A139267 -id:A139267 - cp's OEIS frontend

A226488 a(n) = n(13n - 9)/2.

0, 2, 17, 45, 86, 140, 207, 287, 380, 486, 605, 737, 882, 1040, 1211, 1395, 1592, 1802, 2025, 2261, 2510, 2772, 3047, 3335, 3636, 3950, 4277, 4617, 4970, 5336, 5715, 6107, 6512, 6930, 7361, 7805, 8262, 8732, 9215, 9711, 10220, 10742, 11277, 11825, 12386, 12960

Offset: 0

Bruno Berselli, Jun 09 2013

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

Sum of reciprocals of a(n), for n>0: 0.629618994194109711163742089971688...

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

Cf. A000567, A001106, A153080 (first differences).

Cf. numbers of the form n*(n*k-k+4)/2 listed in A005843 (k=0), A000096 (k=1), A002378 (k=2), A005449 (k=3), A001105 (k=4), A005476 (k=5), A049450 (k=6), A218471 (k=7), A002939 (k=8), A062708 (k=9), A135706 (k=10), A180223 (k=11), A139267 (n=12), this sequence (k=13), A139268 (k=14), A226489 (k=15), A139271 (k=16), A180232 (k=17), A152995 (k=18), A226490 (k=19), A152965 (k=20), A226491 (k=21), A152997 (k=22).

List([0..50], n-> n*(13*n-9)/2); # G. C. Greubel, Aug 30 2019

Magma
```
[n*(13*n-9)/2: n in [0..50]];
```

I:=[0,2,17]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2) +Self(n-3): n in [1..50]]; // Vincenzo Librandi, Aug 18 2013

A226488:=n->n*(13*n - 9)/2; seq(A226488(n), n=0..50); # Wesley Ivan Hurt, Feb 25 2014

Table[n(13n-9)/2, {n, 0, 50}]
LinearRecurrence[{3, -3, 1}, {0, 2, 17}, 50] (* Harvey P. Dale, Jun 19 2013 *)
CoefficientList[Series[x(2+11x)/(1-x)^3, {x, 0, 45}], x] (* Vincenzo Librandi, Aug 18 2013 *)

a(n)=n*(13*n-9)/2 \\ Charles R Greathouse IV, Sep 24 2015

[n*(13*n-9)/2 for n in (0..50)] # G. C. Greubel, Aug 30 2019

G.f.: x*(2+11*x)/(1-x)^3.
a(n) + a(-n) = A152742(n).
a(0)=0, a(1)=2, a(2)=17; for n>2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Jun 19 2013
E.g.f.: x*(4 + 13*x)*exp(x)/2. - G. C. Greubel, Aug 30 2019
a(n) = A000567(n) + A001106(n). - Michel Marcus, Aug 31 2019

A152751 3 times octagonal numbers: a(n) = 3n(3*n-2).

Original entry on oeis.org

0, 3, 24, 63, 120, 195, 288, 399, 528, 675, 840, 1023, 1224, 1443, 1680, 1935, 2208, 2499, 2808, 3135, 3480, 3843, 4224, 4623, 5040, 5475, 5928, 6399, 6888, 7395, 7920, 8463, 9024, 9603, 10200, 10815, 11448, 12099, 12768, 13455, 14160, 14883, 15624, 16383, 17160

Offset: 0

Omar E. Pol, Dec 12 2008

a(n) also can be represented as n concentric triangles (see example). - Omar E. Pol, Aug 21 2011

			From _Omar E. Pol_, Aug 21 2011: (Start)
Illustration of initial terms as concentric triangles:
.
.                                          o
.                                         o o
.                                        o   o
.                                       o     o
.                 o                    o   o   o
.                o o                  o   o o   o
.               o   o                o   o   o   o
.              o     o              o   o     o   o
.    o        o   o   o            o   o   o   o   o
.   o o      o   o o   o          o   o   o o   o   o
.           o           o        o   o           o   o
.          o o o o o o o o      o   o o o o o o o o   o
.                              o                       o
.                             o o o o o o o o o o o o o o
.
.    3            24                       63
(End)

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

Cf. A000567, A064201, A139267, A152995.
3 times n-gonal numbers: A045943, A033428, A062741, A094159, A152773, A152759, A152767, A153783, A153448, A153875.
Cf. A033581, A085250, A152734, A194273. - Omar E. Pol, Aug 21 2011
Cf. numbers of the form n*(n*k - k + 6)/2, this sequence is the case k=18: see Comments lines of A226492.

s=0;lst={s};Do[s+=n;AppendTo[lst,s],{n,3,6!,18}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 02 2009 *)
3*PolygonalNumber[8,Range[0,40]] (* Harvey P. Dale, May 08 2022 *)

a(n)=3*n*(3*n-2) \\ Charles R Greathouse IV, Sep 24 2015

a(n) = 9*n^2 - 6*n = 3*A000567(n) = A064201(n)/3.
a(n) = a(n-1) + 18*n - 15 with n > 0, a(0)=0. - Vincenzo Librandi, Nov 26 2010
G.f.: 3*x*(1+5*x)/(1-x)^3. - Bruno Berselli, Jan 21 2011
From Elmo R. Oliveira, Dec 25 2024: (Start)
E.g.f.: 3*exp(x)*x*(1 + 3*x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3.
a(n) = n + A152995(n). (End)

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

Original entry on oeis.org

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)

A153794 4 times octagonal numbers: a(n) = 4n(3*n-2).

Original entry on oeis.org

0, 4, 32, 84, 160, 260, 384, 532, 704, 900, 1120, 1364, 1632, 1924, 2240, 2580, 2944, 3332, 3744, 4180, 4640, 5124, 5632, 6164, 6720, 7300, 7904, 8532, 9184, 9860, 10560, 11284, 12032, 12804, 13600, 14420, 15264, 16132, 17024

Offset: 0

Omar E. Pol, Jan 19 2009

Sequence found by reading the segment (0, 4) together with the line from 4, in the direction 4, 32, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. - Omar E. Pol, Jul 18 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. A000567, A139267, A152751, A153808.

s=0;lst={s};Do[s+=n;AppendTo[lst,s],{n,4,7!,24}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 02 2009 *)
Table[4n(3n-2),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{0,4,32},41] (* Harvey P. Dale, Jul 14 2011 *)
4*PolygonalNumber[8,Range[0,40]] (* Harvey P. Dale, Dec 04 2022 *)

a(n) = 12*n^2 - 8*n; \\ Altug Alkan, Aug 29 2016

a(n) = 12*n^2 - 8*n = 4*A000567(n) = 2*A139267(n).
a(n) = 24*n + a(n-1) - 20 (with a(0)=0). - Vincenzo Librandi, Aug 03 2010
a(0)=0, a(1)=4, a(2)=32, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Jul 14 2011
G.f.: 4*(x + 5*x^2)/(1-x)^3. - Harvey P. Dale, Jul 14 2011
E.g.f.: 4*x*(1 + 3*x)*exp(x). - G. C. Greubel, Aug 29 2016

A153796 6 times octagonal numbers: a(n) = 6n(3*n-2).

Original entry on oeis.org

0, 6, 48, 126, 240, 390, 576, 798, 1056, 1350, 1680, 2046, 2448, 2886, 3360, 3870, 4416, 4998, 5616, 6270, 6960, 7686, 8448, 9246, 10080, 10950, 11856, 12798, 13776, 14790, 15840, 16926, 18048, 19206, 20400, 21630, 22896, 24198, 25536, 26910, 28320, 29766

Offset: 0

Omar E. Pol, Jan 19 2009

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, A139267, A152751.

[6*n*(3*n-2): n in [0..60]]; // Wesley Ivan Hurt, Aug 29 2016

A153796:=n->6*n*(3*n-2): seq(A153796(n), n=0..60); # Wesley Ivan Hurt, Aug 29 2016

s=0;lst={s};Do[s+=n;AppendTo[lst,s],{n,6,8!,36}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 03 2009 *)
Table[6*n*(3*n-2), {n,0,25}] (* or *) LinearRecurrence[{3,-3,1}, {0,6,48}, 25] (* G. C. Greubel, Aug 29 2016 *)
6*PolygonalNumber[8,Range[0,50]] (* Harvey P. Dale, Dec 17 2023 *)

a(n)=6*n*(3*n-2) \\ Charles R Greathouse IV, Aug 29 2016

a(n) = 18*n^2 - 12*n = 6*A000567(n) = 3*A139267(n) = 2*A152751(n).
a(n) = a(n-1) + 36*n - 30 (with a(0)=0). - Vincenzo Librandi, Dec 15 2010
From G. C. Greubel, Aug 29 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2.
G.f.: 6*x*(1 + 5*x)/(1 - x)^3.
E.g.f.: 6*x*(1 + 3*x)*exp(x). (End)

A153808 8 times octagonal numbers: 8n(3*n-2).

Original entry on oeis.org

0, 8, 64, 168, 320, 520, 768, 1064, 1408, 1800, 2240, 2728, 3264, 3848, 4480, 5160, 5888, 6664, 7488, 8360, 9280, 10248, 11264, 12328, 13440, 14600, 15808, 17064, 18368, 19720, 21120, 22568, 24064, 25608, 27200, 28840, 30528, 32264

Offset: 0

Omar E. Pol, Jan 19 2009

G. C. Greubel, Table of n, a(n) for n = 0..999
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000567 (octagonal numbers), A064201 (9 times octagonal numbers), A139267 (twice octagonal numbers), A152751 (3 times octagonal numbers), A153794 (4 times octagonal numbers).

Magma
```
[ 8*n*(3*n-2): n in [0..40] ];
```

Table[8*n*(3*n-2), {n,0,25}] (* or *) LinearRecurrence[{3,-3,1},{0,8,64}, 25] (* G. C. Greubel, Aug 29 2016 *)
8*PolygonalNumber[8,Range[0,40]] (* Harvey P. Dale, Nov 22 2023 *)

a(n)=24*n^2-16*n \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 24*n^2 - 16*n = 8*A000567(n) = 4*A139267(n) = 2*A153794(n).
a(n) = a(n-1) + 48*n - 40 (with a(0)=0). - Vincenzo Librandi, Nov 27 2010
From G. C. Greubel, Aug 29 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: 8*x*(1 + 5*x)/(1 - x)^3.
E.g.f.: 8*x*(1 + 3*x)*exp(x). (End)

A221912 Partial sums of floor(n/12).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 125, 130, 135, 140, 145, 150, 155

Offset: 0

Philippe Deléham, Mar 27 2013

Apart from the initial zeros, the same as A008730.

			..0....0....0....0....0....0....0....0....0....0....0....0
..1....2....3....4....5....6....7....8....9...10...11...12
.14...16...18...20...22...24...26...28...30...32...34...36
.39...42...45...48...51...54...57...60...63...66...69...72
.76...80...84...88...92...96..100..104..108..112..116..120
125..130..135..140..145..150..155..160..165..170..175..180
186..192..198..204..210..216..222..228..234..240..246..252
259..266..273..280..287..294..301..308..315..322..329..336
344..352..360..368..376..384..392..400..408..416..424..432
441..450..459..468..477..486..495..504..513..522..531..540
...

Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,0,0,0,0,1,-2,1).

Cf. A000217, A002620, A130518, A130519, A130520, A174709, A174738, A118729, A218470, A131242, A218530.

Accumulate[Floor[Range[0,70]/12]] (* or *) LinearRecurrence[{2,-1,0,0,0,0,0,0,0,0,0,1,-2,1},{0,0,0,0,0,0,0,0,0,0,0,0,1,2},70] (* Harvey P. Dale, Mar 23 2015 *)

a(12n) = A051866(n).
a(12n+1) = A139267(n).
a(12n+2) = A094159(n).
a(12n+3) = A033579(n).
a(12n+4) = A049452(n).
a(12n+5) = A033581(n).
a(12n+6) = A049453(n).
a(12n+7) = A033580(n).
a(12n+8) = A195319(n).
a(12n+9) = A202804(n).
a(12n+10) = A211014(n).
a(12n+11) = A049598(n).
G.f.: x^12/((1-x)^2*(1-x^12)).
a(0)=0, a(1)=0, a(2)=0, a(3)=0, a(4)=0, a(5)=0, a(6)=0, a(7)=0, a(8)=0, a(9)=0, a(10)=0, a(11)=0, a(12)=1, a(13)=2, a(n)=2*a(n-1)- a(n-2)+ a(n-12)- 2*a(n-13)+ a(n-14). - Harvey P. Dale, Mar 23 2015

A008730 Molien series 1/((1-x)^2(1-x^12)) for 3-dimensional group [2,n] = 22n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175, 180, 186, 192, 198, 204

Offset: 0

N. J. A. Sloane

			..1....2....3....4....5....6....7....8....9...10...11...12
.14...16...18...20...22...24...26...28...30...32...34...36
.39...42...45...48...51...54...57...60...63...66...69...72
.76...80...84...88...92...96..100..104..108..112..116..120
125..130..135..140..145..150..155..160..165..170..175..180
186..192..198..204..210..216..222..228..234..240..246..252
259..266..273..280..287..294..301..308..315..322..329..336
344..352..360..368..376..384..392..400..408..416..424..432
441..450..459..468..477..486..495..504..513..522..531..540
550..560..570..580..590..600..610..620..630..640..650..660
...
The columns are: A051866, A139267, A094159, A033579, A049452, A033581, A049453, A033580, A195319, A202804, A211014, A049598
- _Philippe Deléham_, Apr 03 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 195
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1).

Cf. A001840, A001972, A008724, A008725, A008726, A008727, A008728, A008729, A008732. - Vladimir Joseph Stephan Orlovsky, Mar 14 2010

Cf. A221912

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/((1-x)^2*(1-x^12)) )); // G. C. Greubel, Jul 30 2019

seq(coeff(series(1/(1-x)^2/(1-x^12), x, n+1), x, n), n=0..80);

CoefficientList[Series[1/((1-x)^2*(1-x^12)), {x,0,70}], x] (* Vincenzo Librandi, Jun 11 2013 *)
LinearRecurrence[{2,-1,0,0,0,0,0,0,0,0,0,1,-2,1},{1,2,3,4,5,6,7,8,9,10,11,12,14,16},70] (* Harvey P. Dale, Jan 01 2024 *)

my(x='x+O('x^70)); Vec(1/((1-x)^2*(1-x^12))) \\ G. C. Greubel, Jul 30 2019

(1/((1-x)^2*(1-x^12))).series(x, 70).coefficients(x, sparse=False) # G. C. Greubel, Jul 30 2019

G.f. 1/( (1-x)^3 * (1+x) *(1+x+x^2) *(1-x+x^2) * (1+x^2) *(1-x^2+x^4)). - R. J. Mathar, Aug 11 2021
From Mitch Harris, Sep 08 2008: (Start)
a(n) = Sum_{j=0..n+12} floor(j/12).
a(n-12) = (1/2)*floor(n/12)*(2*n - 10 - 12*floor(n/12)). (End)
a(n) = A221912(n+12). - Philippe Deléham, Apr 03 2013

More terms from Vladimir Joseph Stephan Orlovsky, Mar 14 2010

A239325 a(n) = 6n^2 + 8n + 1.

Original entry on oeis.org

1, 15, 41, 79, 129, 191, 265, 351, 449, 559, 681, 815, 961, 1119, 1289, 1471, 1665, 1871, 2089, 2319, 2561, 2815, 3081, 3359, 3649, 3951, 4265, 4591, 4929, 5279, 5641, 6015, 6401, 6799, 7209, 7631, 8065, 8511, 8969, 9439, 9921, 10415, 10921, 11439, 11969

Offset: 0

Philippe Deléham, Mar 16 2014

Binomial transform of 1, 14, 12, 0, 0, 0 (0 continued).

Sum_{n>=0} 1/a(n) = (Psi(0,(4+sqrt(10))/6) - Psi(0,(4-sqrt(10))/6))/(2*sqrt(10)) = 1.14373625509612753878..., where Psi(n,k) is the n^th derivative of the digamma function. - Bruno Berselli, Mar 16 2014

			a(0) = 1*1 = 1;
a(1) = 1*1 + 14*1 = 15;
a(2) = 1*1 + 14*2 + 12*1 = 41;
a(3) = 1*1 + 14*3 + 12*3 = 79;
a(4) = 1*1 + 14*4 + 12*6 = 129; etc.

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

Cf. A069133, A100536, A139267.

Table[6 n^2 + 8 n + 1, {n, 0, 44}] (* or *)
CoefficientList[Series[(1 + 12 x - x^2)/(1 - x)^3, {x, 0, 44}], x] (* Michael De Vlieger, Oct 04 2016 *)
LinearRecurrence[{3,-3,1},{1,15,41},50] (* Harvey P. Dale, May 11 2019 *)

a(n)=6*n^2+8*n+1 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: (1 + 12*x - x^2)/(1-x)^3.
a(0) = 1, a(1) = 15, a(2) = 41; for n>2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = C(n,0) + 14*C(n,1) + 12*C(n,2).
a(n) = (A069133(n+1) + A100536(n+1) - A000290(n))/2.
a(n) = A139267(n+1) - 1. - Yuriy Sibirmovsky, Oct 04 2016

A290166 Number of complete non-collateral matches with lattice points on the edges of an n X n square.

Original entry on oeis.org

1, 13, 684, 73980, 13376448, 3627115200, 1376014521600, 695592156268800, 451867517982720000, 366777996951376281600, 363753784968105369600000, 432795572570448228556800000, 608442975450529801872998400000, 997771862620790990336507904000000

Offset: 1

César Eliud Lozada, Jul 22 2017

nonn

It appears that an n X n square here refers to a square array of points with n+1 points along each side, so that there are n edges on each side. - N. J. A. Sloane, Aug 22 2017

			Points on the sides of a 2 X 2 square can be matched in 13 different ways, if matching two points on the same side is not allowed. Therefore a(2)=13.

César Eliud Lozada, Complete matches

Cf. A139267 (number of distinct matches), A290167.

\\ s is without corners and left:m-a, right:m-b, top:m-c, bottom:m-d.
s(m,a,b,c,d) = {sum(k=0, m, my(j = k+(a+b-c-d)/2); if(j<0||k<0||2*(m-k)Andrew Howroyd, Sep 05 2017

a(5)-a(14) from Andrew Howroyd, Sep 05 2017

cp's OEIS Frontend

A226488 a(n) = n*(13*n - 9)/2.

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A152751 3 times octagonal numbers: a(n) = 3*n*(3*n-2).

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

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A153794 4 times octagonal numbers: a(n) = 4*n*(3*n-2).

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A153796 6 times octagonal numbers: a(n) = 6*n*(3*n-2).

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A153808 8 times octagonal numbers: 8*n*(3*n-2).

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A221912 Partial sums of floor(n/12).

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A226488 a(n) = n(13n - 9)/2.

A152751 3 times octagonal numbers: a(n) = 3n(3*n-2).

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

A153794 4 times octagonal numbers: a(n) = 4n(3*n-2).

A153796 6 times octagonal numbers: a(n) = 6n(3*n-2).

A153808 8 times octagonal numbers: 8n(3*n-2).

A008730 Molien series 1/((1-x)^2(1-x^12)) for 3-dimensional group [2,n] = 22n.

A239325 a(n) = 6n^2 + 8n + 1.