A226488 -id:A226488 - cp's OEIS frontend

A139267 Twice octagonal numbers: 2n(3*n-2).

0, 2, 16, 42, 80, 130, 192, 266, 352, 450, 560, 682, 816, 962, 1120, 1290, 1472, 1666, 1872, 2090, 2320, 2562, 2816, 3082, 3360, 3650, 3952, 4266, 4592, 4930, 5280, 5642, 6016, 6402, 6800, 7210, 7632, 8066, 8512, 8970, 9440, 9922

Offset: 0

Omar E. Pol, May 14 2008, May 19 2008

Sequence found by reading the line from 0, in the direction 0, 2,..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. Opposite numbers to the members of A033580 in the same spiral. - Omar E. Pol, Sep 09 2011

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. A000567, A017545.

Cf. numbers of the form n*(n*k-k+4)/2 listed in A226488 (this sequence is the case k=12).

List([0..50], n-> 2*n*(3*n-2)); # G. C. Greubel, Sep 18 2019

[2*n*(3*n-2): n in [0..50]]; // G. C. Greubel, Sep 18 2019

seq(2*n*(3*n-2), n=0..50); # G. C. Greubel, Sep 18 2019

Table[2*n*(3*n-2), {n,0,50}] (* G. C. Greubel, Jun 07 2017 *)

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

[2*n*(3*n-2) for n in (0..50)] # G. C. Greubel, Sep 18 2019

a(n) = 2*A000567(n) = 6*n^2 - 4*n = 2*n*(3*n - 2).
a(n) = a(n-1) + 12*n - 10, with n>0, a(0)=0. - Vincenzo Librandi, Aug 03 2010
G.f.: x*(2+10*x)/(1-3*x+3*x^2-x^3). - Colin Barker, Jan 06 2012
After 0, a(n) = Sum_{i=0..n-1} (12*i + 2). - Bruno Berselli, Sep 11 2013
E.g.f.: 2*x*(1 + 3*x)*exp(x). - G. C. Greubel, Sep 18 2019

A180223 a(n) = (11n^2 - 7n)/2.

Original entry on oeis.org

0, 2, 15, 39, 74, 120, 177, 245, 324, 414, 515, 627, 750, 884, 1029, 1185, 1352, 1530, 1719, 1919, 2130, 2352, 2585, 2829, 3084, 3350, 3627, 3915, 4214, 4524, 4845, 5177, 5520, 5874, 6239, 6615, 7002, 7400, 7809, 8229, 8660

Offset: 0

Graziano Aglietti (mg5055(AT)mclink.it), Aug 16 2010

This sequence is related to A050441 by n*a(n) - Sum_{i=0..n-1} a(i) = 2*A050441(n). - Bruno Berselli, Aug 19 2010
Sum of n-th heptagonal number (A000566) and n-th octagonal number (A000567). - Bruno Berselli, Jun 11 2013
Create a triangle with T(r,1) = r^2 and T(r,c) = r^2 + r*c + c^2. The difference of the sum of the terms in row n and those in row n-1 is a(n). - J. M. Bergot, Jun 17 2013

B. Berselli, Table of n, a(n) for n = 0..10000.
B. Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian).
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A050441, A000566, A000567, A226492.

Cf. numbers of the form n*(n*k-k+4)/2 listed in A226488 (this sequence is the case k=11). - Bruno Berselli, Jun 10 2013

List([0..30], n-> n*(11*n-7)/2); # G. C. Greubel, Sep 18 2019

[(11*n^2 - 7*n)/2: n in [0..30]]; // Vincenzo Librandi, Apr 18 2011

A180223:=n->(11*n^2 - 7*n)/2; seq(A180223(n), n=0..30); # Wesley Ivan Hurt, Feb 25 2014

Table[(11*n^2 - 7*n)/2, {n, 0, 30}] (* Wesley Ivan Hurt, Feb 25 2014 *)
LinearRecurrence[{3,-3,1},{0,2,15},50] (* Harvey P. Dale, Oct 10 2020 *)

PARI
```
a(n)=1/2*(11*n^2 - 7*n);
```

[n*(11*n-7)/2 for n in (0..30)] # G. C. Greubel, Sep 18 2019

G.f.: x*(2+9*x)/(1-x)^3. - Bruno Berselli, Aug 19 2010 - corrected in Apr 18 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) with n>2. - Bruno Berselli, Aug 19 2010
a(n) = n + A226492(n). - Bruno Berselli, Jun 11 2013
E.g.f.: x*(4 + 11*x)*exp(x)/2. - G. C. Greubel, Aug 24 2015

A218471 a(n) = n(7n-3)/2.

Original entry on oeis.org

0, 2, 11, 27, 50, 80, 117, 161, 212, 270, 335, 407, 486, 572, 665, 765, 872, 986, 1107, 1235, 1370, 1512, 1661, 1817, 1980, 2150, 2327, 2511, 2702, 2900, 3105, 3317, 3536, 3762, 3995, 4235, 4482, 4736, 4997, 5265, 5540, 5822, 6111, 6407, 6710, 7020, 7337

Offset: 0

Philippe Deléham, Mar 26 2013

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. A001106, A022264.

Cf. numbers of the form n*(n*k-k+4)/2 listed in A226488 (this sequence is the case k=7). - Bruno Berselli, Jun 10 2013

List([0..50], n-> n*(7*n-3)/2); # G. C. Greubel, Aug 31 2019

[n*(7*n-3)/2: n in [0..50]]; // G. C. Greubel, Aug 31 2019

seq(n*(7*n-3)/2, n=0..50); # G. C. Greubel, Aug 31 2019

Table[n*(7*n-3)/2, {n,0,50}] (* G. C. Greubel, Aug 23 2017 *)

a(n)=n*(7*n-3)/2 \\ Charles R Greathouse IV, Jun 17 2017

[n*(7*n-3)/2 for n in (0..50)] # G. C. Greubel, Aug 31 2019

G.f.: x*(2+5*x)/(1-x)^3.
a(n) = 3*a(n-1) -3*a(n-2) +a(n-3) with a(0)=0, a(1)=2, a(2)=11.
a(n) = A001106(n) +  n.
a(n) = A022264(n) -  n.
a(n) = A022265(n) - 2*n.
a(n) = A186029(n) - 3*n.
a(n) = A179986(n) - 4*n.
a(n) = A024966(n) - 5*n.
a(n) = A174738(7*n+1).
E.g.f.: (x/2)*(7*x + 4)*exp(x). - G. C. Greubel, Aug 23 2017

A152965 Twice 12-gonal numbers: a(n) = 2n(5*n-4).

Original entry on oeis.org

0, 2, 24, 66, 128, 210, 312, 434, 576, 738, 920, 1122, 1344, 1586, 1848, 2130, 2432, 2754, 3096, 3458, 3840, 4242, 4664, 5106, 5568, 6050, 6552, 7074, 7616, 8178, 8760, 9362, 9984, 10626, 11288, 11970, 12672, 13394, 14136, 14898, 15680, 16482, 17304, 18146, 19008

Offset: 0

Omar E. Pol, Dec 21 2008

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

Cf. A051624 (12-gonal numbers), A051874.

Cf. numbers of the form n*(n*k - k + 4)/2 listed in A226488 (this sequence is the case k=20). - Bruno Berselli, Jun 10 2013

[10*n^2-8*n: n in [0..50]]; // Klaus Brockhaus, Nov 27 2010

Table[10*n^2-8*n,{n,0,50}] (* Vincenzo Librandi, Jul 10 2012 *)
LinearRecurrence[{3,-3,1},{0,2,24},60] (* Harvey P. Dale, Apr 18 2016 *)

a(n)=2*n*(5*n-4) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 2*A051624(n).
From Vincenzo Librandi, Jul 10 2012: (Start)
G.f.: 2*x*(1+9*x)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
From Elmo R. Oliveira, Dec 27 2024: (Start)
E.g.f.: 2*exp(x)*x*(1 + 5*x).
a(n) = n + A051874(n). (End)

A226489 a(n) = n(15n-11)/2.

Original entry on oeis.org

0, 2, 19, 51, 98, 160, 237, 329, 436, 558, 695, 847, 1014, 1196, 1393, 1605, 1832, 2074, 2331, 2603, 2890, 3192, 3509, 3841, 4188, 4550, 4927, 5319, 5726, 6148, 6585, 7037, 7504, 7986, 8483, 8995, 9522, 10064, 10621, 11193, 11780, 12382, 12999, 13631, 14278, 14940

Offset: 0

Bruno Berselli, Jun 09 2013

Sum of n-th 9-gonal (nonagonal) number and n-th 10-gonal (decagonal) number.

Sum of reciprocals of a(n), for n > 0: 0.614629940137818703272919217222307...

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

Cf. A001106, A001107, A064761.

Cf. numbers of the form n*(n*k - k + 4)/2, this sequence is the case k=15: see list in A226488.

Magma
```
[n*(15*n-11)/2: n in [0..50]];
```

I:=[0,2,19]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..45]]; // Vincenzo Librandi, Aug 18 2013

Table[n (15 n - 11)/2, {n, 0, 50}]
CoefficientList[Series[x (2 + 13 x) / (1 - x)^3, {x, 0, 45}], x] (* Vincenzo Librandi, Aug 18 2013 *)

a(n)=n*(15*n-11)/2 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: x*(2+13*x)/(1-x)^3.
a(n) + a(-n) = A064761(n).
From Elmo R. Oliveira, Jan 12 2025: (Start)
E.g.f.: exp(x)*x*(4 + 15*x)/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A152995 Twice 11-gonal numbers: a(n) = n(9n-7).

Original entry on oeis.org

0, 2, 22, 60, 116, 190, 282, 392, 520, 666, 830, 1012, 1212, 1430, 1666, 1920, 2192, 2482, 2790, 3116, 3460, 3822, 4202, 4600, 5016, 5450, 5902, 6372, 6860, 7366, 7890, 8432, 8992, 9570, 10166, 10780, 11412, 12062, 12730, 13416, 14120

Offset: 0

Omar E. Pol, Dec 22 2008

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. A051682 (11-gonal numbers).

Cf. A226488.

List([0..50], n-> n*(9*n-7)); # G. C. Greubel, Sep 01 2019

Magma
```
[n*(9*n-7): n in [0..50]];
```

seq(n*(9*n-7), n=0..50); # G. C. Greubel, Sep 01 2019

Table[n(9n-7),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{0,2,22},40] (* Harvey P. Dale, Nov 02 2011 *)
2*PolygonalNumber[11,Range[0,40]] (* Harvey P. Dale, May 31 2024 *)

a(n)=n*(9*n-7) \\ Charles R Greathouse IV, Jun 17 2017

[n*(9*n-7) for n in (0..50)] # G. C. Greubel, Sep 01 2019

a(n) = 9*n^2 - 7*n = A051682(n)*2.
a(n) = a(n-1) + 18*n - 16 (with a(0)=0). - Vincenzo Librandi, Nov 27 2010
a(0)=0, a(1)=2, a(2)=22, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Nov 02 2011
From G. C. Greubel, Sep 01 2019: (Start)
G.f.: 2*x*(1+8*x)/(1-x)^3.
E.g.f.: x*(2+9*x)*exp(x). (End)

A226490 a(n) = n(19n-15)/2.

Original entry on oeis.org

0, 2, 23, 63, 122, 200, 297, 413, 548, 702, 875, 1067, 1278, 1508, 1757, 2025, 2312, 2618, 2943, 3287, 3650, 4032, 4433, 4853, 5292, 5750, 6227, 6723, 7238, 7772, 8325, 8897, 9488, 10098, 10727, 11375, 12042, 12728, 13433, 14157, 14900, 15662, 16443, 17243, 18062

Offset: 0

Bruno Berselli, Jun 09 2013

Sum of n-th hendecagonal number and n-th dodecagonal number.

Sum of reciprocals of a(n), for n > 0: 0.59314195720519963010713286193275...

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

Cf. A051624, A051682, A051873.

Cf. numbers of the form n*(n*k - k + 4)/2, this sequence is the case k=19: see list in A226488.

Magma
```
[n*(19*n-15)/2: n in [0..50]];
```

I:=[0,2,23]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..45]]; // Vincenzo Librandi, Aug 18 2013

Table[n (19 n - 15)/2, {n, 0, 50}]
CoefficientList[Series[x (2 + 17 x) / (1 - x)^3, {x, 0, 45}], x] (* Vincenzo Librandi, Aug 18 2013 *)
LinearRecurrence[{3,-3,1},{0,2,23},50] (* Harvey P. Dale, Aug 17 2017 *)

a(n)=n*(19*n-15)/2 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: x*(2+17*x)/(1-x)^3.
From Elmo R. Oliveira, Dec 27 2024: (Start)
E.g.f.: exp(x)*x*(4 + 19*x)/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
a(n) = n + A051873(n). (End)

A226491 a(n) = n(21n-17)/2.

Original entry on oeis.org

0, 2, 25, 69, 134, 220, 327, 455, 604, 774, 965, 1177, 1410, 1664, 1939, 2235, 2552, 2890, 3249, 3629, 4030, 4452, 4895, 5359, 5844, 6350, 6877, 7425, 7994, 8584, 9195, 9827, 10480, 11154, 11849, 12565, 13302, 14060, 14839, 15639, 16460, 17302, 18165, 19049, 19954

Offset: 0

Bruno Berselli, Jun 09 2013

Sum of n-th dodecagonal number and n-th tridecagonal number.

Sum of reciprocals of a(n), for n > 0: 0.58517199913243139233033474262449...

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

Cf. A051624, A051865, A064762.

Cf. numbers of the form n*(n*k - k + 4)/2, this sequence is the case k=21: see list in A226488.

Magma
```
[n*(21*n-17)/2: n in [0..50]];
```

I:=[0,2,25]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..45]]; // Vincenzo Librandi, Aug 18 2013

Table[n (21 n - 17)/2, {n, 0, 50}]
CoefficientList[Series[x (2 + 19 x) / (1 - x)^3, {x, 0, 45}], x] (* Vincenzo Librandi, Aug 18 2013 *)
LinearRecurrence[{3,-3,1},{0,2,25},50] (* Harvey P. Dale, Feb 01 2023 *)

a(n)=n*(21*n-17)/2 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: x*(2+19*x)/(1-x)^3.
a(n) + a(-n) = A064762(n).
From Elmo R. Oliveira, Jan 12 2025: (Start)
E.g.f.: exp(x)*x*(4 + 21*x)/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A139268 Twice nonagonal numbers (or twice 9-gonal numbers): a(n) = n(7n-5).

Original entry on oeis.org

0, 2, 18, 48, 92, 150, 222, 308, 408, 522, 650, 792, 948, 1118, 1302, 1500, 1712, 1938, 2178, 2432, 2700, 2982, 3278, 3588, 3912, 4250, 4602, 4968, 5348, 5742, 6150, 6572, 7008, 7458, 7922, 8400, 8892, 9398, 9918, 10452, 11000

Offset: 0

Omar E. Pol, May 15 2008

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

Cf. A001106, A051868.

Cf. numbers of the form n*(n*k - k + 4)/2 listed in A226488 (this sequence is the case k=14). - Bruno Berselli, Jun 10 2013

s=0;lst={s};Do[s+=n;AppendTo[lst,s],{n,2,6!,14}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 02 2009 *)
2*PolygonalNumber[9,Range[0,40]] (* or *) LinearRecurrence[{3,-3,1},{0,2,18},50] (* Harvey P. Dale, Feb 08 2024 *)

a(n)=n*(7*n-5) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 2*A001106(n) = 7*n^2 - 5*n = n*(7*n-5).
a(n) = 14*n + a(n-1) - 12, with a(0)=0. - Vincenzo Librandi, Aug 03 2010
G.f.: 2*x*(1 + 6*x)/(1 - x)^3. - Philippe Deléham, Apr 03 2013
From Elmo R. Oliveira, Dec 27 2024: (Start)
E.g.f.: exp(x)*x*(2 + 7*x).
a(n) = n + A051868(n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A152997 Twice 13-gonal numbers: a(n) = n(11n - 9).

Original entry on oeis.org

0, 2, 26, 72, 140, 230, 342, 476, 632, 810, 1010, 1232, 1476, 1742, 2030, 2340, 2672, 3026, 3402, 3800, 4220, 4662, 5126, 5612, 6120, 6650, 7202, 7776, 8372, 8990, 9630, 10292, 10976, 11682, 12410, 13160, 13932, 14726, 15542, 16380, 17240, 18122, 19026, 19952, 20900, 21870, 22862, 23876

Offset: 0

Omar E. Pol, Dec 22 2008

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. A051865 (13-gonal numbers).

Cf. numbers of the form n*(n*k - k + 4)/2 listed in A226488 (this sequence is the case k=22). - Bruno Berselli, Jun 10 2013

List([0..50], n-> n*(11*n-9)); # G. C. Greubel, Sep 01 2019

Magma
```
[n*(11*n-9): n in [0..50]];
```

seq(n*(11*n-9), n=0..50); # G. C. Greubel, Sep 01 2019

Table[n*(11*n-9), {n,0,50}] (* G. C. Greubel, Sep 01 2019 *)

a(n)=n*(11*n-9) \\ Charles R Greathouse IV, Jun 17 2017

[n*(11*n-9) for n in (0..50)] # G. C. Greubel, Sep 01 2019

a(n) = 11*n^2 - 9*n = A051865(n)*2.
a(n) = a(n-1) + 22*n - 20 (with a(0)=0). - Vincenzo Librandi, Nov 27 2010
From G. C. Greubel, Sep 01 2019: (Start)
G.f.: 2*x*(1 + 10*x)/(1-x)^3.
E.g.f.: x*(2 + 11*x)*exp(x). (End)

Terms a(39) onward added by G. C. Greubel, Sep 01 2019

cp's OEIS Frontend

A139267 Twice octagonal numbers: 2*n*(3*n-2).

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A180223 a(n) = (11*n^2 - 7*n)/2.

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A218471 a(n) = n*(7*n-3)/2.

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A152965 Twice 12-gonal numbers: a(n) = 2*n*(5*n-4).

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Magma

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A226489 a(n) = n*(15*n-11)/2.

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Magma

Magma

Mathematica

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A152995 Twice 11-gonal numbers: a(n) = n*(9*n-7).

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Magma

Maple

Mathematica

A139267 Twice octagonal numbers: 2n(3*n-2).

A180223 a(n) = (11n^2 - 7n)/2.

A218471 a(n) = n(7n-3)/2.

A152965 Twice 12-gonal numbers: a(n) = 2n(5*n-4).

A226489 a(n) = n(15n-11)/2.

A152995 Twice 11-gonal numbers: a(n) = n(9n-7).

A226490 a(n) = n(19n-15)/2.

A226491 a(n) = n(21n-17)/2.

A139268 Twice nonagonal numbers (or twice 9-gonal numbers): a(n) = n(7n-5).

A152997 Twice 13-gonal numbers: a(n) = n(11n - 9).