A017209 -id:A017209 - cp's OEIS frontend

A247678 Odd composite numbers congruent to 4 modulo 9.

49, 85, 121, 175, 247, 265, 301, 319, 355, 391, 427, 445, 481, 517, 535, 553, 589, 625, 679, 697, 715, 805, 841, 895, 913, 931, 949, 985, 1003, 1057, 1075, 1111, 1147, 1165, 1183, 1219, 1255, 1273, 1309, 1345, 1363, 1417, 1435, 1507, 1525, 1561, 1615, 1633

Offset: 1

Odimar Fabeny, Sep 22 2014

Odimar Fabeny, Table of n, a(n) for n = 1..10000

Cf. A017209 (9n + 4, supersequence of this sequence), A247676, A247679, A247681, A247682, A247683.

Select[18Range[125] + 13, Not[PrimeQ[#]] &] (* Alonso del Arte, Sep 24 2014 *)
Select[Range[13,1700,18],CompositeQ] (* Harvey P. Dale, Aug 21 2024 *)

lista(nn) = {forcomposite(n=1, nn, if ((n % 2) && ((n % 9) == 4), print1(n, ", ")); ); } \\ Michel Marcus, Sep 22 2014

A154266 a(n) = 27*n + 12.

Original entry on oeis.org

12, 39, 66, 93, 120, 147, 174, 201, 228, 255, 282, 309, 336, 363, 390, 417, 444, 471, 498, 525, 552, 579, 606, 633, 660, 687, 714, 741, 768, 795, 822, 849, 876, 903, 930, 957, 984, 1011, 1038, 1065, 1092, 1119, 1146, 1173, 1200, 1227, 1254, 1281, 1308, 1335

Offset: 0

Vincenzo Librandi, Jan 06 2009

The identity (81*n^2 + 72*n + 17)^2 - (9*n^2 + 8*n + 2)*(27*n + 12)^2 = 1 can be written as A154295(n+1)^2 - A154262(n+1)*a(n)^2 = 1. - Vincenzo Librandi, Feb 03 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008585, A017197, A017209, A154262, A154295.

I:=[12, 39]; [n le 2 select I[n] else 2*Self(n-1)-Self(n-2): n in [1..40]]; // Vincenzo Librandi, Feb 02 2012

Range[12, 7000, 27] (* Vladimir Joseph Stephan Orlovsky, Jul 13 2011 *)
LinearRecurrence[{2, -1}, {12, 39}, 50] (* Vincenzo Librandi, Feb 02 2012 *)

a(n)=27*n+12 \\ Charles R Greathouse IV, Dec 28 2011

From R. J. Mathar, Jan 05 2011: (Start)
G.f.: 3*(4 + 5*x)/(1-x)^2.
a(n) = 3*A017209(n). (End)
a(n) = 2*a(n-1) - a(n-2). - Vincenzo Librandi, Feb 02 2012
E.g.f.: (27*x + 12)*exp(x). - G. C. Greubel, Sep 08 2016
a(n) = A017197(3*n+1) = A008585(9*n+4). - Elmo R. Oliveira, Apr 12 2025

119 replaced by 1119 - R. J. Mathar, Jan 07 2009

A155551 Triangle read by rows where T(m,n)=2mn + m + n - 9.

Original entry on oeis.org

-5, -2, 3, 1, 8, 15, 4, 13, 22, 31, 7, 18, 29, 40, 51, 10, 23, 36, 49, 62, 75, 13, 28, 43, 58, 73, 88, 103, 16, 33, 50, 67, 84, 101, 118, 135, 19, 38, 57, 76, 95, 114, 133, 152, 171, 22, 43, 64, 85, 106, 127, 148, 169, 190, 211, 25, 48, 71, 94, 117, 140, 163, 186, 209

Offset: 1

Vincenzo Librandi, Jan 24 2009

The numbers 2*T(m,n)+19 =(2*n+1)*(2*m+1) are not prime.

First column: A016777, second column: A016885, third column: A016993, fourth column: A017209. - Vincenzo Librandi, Nov 20 2012

			Triangle begins:
-5;
-2, 3;
1,  8,  15;
4,  13, 22, 31;
7,  18, 29, 40, 51;
10, 23, 36, 49, 62,  75;
13, 28, 43, 58, 73,  88,  103;
16, 33, 50, 67, 84,  101, 118, 135;
19, 38, 57, 76, 95,  114, 133, 152, 171;
22, 43, 64, 85, 106, 127, 148, 169, 190, 211; etc.

Vincenzo Librandi, Rows n = 1..100, flattened

Cf. A153143, A153144, A016777, A016885, A016993, A017209.

[2*n*k + n + k - 9: k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 20 2012

t[n_,k_]:=2 n*k + n + k - 9; Table[t[n, k], {n, 15}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 20 2012 *)

A163674 Triangle T(n,m) = 2mn + m + n + 9 read by rows.

Original entry on oeis.org

13, 16, 21, 19, 26, 33, 22, 31, 40, 49, 25, 36, 47, 58, 69, 28, 41, 54, 67, 80, 93, 31, 46, 61, 76, 91, 106, 121, 34, 51, 68, 85, 102, 119, 136, 153, 37, 56, 75, 94, 113, 132, 151, 170, 189, 40, 61, 82, 103, 124, 145, 166, 187, 208, 229, 43, 66, 89, 112, 135, 158, 181

Offset: 1

Vincenzo Librandi, Aug 03 2009

2*T(m,n) - 17 =(2*n+1)*(2*m+1) and 2*T(n,n) - 17 is a square. Also:
first column:  A112414;
second column: A016861;
third column:  A017041;
fourth column: A017209. [Vincenzo Librandi, Nov 20 2012]

			Triangle begins:
  13;
  16,  21;
  19,  26,  33;
  22,  31,  40,  49;
  25,  36,  47,  58,  69;
  28,  41,  54,  67,  80,  93;
  31,  46,  61,  76,  91, 106, 121;
  34,  51,  68,  85, 102, 119, 136, 153;

Vincenzo Librandi, Rows n = 1..100, flattened

Cf. A016861, A017041, A017209, A112414, A153051, A163657.

[2*n*k + n + k + 9: k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 20 2012

t[n_,k_]:=2 n*k + n + k + 9; Table[t[n, k], {n, 15}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 20 2012 *)

for(n=1,10, for(m=1,n, print1(2*m*n + m + n + 9, ", "))) \\ G. C. Greubel, Aug 02 2017

T(n,m) = A163657(n,m) + 1.

Edited by R. J. Mathar, Oct 12 2009

A155156 Triangle T(n, k) = 4nk + 2n + 2k, read by rows.

Original entry on oeis.org

8, 14, 24, 20, 34, 48, 26, 44, 62, 80, 32, 54, 76, 98, 120, 38, 64, 90, 116, 142, 168, 44, 74, 104, 134, 164, 194, 224, 50, 84, 118, 152, 186, 220, 254, 288, 56, 94, 132, 170, 208, 246, 284, 322, 360, 62, 104, 146, 188, 230, 272, 314, 356, 398, 440, 68, 114, 160, 206, 252, 298, 344, 390, 436, 482, 528

Offset: 1

Vincenzo Librandi, Jan 21 2009

First column: A016933, second column: A017317, third column: A063151, fourth column: 2*A017209. - Vincenzo Librandi, Nov 21 2012

			Triangle begins:
   8;
  14,  24;
  20,  34,  48;
  26,  44,  62,  80;
  32,  54,  76,  98, 120;
  38,  64,  90, 116, 142, 168;
  44,  74, 104, 134, 164, 194, 224;
  50,  84, 118, 152, 186, 220, 254, 288;
  56,  94, 132, 170, 208, 246, 284, 322, 360;
  62, 104, 146, 188, 230, 272, 314, 356, 398, 440;

Vincenzo Librandi, Rows n = 1..100, flattened

Cf. A016933, A017317, A063151, A017209. A083487, A162254, A163832 (row sums).

[4*n*k + 2*n + 2*k : k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 21 2012

seq(seq( 2*(2*n*k +n+k), k=1..n), n=1..15); # G. C. Greubel, Mar 20 2021

T[n_,k_]:=4*n*k +2*n +2*k; Table[T[n, k], {n, 15}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 21 2012 *)

flatten([[2*(2*n*k +n+k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 20 2021

T(n, k) = 2*A083487(n, k). - R. J. Mathar, Jan 05 2011

Sum_{k=0..n} T(n,k) = n*(2*n^2 + 5*n + 1) = 2*A162254(n) = A163832(n). - G. C. Greubel, Mar 20 2021

Edited by Robert Hochberg, Jun 21 2010

A276819 a(n) = (9*n^2 - n)/2 + 1.

Original entry on oeis.org

1, 5, 18, 40, 71, 111, 160, 218, 285, 361, 446, 540, 643, 755, 876, 1006, 1145, 1293, 1450, 1616, 1791, 1975, 2168, 2370, 2581, 2801, 3030, 3268, 3515, 3771, 4036, 4310, 4593, 4885, 5186, 5496, 5815, 6143, 6480, 6826, 7181, 7545, 7918, 8300, 8691, 9091, 9500, 9918, 10345, 10781, 11226, 11680, 12143, 12615

Offset: 0

Yuriy Sibirmovsky, Sep 18 2016

Diagonal of triangular spiral in A051682. The other 5 diagonals are given by A140064, A117625, A081267, A064225, A006137. See the link as well.
First differences are given by A017209.
72*a(n) - 71 is a perfect square. - Klaus Purath, Jan 14 2022

Colin Barker, Table of n, a(n) for n = 0..1000
Yuriy Sibirmovsky, Six diagonals of the triangular spiral.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A002378, A003215, A006137, A017209, A051682, A064225, A081267, A117625, A140064.

Mathematica
```
Table[(9*n^2-n)/2+1, {n,0,100}]
```

Vec((1+2*x+6*x^2)/(1-x)^3 + O(x^60)) \\ Colin Barker, Sep 18 2016

a(n) = (9*n^2 - n)/2 + 1; \\ Altug Alkan, Sep 18 2016

a(n) = (9*n^2 - n)/2 + 1.
a(n) = a(n-1) + 9*n - 5 with a(0) = 1.
From Colin Barker, Sep 18 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
G.f.: (1 + 2*x + 6*x^2)/(1 - x)^3. (End)
From Klaus Purath, Jan 14 2022: (Start)
a(n) = A006137(n) - n.
A003215(a(n)) - A003215(a(n)-3) = A002378(9*n-1). (End)
E.g.f.: exp(x)*(2 + 8*x + 9*x^2)/2. - Stefano Spezia, Dec 25 2022

A162245 Triangle T(n,m) = 6mn + 3m + 3n + 1 read by rows.

Original entry on oeis.org

13, 22, 37, 31, 52, 73, 40, 67, 94, 121, 49, 82, 115, 148, 181, 58, 97, 136, 175, 214, 253, 67, 112, 157, 202, 247, 292, 337, 76, 127, 178, 229, 280, 331, 382, 433, 85, 142, 199, 256, 313, 370, 427, 484, 541, 94, 157, 220, 283, 346, 409, 472, 535, 598, 661

Offset: 1

Vincenzo Librandi, Jun 28 2009

If h belongs to the main diagonal of the triangle then 6*h+3 is a square since T(n,n) = (3/2)*(2*n+1)^2-1/2 and 6*T(n,n)+3 = 9*(2*n+1)^2. Also, the first column is A017209 (after 4). - Vincenzo Librandi, Nov 20 2012

			Triangle begins:
13;
22, 37;
31, 52,  73;
40, 67,  94,  121;
49, 82,  115, 148, 181;
58, 97,  136, 175, 214, 253;
67, 112, 157, 202, 247, 292, 337;
76, 127, 178, 229, 280, 331, 382, 433; etc.

Vincenzo Librandi, Rows n = 1..100, flattened

Cf. A003154, A017209, A083487, A163433.

[6*n*k + 3*n + 3*k + 1:  k in [1..n],  n in [1..11]]; // Vincenzo Librandi, Nov 20 2012

Flatten@Table[6*m*n + 3*m + 3*n + 1, {n, 20}, {m, n}] (* Vincenzo Librandi, Mar 03 2012 *)

Row sums: Sum_{m=1..n} T(n,m) = n*(5+6*n^2+15*n)/2. - R. J. Mathar, Jul 26 2009
T(n,m) = 3*A083487(n,m)+1. - R. J. Mathar, Jul 26 2009
T(k,k) = A003154(k+1) and T(k+1,k) = A163433(k+2). - Avi Friedlich, May 22 2015

Edited by R. J. Mathar, Jul 26 2009

A177073 a(n) = (9n+4)(9*n+5).

Original entry on oeis.org

20, 182, 506, 992, 1640, 2450, 3422, 4556, 5852, 7310, 8930, 10712, 12656, 14762, 17030, 19460, 22052, 24806, 27722, 30800, 34040, 37442, 41006, 44732, 48620, 52670, 56882, 61256, 65792, 70490, 75350, 80372, 85556, 90902, 96410, 102080, 107912, 113906, 120062

Offset: 0

Vincenzo Librandi, May 31 2010

Cf. comment of Reinhard Zumkeller in A177059: in general, (h*n+h-k)*(h*n+k) = h^2*A002061(n+1) + (h-k)*k - h^2; therefore a(n) = 81*A002061(n+1) - 61. - Bruno Berselli, Aug 24 2010

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

Cf. A002061, A017209, A017221, A177059.

[(9*n+4)*(9*n+5): n in [0..50]]; // Vincenzo Librandi, Apr 08 2013

f[n_] := Module[{c = 9n}, (c+4)(c+5)]; Array[f, 40, 0] (* or *) LinearRecurrence[{3, -3, 1}, {20, 182, 506}, 40] (* Harvey P. Dale, Jun 24 2011 *)

a(n)=(9*n+4)*(9*n+5) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 162*n + a(n-1) with n > 0, a(0)=20.
From Harvey P. Dale, Jun 24 2011: (Start)
a(0)=20, a(1)=182, a(2)=506, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: -2*(x*(10*x+61)+10)/(x-1)^3. (End)
From Amiram Eldar, Feb 19 2023: (Start)
a(n) = A017209(n)*A017221(n).
Sum_{n>=0} 1/a(n) = tan(Pi/18)*Pi/9.
Product_{n>=0} (1 - 1/a(n)) = sec(Pi/18)*cos(sqrt(5)*Pi/18).
Product_{n>=0} (1 + 1/a(n)) = sec(Pi/18)*cosh(sqrt(3)*Pi/18). (End)
E.g.f.: exp(x)*(20 + 81*x*(2 + x)). - Elmo R. Oliveira, Oct 18 2024

Edited by N. J. A. Sloane, Jun 22 2010

A304503 a(n) = 3(n+1)(9*n+4).

Original entry on oeis.org

12, 78, 198, 372, 600, 882, 1218, 1608, 2052, 2550, 3102, 3708, 4368, 5082, 5850, 6672, 7548, 8478, 9462, 10500, 11592, 12738, 13938, 15192, 16500, 17862, 19278, 20748, 22272, 23850, 25482, 27168, 28908, 30702, 32550, 34452, 36408, 38418, 40482, 42600, 44772

Offset: 0

Emeric Deutsch, May 13 2018

The first Zagreb index of the single-defect 3-gonal nanocone CNC(3,n) (see definition in the Doslic et al. reference, p. 27).
The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternatively, it is the sum of the degree sums d(i) + d(j) over all edges ij of the graph.
The M-polynomial of CNC(3,n) is M(CNC(3,n);x,y) = 3*x^2*y^2 + 6*n*x^2*y^3 + 3*n*(3*n+1)*x^3*y^3/2.
More generally, the M-polynomial of CNC(k,n) is M(CNC(k,n);x,y) = k*x^2*y^2 + 2*k*n*x^2*y^3 + k*n*(3*n + 1)*x^3*y^3/2.
12*a(n) + 25 is a square. - Bruno Berselli, May 14 2018

Colin Barker, Table of n, a(n) for n = 0..1000
Emeric Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, Vol. 6, No. 2, 2015, pp. 93-102.
T. Doslic and M. Saheli, Augmented eccentric connectivity index of single-defect nanocones, J. of Mathematical Nanoscience, Vol. 1, No. 1, 2011, pp. 25-31.
A. Khaksar, M. Ghorbani, and H. R. Maimani, On atom bond connectivity and GA indices of nanocones, Optoelectronics and Advanced Materials - Rapid Communications, Vol. 4, No. 11, 2010, pp. 1868-1870.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A008585, A017209, A062708, A304504.

Maple
```
seq((3*(n+1))*(9*n+4), n = 0 .. 40);
```

Vec(6*(2 + 7*x) / (1 - x)^3 + O(x^40)) \\ Colin Barker, May 14 2018

From Colin Barker, May 14 2018: (Start)
G.f.: 6*(2 + 7*x)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
From Elmo R. Oliveira, Nov 15 2024: (Start)
E.g.f.: 3*exp(x)*(4 + 22*x + 9*x^2).
a(n) = 6*A062708(n+1) = A017209(n)*A008585(n+1). (End)

A304505 a(n) = 4(n+1)(9*n+4).

Original entry on oeis.org

16, 104, 264, 496, 800, 1176, 1624, 2144, 2736, 3400, 4136, 4944, 5824, 6776, 7800, 8896, 10064, 11304, 12616, 14000, 15456, 16984, 18584, 20256, 22000, 23816, 25704, 27664, 29696, 31800, 33976, 36224, 38544, 40936, 43400, 45936, 48544, 51224, 53976, 56800, 59696

Offset: 0

Emeric Deutsch, May 14 2018

a(n) is the first Zagreb index of the single-defect 4-gonal nanocone CNC(4,n) (see definition in the Doslic et al. reference, p. 27).
The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternatively, it is the sum of the degree sums d(i) + d(j) over all edges ij of the graph.
The M-polynomial of CNC(4,n) is M(CNC(4,n); x,y) = 4*x^2*y^2 + 8*n*x^2*y^3 + 2*n*(3*n+1)*x^3*y^3.
More generally, the M-polynomial of CNC(k,n) is M(CNC(k,n); x,y) = k*x^2*y^2 + 2*k*n*x^2*y^3 + k*n*(3*n + 1)*x^3*y^3/2.
9*a(n) + 25 is a square. - Bruno Berselli, May 14 2018

Colin Barker, Table of n, a(n) for n = 0..1000
Emeric Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, Vol. 6, No. 2, 2015, pp. 93-102.
T. Doslic and M. Saheli, Augmented eccentric connectivity index of single-defect nanocones, J. of Mathematical Nanoscience, Vol. 1, No. 1, 2011, pp. 25-31.
A. Khaksar, M. Ghorbani, and H. R. Maimani, On atom bond connectivity and GA indices of nanocones, Optoelectronics and Advanced Materials - Rapid Communications, Vol. 4, No. 11, 2010, pp. 1868-1870.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A008586, A017209, A062708, A304506.

List([0..50],n->4*(n+1)*(9*n+4)); # Muniru A Asiru, May 14 2018

Maple
```
seq((4*(n+1))*(9*n+4), n = 0 .. 40);
```

a(n) = 4*(n+1)*(9*n+4); \\ Altug Alkan, May 14 2018

Vec(8*(2 + 7*x) / (1 - x)^3 + O(x^40)) \\ Colin Barker, May 14 2018

From Colin Barker, May 14 2018: (Start)
G.f.: 8*(2 + 7*x)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
From Elmo R. Oliveira, Nov 15 2024: (Start)
E.g.f.: 4*exp(x)*(4 + 22*x + 9*x^2).
a(n) = 8*A062708(n+1) = A017209(n)*A008586(n+1). (End)

cp's OEIS Frontend

A247678 Odd composite numbers congruent to 4 modulo 9.

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PARI

A154266 a(n) = 27*n + 12.

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A155551 Triangle read by rows where T(m,n)=2*m*n + m + n - 9.

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A163674 Triangle T(n,m) = 2*m*n + m + n + 9 read by rows.

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A155156 Triangle T(n, k) = 4*n*k + 2*n + 2*k, read by rows.

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A276819 a(n) = (9*n^2 - n)/2 + 1.

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A162245 Triangle T(n,m) = 6*m*n + 3*m + 3*n + 1 read by rows.

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A155551 Triangle read by rows where T(m,n)=2mn + m + n - 9.

A163674 Triangle T(n,m) = 2mn + m + n + 9 read by rows.

A155156 Triangle T(n, k) = 4nk + 2n + 2k, read by rows.

A162245 Triangle T(n,m) = 6mn + 3m + 3n + 1 read by rows.

A177073 a(n) = (9n+4)(9*n+5).

A304503 a(n) = 3(n+1)(9*n+4).

A304505 a(n) = 4(n+1)(9*n+4).