A185019 -id:A185019 - cp's OEIS frontend

A033429 a(n) = 5*n^2.

0, 5, 20, 45, 80, 125, 180, 245, 320, 405, 500, 605, 720, 845, 980, 1125, 1280, 1445, 1620, 1805, 2000, 2205, 2420, 2645, 2880, 3125, 3380, 3645, 3920, 4205, 4500, 4805, 5120, 5445, 5780, 6125, 6480, 6845, 7220, 7605, 8000, 8405, 8820, 9245, 9680, 10125, 10580, 11045, 11520, 12005, 12500

Offset: 0

Jeff Burch

Number of edges of the complete bipartite graph of order 6n, K_n,5n. - Roberto E. Martinez II, Jan 07 2002
Number of edges of the complete tripartite graph of order 4n, K_n,n,2n. - Roberto E. Martinez II, Jan 07 2002
a(n+1)-a(n) : 5, 15, 25, 35, 45, ... (see A017329). - Philippe Deléham, Dec 08 2011
From Larry J Zimmermann, Feb 21 2013: (Start)
The sum of the areas of 2 squares that equals the area of a rectangle with whole number sides using the formula x^2 + y^2 = (x+y+sqrt(2*x*y))(x+y-sqrt(2*x*y)), where the substitution y=2*x obtains the whole number sides of the rectangle. So x^2+(2*x)^2=5x(x).
  x     squares sum      rectangle (l,w)   area
  1     1,4     5                   5,1    5
  2     4,16    20                  10,2   20  (End)

G. C. Greubel, Table of n, a(n) for n = 0..5000
Lancelot Hogben, Choice and Chance by Cardpack and Chessboard, Vol. 1, Max Parrish and Co, London, 1950, p. 36.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Central column of A055096.
Cf. A000290.
Cf. numbers of the form  n*(d*n+10-d)/2:  A008587, A056000, A028347, A140090, A014106, A028895, A045944, A186029, A007742, A022267, A022268, A049452, A186030, A135703, A152734, A139273.
Cf. A185019.
Cf. A001105, A033428.
Similar sequences are listed in A316466.

5*Range[50]^2 (* Alonso del Arte, May 23 2012 *)

PARI
```
a(n)=5*n^2
```

a(n) = 5*A000290(n). - Omar E. Pol, Dec 11 2008
From Bruno Berselli, Feb 11 2011: (Start)
G.f.: 5*x*(1+x)/(1-x)^3.
a(n) = 4*A000217(n) + A000567(n). (End)
a(n) = a(n-1)+5*(2*n-1) (with a(0)=0). - Vincenzo Librandi, Nov 17 2010
a(n) = A131242(10*n+4). - Philippe Deléham, Mar 27 2013
a(n) = a(n-1) + 10*n - 5, with a(0)=0. - Jean-Bernard François, Oct 04 2013
a(n) = A001105(n) + A033428(n). - Altug Alkan, Sep 28 2015
E.g.f.: 5*x*(x+1)*exp(x). - G. C. Greubel, Jul 17 2017
a(n) = Sum_{i = 2..6} P(i,n), where P(i,m) = m*((i-2)*m-(i-4))/2. - Bruno Berselli, Jul 04 2018
From Amiram Eldar, Feb 03 2021: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/30.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/60.
Product_{n>=1} (1 + 1/a(n)) = sqrt(5)*sinh(Pi/sqrt(5))/Pi.
Product_{n>=1} (1 - 1/a(n)) = sqrt(5)*sin(Pi/sqrt(5))/Pi. (End)

Better description from N. J. A. Sloane, May 15 1998

A049452 Pentagonal numbers with even index.

Original entry on oeis.org

0, 5, 22, 51, 92, 145, 210, 287, 376, 477, 590, 715, 852, 1001, 1162, 1335, 1520, 1717, 1926, 2147, 2380, 2625, 2882, 3151, 3432, 3725, 4030, 4347, 4676, 5017, 5370, 5735, 6112, 6501, 6902, 7315, 7740, 8177, 8626, 9087, 9560, 10045, 10542

Offset: 0

Joe Keane (jgk(AT)jgk.org)

If Y is a 3-subset of an (2n+1)-set X then, for n>=4, a(n-1) is the number of 4-subsets of X having at least two elements in common with Y. - Milan Janjic, Dec 16 2007
Sequence found by reading the line (one of the diagonal axes) from 0, in the direction 0, 5,..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - Omar E. Pol, Sep 08 2011
a(n) is the sum of 2*n consecutive integers starting from 2*n. - Bruno Berselli, Jan 16 2018

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. A000326, A033570, A049453, A001318, A033568, A185019.

See index to sequences with numbers of the form n*(d*n+10-d)/2 in A140090.

seq(n*(6*n-1),n=0..42); # Zerinvary Lajos, Jun 12 2007

Table[n(6n-1),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,5,22},50] (* Harvey P. Dale, Mar 07 2012 *)

a(n)=n*(6*n-1) \\ Charles R Greathouse IV, Sep 24 2015

a(n) = n*(6*n-1).
G.f.: x*(5+7*x)/(1-x)^3.
a(n) = C(6*n,2)/3. - Zerinvary Lajos, Jan 02 2007
a(n) = A001105(n) + A033991(n) = A033428(n) + A049450(n) = A022266(n) + A000326(n). - Zerinvary Lajos, Jun 12 2007
a(n) = 12*n + a(n-1) - 7 for n>0, a(0)=0. - Vincenzo Librandi, Aug 06 2010
a(n) = 4*A000217(n) + A001107(n). - Bruno Berselli, Feb 11 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2, a(0)=0, a(1)=5, a(2)=22. - Harvey P. Dale, Mar 07 2012
E.g.f.: (6*x^2 + 5*x)*exp(x). - G. C. Greubel, Jul 17 2017
From Amiram Eldar, Jul 03 2020: (Start)
Sum_{n>=1} 1/a(n) = 2*log(2) + 3*log(3)/2 - sqrt(3)*Pi/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi - log(2) - 2*sqrt(3)*arccoth(sqrt(3)). (End)

A049453 Second pentagonal numbers with even index: a(n) = n(6n+1).

Original entry on oeis.org

0, 7, 26, 57, 100, 155, 222, 301, 392, 495, 610, 737, 876, 1027, 1190, 1365, 1552, 1751, 1962, 2185, 2420, 2667, 2926, 3197, 3480, 3775, 4082, 4401, 4732, 5075, 5430, 5797, 6176, 6567, 6970, 7385, 7812, 8251, 8702, 9165, 9640, 10127, 10626, 11137, 11660, 12195

Offset: 0

Joe Keane (jgk(AT)jgk.org)

Number of edges in the join of the complete tripartite graph of order 3n and the cycle graph of order n, K_n,n,n * C_n. - Roberto E. Martinez II, Jan 07 2002
Sequence found by reading the line (one of the diagonal axes) from 0, in the direction 0, 7, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - Omar E. Pol, Sep 08 2011
First bisection of A036498. - Bruno Berselli, Nov 25 2012

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

Cf. A001318, A005449, A033568, A033570, A036498, A049452, A185019, A194454.

Cf. A254963 (comment).

seq(binomial(6*n+1,2)/3, n=0..42); # Zerinvary Lajos, Jan 21 2007

s=0;lst={s};Do[s+=n++ +7;AppendTo[lst, s], {n, 0, 7!, 12}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 16 2008 *)
Table[n*(6*n + 1), {n,0,50}] (* G. C. Greubel, Jun 07 2017 *)

x='x+O('x^50); concat([0], Vec(x*(7+5*x)/(1-x)^3)) \\ G. C. Greubel, Jun 07 2017

G.f.: x*(7+5*x)/(1-x)^3.
a(n) = 12*n + a(n-1) - 5 with n > 0, a(0)=0. - Vincenzo Librandi, Aug 06 2010
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - G. C. Greubel, Jun 07 2017
From Amiram Eldar, Feb 18 2022: (Start)
Sum_{n>=1} 1/a(n) = 6 - sqrt(3)*Pi/2 - 2*log(2) - 3*log(3)/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi + log(2) + sqrt(3)*log(2 + sqrt(3)) - 6. (End)
E.g.f.: exp(x)*x*(7 + 6*x). - Elmo R. Oliveira, Dec 12 2024

A193053 a(n) = (14n(n+3) + (2n-5)(-1)^n + 21)/16.

Original entry on oeis.org

1, 5, 10, 17, 26, 36, 49, 62, 79, 95, 116, 135, 160, 182, 211, 236, 269, 297, 334, 365, 406, 440, 485, 522, 571, 611, 664, 707, 764, 810, 871, 920, 985, 1037, 1106, 1161, 1234, 1292, 1369, 1430, 1511, 1575, 1660, 1727, 1816, 1886, 1979, 2052, 2149, 2225, 2326

Offset: 0

Bruno Berselli, Oct 20 2011 - based on remarks and sequences by Omar E. Pol

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

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. A195020 (vertices of the numerical spiral in link).

Cf. A001106, A022264, A033572, A144555, A152760, A158482, A158485, A185019, A195021, A195023-A195030, A195320, A198017 [incomplete list].

[(14*n*(n+3)+(2*n-5)*(-1)^n+21)/16: n in [0..50]];

Table[(14*n*(n + 3) + (2*n - 5)*(-1)^n + 21)/16, {n, 0, 50}] (* Vincenzo Librandi, Mar 26 2013 *)
LinearRecurrence[{1,2,-2,-1,1},{1,5,10,17,26},60] (* Harvey P. Dale, Jun 19 2020 *)

for(n=0, 50, print1((14*n*(n+3)+(2*n-5)*(-1)^n+21)/16", "));

O.g.f.: (1 + 4*x + 3*x^2 - x^3)/((1 + x)^2*(1 - x)^3).
E.g.f.: (1/16)*((21 + 56*x + 14*x^2)*exp(x) - (5 + 2*x)*exp(-x)). - G. C. Greubel, Aug 19 2017
a(n) = A195020(n) + n + 1.
a(n)   - a(-n-1)  = A047336(n+1).
a(n+1) - a(-n)    = A113804(n+1).
a(n+2) - a(n)     = A047385(n+3).
a(n+4) - a(n)     = A113803(n+4).
a(2*n) + a(2*n-1) = A069127(n+1).
a(2*n) - a(2*n-1) = A016813(n).
a(2*n+1) - a(2*n) = A016777(n+1).
a(n+2) + 2*a(n+1) + a(n) = A024966(n+2).

A195320 7 times hexagonal numbers: a(n) = 7n(2*n-1).

Original entry on oeis.org

0, 7, 42, 105, 196, 315, 462, 637, 840, 1071, 1330, 1617, 1932, 2275, 2646, 3045, 3472, 3927, 4410, 4921, 5460, 6027, 6622, 7245, 7896, 8575, 9282, 10017, 10780, 11571, 12390, 13237, 14112, 15015, 15946, 16905, 17892, 18907, 19950, 21021, 22120, 23247, 24402, 25585

Offset: 0

Omar E. Pol, Sep 18 2011

Sequence found by reading the line from 0, in the direction 0, 7, ..., in the square spiral whose vertices are the generalized enneagonal numbers A118277.

Also sequence found by reading the same line (mentioned above) in the Pythagorean spiral whose edges have length A195019 and whose vertices are the numbers A195020. This is the one of the semi-diagonals of the square spiral, which is related to the primitive Pythagorean triple [3, 4, 5]. - Omar E. Pol, Oct 13 2011

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 A024966.

Cf. A000384, A118277, A152746, A152750, A185019, A193053, A195019, A195020, A198017, A316466.

[7*n*(2*n-1): n in [0..50]]; // Vincenzo Librandi, Sep 28 2011

Table[7*n*(2*n - 1), {n, 0, 50}] (* Paolo Xausa, Jan 29 2025 *)
7*PolygonalNumber[6, Range[0, 50]] (* Paolo Xausa, Jan 29 2025 *)

a(n)=7*n*(2*n-1) \\ Charles R Greathouse IV, Sep 28 2015

a(n) = 14*n^2 - 7*n = 7*A000384(n).
G.f.: -7*x*(1+3*x)/(x-1)^3. - R. J. Mathar, Sep 27 2011
From Elmo R. Oliveira, Dec 27 2024: (Start)
E.g.f.: 7*exp(x)*x*(2*x + 1).
a(n) = A316466(n) - n = A024966(2*n+1).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A198017 a(n) = n(7n + 11)/2 + 1.

Original entry on oeis.org

1, 10, 26, 49, 79, 116, 160, 211, 269, 334, 406, 485, 571, 664, 764, 871, 985, 1106, 1234, 1369, 1511, 1660, 1816, 1979, 2149, 2326, 2510, 2701, 2899, 3104, 3316, 3535, 3761, 3994, 4234, 4481, 4735, 4996, 5264, 5539, 5821, 6110, 6406, 6709, 7019, 7336, 7660, 7991

Offset: 0

Bruno Berselli, Oct 21 2011 - based on remarks and sequences by Omar E. Pol

First bisection of A193053 (see also the numerical spiral illustrated in the Links section).

The inverse binomial transform yields 1, 9, 7, 0, 0 (0 continued).

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 (3,-3,1).

Cf. A195020 (vertices of the numerical spiral in link).
Cf. A017005 (first differences).
Cf. A069099, A001106.
Cf. A001106, A022264, A033572, A144555, A152760, A158482, A158485, A185019, A193053, A195021, A195023-A195030, A195320 [incomplete list].

Magma
```
[n*(7*n+11)/2+1: n in [0..47]];
```

Table[(n(7n+11))/2+1,{n,0,60}] (* or *) LinearRecurrence[{3,-3,1},{1,10,26},60] (* Harvey P. Dale, Mar 03 2013 *)

for(n=0, 47, print1(n*(7*n+11)/2+1", "));

G.f.: (1 + 7*x - x^2)/(1-x)^3.
a(n) = A195020(2*n) + 2*n + 1.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) = 2*a(n-1) - a(n-2) + 7.
From Elmo R. Oliveira, Dec 24 2024: (Start)
E.g.f.: exp(x)*(2 + 18*x + 7*x^2)/2.
a(n) = n + A001106(n+1). (End)

A135703 a(n) = n(7n-2).

Original entry on oeis.org

0, 5, 24, 57, 104, 165, 240, 329, 432, 549, 680, 825, 984, 1157, 1344, 1545, 1760, 1989, 2232, 2489, 2760, 3045, 3344, 3657, 3984, 4325, 4680, 5049, 5432, 5829, 6240, 6665, 7104, 7557, 8024, 8505, 9000, 9509, 10032, 10569, 11120, 11685, 12264, 12857, 13464

Offset: 0

N. J. A. Sloane, Mar 04 2008

G. C. Greubel, Table of n, a(n) for n = 0..2500
L. Hogben, Choice and Chance by Cardpack and Chessboard, Vol. 1, Max Parrish and Co, London, 1950, p. 36.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. index to numbers of the form n*(d*n+10-d)/2 in A014106.

Cf. A185019.

List([0..50], n-> n*(7*n-2)); # G. C. Greubel, Jul 04 2019

[n*(7*n-2): n in [0..50]]; // G. C. Greubel, Jul 04 2019

Array[ #*(7*# - 2) &, 50, 0] (* Zerinvary Lajos, Jul 10 2009 *)

a(n)=n*(7*n-2) \\ Charles R Greathouse IV, Oct 07 2015

[n*(7*n-2) for n in (0..50)] # G. C. Greubel, Jul 04 2019

a(n) = 5*n + 14*binomial(n,2).
From R. J. Mathar, Apr 21 2008: (Start)
O.g.f. x*(5+9*x)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
a(n) = a(n-1) + 14*n - 9 (with a(0)=0). - Vincenzo Librandi, Nov 24 2010
a(n) = 4*A000217(n) + A051624(n). - Bruno Berselli, Feb 11 2011
E.g.f.: x*(5 + 7*x)*exp(x). - G. C. Greubel, Oct 29 2016

A195021 a(n) = n(14n - 11).

Original entry on oeis.org

0, 3, 34, 93, 180, 295, 438, 609, 808, 1035, 1290, 1573, 1884, 2223, 2590, 2985, 3408, 3859, 4338, 4845, 5380, 5943, 6534, 7153, 7800, 8475, 9178, 9909, 10668, 11455, 12270, 13113, 13984, 14883, 15810, 16765, 17748, 18759, 19798, 20865, 21960, 23083, 24234, 25413

Offset: 0

Omar E. Pol, Sep 07 2011

Sequence found by reading the first two vertices [0, 3] together with the line from 34, in the direction 34, 93, ..., in the Pythagorean spiral whose edges have length A195019 and whose vertices are the numbers A195020, which is related to the primitive Pythagorean triple [3, 4, 5]. For another version see A195030.

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

Cf. A144555, A185019, A193053, A195019, A195020, A195023, A195024, A195025, A195030, A195320, A198017.

Cf. numbers of the form n*(n*k - k + 6)/2, this sequence is the case k=28: see Comments lines of A226492.

[14*n^2 - 11*n: n in [0..50]]; // Vincenzo Librandi, Oct 14 2011

Table[n*(14*n - 11), {n, 0, 50}] (* Paolo Xausa, Jan 15 2025 *)

a(n)=n*(14*n-11) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 14*n^2 - 11*n.
From Colin Barker, Apr 09 2012: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: x*(3+25*x)/(1-x)^3. (End)
E.g.f.: exp(x)*x*(3 + 14*x). - Elmo R. Oliveira, Dec 30 2024

Edited by Bruno Berselli, Oct 18 2011

A195024 a(n) = n(14n - 1).

Original entry on oeis.org

0, 13, 54, 123, 220, 345, 498, 679, 888, 1125, 1390, 1683, 2004, 2353, 2730, 3135, 3568, 4029, 4518, 5035, 5580, 6153, 6754, 7383, 8040, 8725, 9438, 10179, 10948, 11745, 12570, 13423, 14304, 15213, 16150, 17115, 18108, 19129, 20178, 21255, 22360, 23493, 24654, 25843

Offset: 0

Omar E. Pol, Oct 13 2011

Related to the primitive Pythagorean triple [3, 4, 5].
Sequence found by reading the line from 0, in the direction 0, 13, ..., in the Pythagorean spiral whose edges have length A195019 and whose vertices are the numbers A195020. This is the one of the semi-diagonals of the square spiral.
Also sequence found by reading the line from 0, in the direction 0, 13, ..., in the square spiral whose vertices are the generalized 9-gonal numbers A118277. - Omar E. Pol, Jul 28 2012

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

Cf. A118277, A144555, A152760, A185019, A193053, A195019, A195020, A195320, A198017.

[14*n^2 - n: n in [0..50]]; // Vincenzo Librandi, Oct 14 2011

Table[n(14n-1),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,13,54},50] (* Harvey P. Dale, Jul 28 2012 *)

a(n)=n*(14*n-1) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 14*n^2 - n.
From Colin Barker, Apr 09 2012: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: x*(13+15*x)/(1-x)^3. (End)
E.g.f.: exp(x)*x*(13 + 14*x). - Elmo R. Oliveira, Jan 12 2025

A195025 a(n) = n(14n + 3).

Original entry on oeis.org

0, 17, 62, 135, 236, 365, 522, 707, 920, 1161, 1430, 1727, 2052, 2405, 2786, 3195, 3632, 4097, 4590, 5111, 5660, 6237, 6842, 7475, 8136, 8825, 9542, 10287, 11060, 11861, 12690, 13547, 14432, 15345, 16286, 17255, 18252, 19277, 20330, 21411, 22520, 23657, 24822, 26015

Offset: 0

Omar E. Pol, Oct 13 2011

Sequence found by reading the line from 0, in the direction 0, 17, ..., in the Pythagorean spiral whose edges have length A195019 and whose vertices are the numbers A195020. This is the one of the semi-axis of the square spiral, which is related to the primitive Pythagorean triple [3, 4, 5].

a(k) is a square for k = (3/56)*((449 + 120*sqrt(14))^n + (449 - 120*sqrt(14))^n - 2). - Bruno Berselli, Oct 18 2011

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

Cf. A144555, A152760, A185019, A193053, A195019, A195020, A195023, A195024, A195320.

[14*n^2 +3*n: n in [0..50]]; // Vincenzo Librandi, Oct 14 2011

Table[n(14n+3),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{0,17,62},50] (* Harvey P. Dale, Jul 17 2023 *)

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

a(n) =  14*n^2 + 3*n.
G.f.: x*(17+11*x)/(1-x)^3. - Bruno Berselli, Oct 18 2011
From Elmo R. Oliveira, Dec 30 2024: (Start)
E.g.f.: exp(x)*x*(17 + 14*x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3. (End)

Name suggested by Bruno Berselli, Oct 13 2011

cp's OEIS Frontend

A033429 a(n) = 5*n^2.

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A049452 Pentagonal numbers with even index.

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A049453 Second pentagonal numbers with even index: a(n) = n*(6*n+1).

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A193053 a(n) = (14*n*(n+3) + (2*n-5)*(-1)^n + 21)/16.

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A195320 7 times hexagonal numbers: a(n) = 7*n*(2*n-1).

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A198017 a(n) = n*(7*n + 11)/2 + 1.

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A135703 a(n) = n*(7*n-2).

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A049453 Second pentagonal numbers with even index: a(n) = n(6n+1).

A193053 a(n) = (14n(n+3) + (2n-5)(-1)^n + 21)/16.

A195320 7 times hexagonal numbers: a(n) = 7n(2*n-1).

A198017 a(n) = n(7n + 11)/2 + 1.

A135703 a(n) = n(7n-2).

A195021 a(n) = n(14n - 11).

A195024 a(n) = n(14n - 1).

A195025 a(n) = n(14n + 3).