Manyama, Mar 25 2017

			From _Peter Bala_, Dec 11 2020: (Start)
n = 20: n is not of the form m*(3*m +- 2), so e(n) = 0 and a(20) = a(19) + a(15) - a(12) - a(4) = 20 + 6 - 19 - 1 = 6;
n = 21: n = m*(3*m - 2) for m = 3, so e(n) = 21 and a(21) = 21 + a(20) + a(16) - a(13) - a(5) = 21 + 6 + 1 - 14 - 6 = 8;
n = 40: n = m*(3*m - 2) for m = 4, so e(n) = -40 and a(4) = -40 + a(39) + a(35) - a(32) - a(24) + a(19) + a(7) = -40 + 14 + 48  - 1 - 43 + 20 + 8 = 6. (End)

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A089802 (f(-x, -x^5)), A195848 (1/f(-x, -x^5)), A222171.

Cf. Sum_{d|n, d = 0, 1, or k-1 mod k} d: A000203 (k=3), A284361 (k=5), this sequence (k=6), A284363 (k=7), A284372 (k=12).

Table[Sum[If[Mod[d, 6] <2 || Mod[d, 6]==5, d, 0], {d, Divisors[n]}], {n, 80}] (* Indranil Ghosh, Mar 25 2017 *)

a(n) = sumdiv(n, d, ((d + 1) % 6 < 3) * d); \\ Amiram Eldar, Apr 12 2024

From Peter Bala, Dec 11 2020: (Start)
O.g.f.: Sum_{k >= 1} ( (6*k)*x^(6*k)/(1 - x^(6*k)) + (6*k-1)*x^(6*k-1)/(1 - x^(6*k-1)) + (6*k-5)*x^(6*k-5)/(1 - x^(6*k-5)) ).
Define a(n) = 0 for n < 1. Then a(n) = e(n) + a(n-1) + a(n-5) - a(n-8) - a(n-16) + + - -, where [1, 5, 8, 16, ...] is the sequence of generalized octagonal numbers A001082, and e(n) = (-1)^(m+1)*n if n is a generalized octagonal number of the form m*(3*m+-2); otherwise e(n) = 0. Examples of this recurrence are given below. (End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/24 = A222171 = 0.411233... . - Amiram Eldar, Apr 12 2024

A057780 Multiples of 3 that are one less than a perfect square.

Original entry on oeis.org

0, 3, 15, 24, 48, 63, 99, 120, 168, 195, 255, 288, 360, 399, 483, 528, 624, 675, 783, 840, 960, 1023, 1155, 1224, 1368, 1443, 1599, 1680, 1848, 1935, 2115, 2208, 2400, 2499, 2703, 2808, 3024, 3135, 3363, 3480, 3720, 3843, 4095, 4224, 4488, 4623, 4899, 5040

Offset: 1

Benjamin Geiger (benjamin_geiger(AT)yahoo.com), Nov 02 2000

Also, numbers of the form 9*m^2+6*m, m an integer. - Jason Kimberley, Nov 08 2012
k is in this list iff k+1 is in the support of A033684. - Jason Kimberley, Nov 13 2012
Exponents in the expansion of Product_{n >= 1} (1 - q^(6*n))^2 * (1 - q ^(9*n)) * (1 - q^(36*n))/((1 - q^(3*n))*(1 - q^(12*n))*(1 - q^(18*n))) = 1 + q^3 + q^15 + q^24 + q^48 + q^63 + q^99 + ... (see Oliver, Theorem 1.1). - Peter Bala, Jan 06 2025

Jason Kimberley, Table of n, a(n) for n = 1..2001
Robert J. Lemke Oliver, Eta quotients and theta functions, Advances in Mathematics, Vol. 241, Jul. 2013, pp. 1-17.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Numbers of the form 9n^2+kn, for integer n: A016766 (k=0), A132355 (k=2), A185039 (k=4), this sequence (k=6), A218864 (k=8). - Jason Kimberley, Nov 08 2012

a:=func;[0]cat[a(n*m):m in[-1, 1],n in[1..24]]; // Jason Kimberley, Nov 09 2012

Select[3*Range[0,2000],IntegerQ[Sqrt[#+1]]&] (* or *) LinearRecurrence[ {1,2,-2,-1,1},{0,3,15,24,48},50] (* Harvey P. Dale, Sep 10 2019 *)

concat(0, Vec(3*x^2*(1+4*x+x^2)/((1-x)^3*(1+x)^2) + O(x^100))) \\ Colin Barker, Dec 26 2015

a(n) = A001651(n)^2 - 1 = 3 * A001082(n).
G.f.: 3*x^2*(1+4*x+x^2) / ((1-x)^3*(1+x)^2). - Colin Barker, Nov 24 2012
From Colin Barker, Dec 26 2015: (Start)
a(n) = 3/8*(6*n^2-2*((-1)^n+3)*n+(-1)^n-1).
a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5) for n>5. (End)

Since this is a list, offset corrected to 1 by Jason Kimberley, Nov 09 2012

cp's OEIS Frontend

A132209 a(0) = 0 and a(n) = 2*n^2 + 2*n - 1, for n>=1.

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A089802 Expansion of q^(-1/3) * (theta_4(q^3) - theta_4(q^(1/3))) / 2 in powers of q.

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A195839 Triangle read by rows which arises from A195829, in the same way as A175003 arises from A195310. Column k starts at row A118277(k).

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A195840 Triangle read by rows which arises from A195830, in the same way as A175003 arises from A195310. Column k starts at row A074377(k).

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A195841 Triangle read by rows which arises from A195831, in the same way as A175003 arises from A195310. Column k starts at row A195160(k).

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A195842 Triangle read by rows which arises from A195832, in the same way as A175003 arises from A195310. Column k starts at row A195162(k).

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A158463 a(n) = 12*n^2 - 1.

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A195843 Triangle read by rows which arises from A195833, in the same way as A175003 arises from A195310. Column k starts at row A195313(k).

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A284362 a(n) = Sum_{d|n, d = 0, 1, or 5 mod 6} d.

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A132209 a(0) = 0 and a(n) = 2n^2 + 2n - 1, for n>=1.