A033428 -id:A033428 - cp's OEIS frontend

A153448 3 times 12-gonal (or dodecagonal) numbers: a(n) = 3n(5*n-4).

0, 3, 36, 99, 192, 315, 468, 651, 864, 1107, 1380, 1683, 2016, 2379, 2772, 3195, 3648, 4131, 4644, 5187, 5760, 6363, 6996, 7659, 8352, 9075, 9828, 10611, 11424, 12267, 13140, 14043, 14976, 15939, 16932, 17955, 19008, 20091, 21204

Offset: 0

Omar E. Pol, Dec 26 2008

This sequence is related to A172117 by 3*A172117(n) = n*a(n) - Sum_{i=0..n-1} a(i) and this is the case d=10 in the identity n*(3*n*(d*n - d + 2)/2) - Sum_{k=0..n-1} 3*k*(d*k - d + 2)/2 = n*(n+1)*(2*d*n - 2*d + 3)/2. - Bruno Berselli, Aug 26 2010

G. C. Greubel, Table of n, a(n) for n = 0..1000
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. A051624, A152965.
3 times n-gonal numbers: A045943, A033428, A062741, A094159, A152773, A152751, A152759, A152767, A153783, A153875, A172117.
Cf. numbers of the form n*(n*k-k+6)/2, this sequence is the case k=30: see Comments lines of A226492.

Table[3n(5n-4),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{0,3,36},40] (* Harvey P. Dale, Jun 18 2014 *)
3*PolygonalNumber[12,Range[0,60]] (* Harvey P. Dale, May 13 2022 *)

a(n)=3*n*(5*n-4) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = 15*n^2 - 12*n = A051624(n)*3.
a(n) = 30*n + a(n-1) - 27 with n>0, a(0)=0. - Vincenzo Librandi, Aug 03 2010
G.f.: 3*x*(1 + 9*x)/(1-x)^3. - Bruno Berselli, Jan 21 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(0)=0, a(1)=3, a(2)=36. - Harvey P. Dale, Jun 18 2014
E.g.f.: 3*x*(1 + 5*x)*exp(x). - G. C. Greubel, Aug 21 2016
a(n) = (4*n-2)^2 - (n-2)^2. In general, if P(k,n) is the k-th n-gonal number, then (2*k+1)*P(8*k+4,n) = ((3*k+1)*n-2*k)^2 - (k*n-2*k)^2. - Charlie Marion, Jul 29 2021

A153783 3 times 11-gonal (or hendecagonal) numbers: a(n) = 3n(9*n-7)/2.

Original entry on oeis.org

0, 3, 33, 90, 174, 285, 423, 588, 780, 999, 1245, 1518, 1818, 2145, 2499, 2880, 3288, 3723, 4185, 4674, 5190, 5733, 6303, 6900, 7524, 8175, 8853, 9558, 10290, 11049, 11835, 12648, 13488, 14355, 15249, 16170, 17118, 18093, 19095

Offset: 0

Omar E. Pol, Jan 02 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. A051682, A152995.
3 times n-gonal numbers: A045943, A033428, A062741, A094159, A152773, A152751, A152759, A152767, A153448, A153875.
Cf. numbers of the form n*(n*k-k+6)/2, this sequence is the case k=27: see Comments lines of A226492.

s=0;lst={s};Do[s+=n;AppendTo[lst,s],{n,3,6!,27}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 02 2009 *)
Table[3*n*(9*n-7)/2, {n,0,25}] (* or *) LinearRecurrence[{3,-3,1},{0,3,33},25] (* G. C. Greubel, Aug 28 2016 *)

a(n)=3*n*(9*n-7)/2 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = (27*n^2 - 21*n)/2 = A051682(n)*3.
a(n) = 27*n + a(n-1) - 24, with n>0, a(0)=0. - Vincenzo Librandi, Aug 03 2010
G.f.: 3*x*(1 + 8*x)/(1-x)^3. - Bruno Berselli, Jan 21 2011
From G. C. Greubel, Aug 28 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
E.g.f.: (3/2)*x*(2 + 9*x)*exp(x). (End)

A153875 3 times 13-gonal (or tridecagonal) numbers: a(n) = 3n(11*n - 9)/2.

Original entry on oeis.org

0, 3, 39, 108, 210, 345, 513, 714, 948, 1215, 1515, 1848, 2214, 2613, 3045, 3510, 4008, 4539, 5103, 5700, 6330, 6993, 7689, 8418, 9180, 9975, 10803, 11664, 12558, 13485, 14445, 15438, 16464, 17523, 18615, 19740, 20898, 22089

Offset: 0

Omar E. Pol, Jan 03 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. A051865, A152997.
3 times n-gonal numbers: A045943, A033428, A062741, A094159, A152773, A152751, A152759, A152767, A153783, A153448.
Cf. numbers of the form n*(n*k-k+6)/2, this sequence is the case k=33: see Comments lines of A226492.

Table[(33*n^2 - 27*n)/2, {n,0,25}] (* G. C. Greubel, Aug 31 2016 *)

a(n)=3*n*(11*n-9)/2 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = (33*n^2 - 27*n)/2 = A051865(n)*3.
a(n) = a(n-1) + 33*n - 30, with n>0, a(0)=0. - Vincenzo Librandi, Dec 14 2010
G.f.: 3*x*(1 + 10*x)/(1-x)^3. - Bruno Berselli, Jan 21 2011
From G. C. Greubel, Aug 31 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
E.g.f.: (3/2)*x*(2 + 11*x)*exp(x). (End)

A195321 a(n) = 18*n^2.

Original entry on oeis.org

0, 18, 72, 162, 288, 450, 648, 882, 1152, 1458, 1800, 2178, 2592, 3042, 3528, 4050, 4608, 5202, 5832, 6498, 7200, 7938, 8712, 9522, 10368, 11250, 12168, 13122, 14112, 15138, 16200, 17298, 18432, 19602, 20808, 22050, 23328, 24642, 25992, 27378, 28800, 30258, 31752

Offset: 0

Omar E. Pol, Sep 16 2011

Sequence found by reading the line from 0, in the direction 0, 18, ..., in the square spiral whose vertices are the generalized hendecagonal numbers A195160. Semi-axis opposite to A195316 in the same spiral.
Area of a square with diagonal 6n. - Wesley Ivan Hurt, Jun 19 2014
Number of identical tessellation tiles that are composed of 48 equilateral edge joined triangles that can be formed into a order n hexagon. The example tiles shown in the link below are tessellated with eight sphinx tiles. See A291582. - Craig Knecht, Sep 02 2017

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

Bisection of A195147.
Cf. A016802, A033581, A033583, A135453, A139098, A144555, A195322, A195323.
Cf. A000290, A001105, A008600, A016766, A033428, A195160, A195316, A291582.

[18*n^2:n in [0..40]]; // Vincenzo Librandi, Sep 20 2011

A195321:=n->18*n^2; seq(A195321(n), n=0..50); # Wesley Ivan Hurt, Jun 19 2014

18 Range[0, 50]^2 (* or *) CoefficientList[Series[18 x*(1 + x)/(1 - x)^3, {x, 0, 30}], x] (* Wesley Ivan Hurt, Jun 20 2014 *)
LinearRecurrence[{3,-3,1},{0,18,72},50] (* Harvey P. Dale, Mar 26 2023 *)

a(n)=18*n^2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 18*A000290(n) = 9*A001105(n) = 6*A033428(n) = 3*A033581(n) = 2*A016766(n).
G.f.: 18*x*(1+x)/(1-x)^3. - Wesley Ivan Hurt, Jun 20 2014
From Elmo R. Oliveira, Dec 01 2024: (Start)
E.g.f.: 18*x*(1 + x)*exp(x).
a(n) = n*A008600(n) = A195147(2*n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A215631 Triangle read by rows: T(n,k) = n^2 + n*k + k^2, 1 <= k <= n.

Original entry on oeis.org

3, 7, 12, 13, 19, 27, 21, 28, 37, 48, 31, 39, 49, 61, 75, 43, 52, 63, 76, 91, 108, 57, 67, 79, 93, 109, 127, 147, 73, 84, 97, 112, 129, 148, 169, 192, 91, 103, 117, 133, 151, 171, 193, 217, 243, 111, 124, 139, 156, 175, 196, 219, 244, 271, 300, 133, 147, 163

Offset: 1

Reinhard Zumkeller, Nov 11 2012

			The triangle begins:
row n   T(n,k), 1 <= k <= n
   1:     3
   2:     7   12
   3:    13   19   27
   4:    21   28   37   48
   5:    31   39   49   61   75
   6:    43   52   63   76   91  108
   7:    57   67   79   93  109  127  147
   8:    73   84   97  112  129  148  169  192
   9:    91  103  117  133  151  171  193  217  243
  10:   111  124  139  156  175  196  219  244  271  300
  11:   133  147  163  181  201  223  247  273  301  331  363
  12:   157  172  189  208  229  252  277  304  333  364  397  432

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened

Cf. A215646 (row sums), A002061 (left edge, shifted), A033428 (right edge), A003215.

a215631 n k = a215631_tabl !! (n-1) !! (k-1)
a215631_row n = a215631_tabl !! (n-1)
a215631_tabl = zipWith3 (zipWith3 (\u v w -> u + v + w))
                        a093995_tabl a075362_tabl a133819_tabl
-- Reinhard Zumkeller, Nov 11 2012

[[i^2+i*j+j^2: j in [1..i]]: i in [1..10]]; // Vincenzo Librandi, Jun 07 2015

seq(seq(i^2+i*j+j^2, j=1..i),i=1..10); # Robert Israel, May 10 2015

Table[n^2 + n*k + k^2, {n, 11}, {k, n}] // Flatten (* Michael De Vlieger, May 12 2015 *)

for(n=1,15,for(k=1,n,print1(n^2+n*k+k^2,", "))) \\ Derek Orr, May 13 2015

T(n,k) = 2*A070216(n,k) - A215630(n,k).
G.f. for triangle: (3-2*x+3*x*y+x^2-11*x^2*y+4*x^3*y+x^3*y^2+x^4*y^2)*x*y/((1-x)^3*(1-x*y)^3). - Robert Israel, May 10 2015
From Avi Friedlich, May 26 2015: (Start)
T(n,k) = A093995(n,k) + A075362(n,k) + A133819(n,k).
T(k+1,k) = A003215(k).
T(k+2,k) = A003215(k)/2 + A003215(k+1)/2.
T(k+3,k) = A003215(k)/3 + A003215(k+1)/3 + A003215(k+2)/3 and so on. (End)

A064761 a(n) = 15*n^2.

Original entry on oeis.org

0, 15, 60, 135, 240, 375, 540, 735, 960, 1215, 1500, 1815, 2160, 2535, 2940, 3375, 3840, 4335, 4860, 5415, 6000, 6615, 7260, 7935, 8640, 9375, 10140, 10935, 11760, 12615, 13500, 14415, 15360, 16335, 17340, 18375, 19440, 20535, 21660, 22815

Offset: 0

Roberto E. Martinez II, Oct 18 2001

Number of edges in a complete 6-partite graph of order 6n, K_n,n,n,n,n,n.

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

Cf. A000217, A000290, A008585, A008587, A022272, A033428, A033429, A033581, A033583.

Cf. A008597, A195046.

Table[15*n^2, {n, 0, 45}] (* Amiram Eldar, Feb 03 2021 *)

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

a(n) = 15*A000290(n) = 5*A033428(n) = 3*A033429(n). - Omar E. Pol, Dec 13 2008
a(n) = A008587(n)*A008585(n). - Reinhard Zumkeller, Apr 12 2010
a(n) = a(n-1) + 30*n - 15 for n > 0, a(0)=0. - Vincenzo Librandi, Dec 15 2010
a(n) = A022272(n) + A022272(-n). - Bruno Berselli, Mar 31 2015
a(n) = t(6*n) - 6*t(n), where t(i) = i*(i+k)/2 for any k. Special case (k=1): a(n) = A000217(6*n) - 6*A000217(n). - Bruno Berselli, Aug 31 2017
From Amiram Eldar, Feb 03 2021: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/90.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/180.
Product_{n>=1} (1 + 1/a(n)) = sqrt(15)*sinh(Pi/sqrt(15))/Pi.
Product_{n>=1} (1 - 1/a(n)) = sqrt(15)*sin(Pi/sqrt(15))/Pi. (End)
From Elmo R. Oliveira, Nov 29 2024: (Start)
G.f.: 15*x*(1 + x)/(1 - x)^3.
E.g.f.: 15*x*(1 + x)*exp(x).
a(n) = n*A008597(n) = A195046(2*n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A073254 Array read by antidiagonals, A(n,k) = n^2 + n*k + k^2.

Original entry on oeis.org

0, 1, 1, 4, 3, 4, 9, 7, 7, 9, 16, 13, 12, 13, 16, 25, 21, 19, 19, 21, 25, 36, 31, 28, 27, 28, 31, 36, 49, 43, 39, 37, 37, 39, 43, 49, 64, 57, 52, 49, 48, 49, 52, 57, 64, 81, 73, 67, 63, 61, 61, 63, 67, 73, 81, 100, 91, 84, 79, 76, 75, 76, 79, 84, 91, 100, 121, 111, 103, 97

Offset: 0

Michael Somos, Jul 23 2002

Norm of elements in planar hexagonal lattice A_2.

Only numbers which appear in A003136 (Loeschian numbers) can appear in this array. - Peter Luschny, Nov 10 2021

			Triangle T(n, k) starts:
[0]               0
[1]              1, 1
[2]            4, 3, 4
[3]           9, 7, 7, 9
[4]       16, 13, 12, 13, 16
[5]     25, 21, 19, 19, 21, 25
[6]   36, 31, 28, 27, 28, 31, 36
[7] 49, 43, 39, 37, 37, 39, 43, 49

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (triangular rows 0 <= n <= 150, flattened)
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2.
Index entries for sequences related to A2 = hexagonal = triangular lattice

A033994 gives antidiagonal sums.
Cf. A003136, A000290, A033428, A003215, A033994, A194595.
Cf. A004016, A057427, A003056, A132111.
Cf. A198063 (m=3), A198064 (m=4), A198065 (m=5).

# Using the triangle formula:
A073254 := (n,k) -> k^2 - k*n + n^2: # Peter Luschny, Oct 26 2011

(* Using the array formula: *)
A[n_, k_] := n^2 + n k + k^2;
Table[A[n - k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 22 2018 *)

PARI
```
{A(n, k) = n^2 + n*k + k^2}
```

From Peter Luschny, Oct 26 2011: (Start)
Let m = 2, for the cases m = 3, 4, and 5 see the cross-references.
T(n,k) = k^2 - k*n + n^2 = A(n-k,k).
T(n,k) = Sum_{j=0..m} Sum_{i=0..m} (-1)^(j+i)*C(i,j)*n^j*k^(m-j) for m = 2.
T(n,0) = T(n,n) = n^m = n^2 = A000290(n).
T(2n,n) = (m+1)*n^m = 3*n^2 = A033428(n).
T(2n+1,n+1) = (n+1)^(m+1) - n^(m+1) = (n+1)^3 - n^3 = A003215(n).
Sum_{k=0..n} T(n,k) = (5*n^3 + 6*n^2 + n)/6 = A033994(n).
T(n+1, k+1)*binomial(n, k)^3/(k+1)^2 = A194595(n,k). (End)

Edited by Peter Luschny, Nov 10 2021

A140677 a(n) = n(3n + 8).

Original entry on oeis.org

0, 11, 28, 51, 80, 115, 156, 203, 256, 315, 380, 451, 528, 611, 700, 795, 896, 1003, 1116, 1235, 1360, 1491, 1628, 1771, 1920, 2075, 2236, 2403, 2576, 2755, 2940, 3131, 3328, 3531, 3740, 3955, 4176, 4403, 4636, 4875, 5120, 5371, 5628

Offset: 0

Omar E. Pol, May 22 2008

			a(1) = 6*1 + 0 + 5 = 11; a(2) = 6*2 + 11 + 5 = 28; a(3) = 6*3 + 28 + 5 = 51. - _Vincenzo Librandi_, Aug 03 2010

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, A140678, A067707, A140679, A140680, A140681, A140689.

Table[n(3n+8),{n,0,45}]  (* Harvey P. Dale, Feb 20 2011 *)

a(n)=n*(3*n+8) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 3*n^2 + 8*n.
a(n) = 6*n + a(n-1) + 5, with a(0)=0. - Vincenzo Librandi, Aug 03 2010
G.f.: x*(11 - 5*x)/(1 - x)^3. - Arkadiusz Wesolowski, Dec 24 2011
E.g.f.: (3*x^2 + 11*x)*exp(x). - G. C. Greubel, Jul 20 2017

A277712 Positions of 2's in A264977; positions of 3's in A277330.

Original entry on oeis.org

2, 10, 26, 58, 82, 122, 170, 250, 346, 418, 506, 562, 626, 698, 842, 1018, 1130, 1258, 1402, 1690, 1858, 2042, 2266, 2522, 2810, 3386, 3722, 4090, 4538, 5050, 5330, 5626, 6242, 6626, 6778, 7450, 7810, 8186, 9082, 9682, 10106, 10418, 10514, 10666, 11258, 11986, 12490, 13258, 13562, 14906, 15626, 16378, 17074, 18170, 19186, 19370, 19810

Offset: 1

Antti Karttunen, Oct 28 2016

nonn

Positions in A260443 of terms that are three times a perfect square (terms in A033428, although not all of them are present in A260443).

Row 2 of A277710.
Cf. A033428, A260443, A264977, A277701, A277330.
Cf. also A277713.

a(n) = 2*A277701(n).

A365929 Number of intersections formed within a triangle by placing n points "in general position" on each of the three sides and connecting each point to each of the points on the other two sides using straight lines.

Original entry on oeis.org

0, 0, 15, 108, 396, 1050, 2295, 4410, 7728, 12636, 19575, 29040, 41580, 57798, 78351, 103950, 135360, 173400, 218943, 272916, 336300, 410130, 495495, 593538, 705456, 832500, 975975, 1137240, 1317708, 1518846, 1742175, 1989270, 2261760, 2561328, 2889711, 3248700, 3640140, 4065930, 4528023

Offset: 0

Vijay Srinivas Balaji, Sep 23 2023

There are n points on each of the three sides (not counting the vertices of the triangle). Each point must be connected to every point on the other two sides. A033428(n) = 3*n^2 gives the number of lines.
Comments from N. J. A. Sloane, Oct 29 2023: (Start)
"In general position" means that all interior intersection points are simple. No three-way or higher intersections are permitted.
If the 3*n+3 boundary points are included in the count, there are 3*A366478 points.
For the configurations where the boundary points are equally spaced and every pair of boundary points is joined by a chord, see A091908, A092098, A331782.
(End)

			a(5) = (3/4) * 5^2 * (3*5^2 - 4*5 + 1) = 1050.

Vijay Srinivas Balaji, Formulating A Conjecture For Intersections Created From Crossing Lines Within Different Polygons, International School of Helsingborg, 2023.

Paolo Xausa, Table of n, a(n) for n = 0..10000
Vijay Srinivas Balaji, Diagram of Intersections for a Triangle.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
Scott R. Shannon, Image for n = 2.
Scott R. Shannon, Image for n = 4.
Scott R. Shannon, Image for n = 7.
Scott R. Shannon, Image for n = 10.

Cf. A367015 (number of regions), A366932 (number of edges), A366478 (vertices including boundary points), A033428 (number of chords).

cp's OEIS Frontend

A153448 3 times 12-gonal (or dodecagonal) numbers: a(n) = 3*n*(5*n-4).

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A153783 3 times 11-gonal (or hendecagonal) numbers: a(n) = 3*n*(9*n-7)/2.

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A153875 3 times 13-gonal (or tridecagonal) numbers: a(n) = 3*n*(11*n - 9)/2.

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A195321 a(n) = 18*n^2.

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A215631 Triangle read by rows: T(n,k) = n^2 + n*k + k^2, 1 <= k <= n.

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A064761 a(n) = 15*n^2.

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A073254 Array read by antidiagonals, A(n,k) = n^2 + n*k + k^2.

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A153448 3 times 12-gonal (or dodecagonal) numbers: a(n) = 3n(5*n-4).

A153783 3 times 11-gonal (or hendecagonal) numbers: a(n) = 3n(9*n-7)/2.

A153875 3 times 13-gonal (or tridecagonal) numbers: a(n) = 3n(11*n - 9)/2.

A140677 a(n) = n(3n + 8).