A152742 -id:A152742 - cp's OEIS frontend

A144555 a(n) = 14*n^2.

0, 14, 56, 126, 224, 350, 504, 686, 896, 1134, 1400, 1694, 2016, 2366, 2744, 3150, 3584, 4046, 4536, 5054, 5600, 6174, 6776, 7406, 8064, 8750, 9464, 10206, 10976, 11774, 12600, 13454, 14336, 15246, 16184, 17150, 18144, 19166, 20216, 21294, 22400, 23534, 24696

Offset: 0

N. J. A. Sloane, Jan 01 2009

Sequence found by reading the line from 0, in the direction 0, 14, ..., in the square spiral whose vertices are the generalized enneagonal numbers A118277. Also sequence found by reading the same line and direction in the square spiral whose edges have length A195019 and whose vertices are the numbers A195020. - Omar E. Pol, Sep 10 2011

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

See also A033428, A033429, A033581, A033582, A033583, A033584, ... and A249327 for the whole table.
Cf. A000290, A001105, A064761, A118277, A152742, A195019, A195020.
Cf. A008596, A195145.

Table[14*n^2, {n, 0, 45}] (* Amiram Eldar, Feb 03 2021 *)

A144555(n)=14*n^2 \\ M. F. Hasler, Oct 31 2014

a(n) = 14*A000290(n) = 7*A001105(n) = 2*A033582(n). - Omar E. Pol, Jan 01 2009
a(n) = a(n-1) + 14*(2*n-1), with a(0) = 0. - Vincenzo Librandi, Nov 25 2010
From Amiram Eldar, Feb 03 2021: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/84.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/168.
Product_{n>=1} (1 + 1/a(n)) = sqrt(14)*sinh(Pi/sqrt(14))/Pi.
Product_{n>=1} (1 - 1/a(n)) = sqrt(14)*sin(Pi/sqrt(14))/Pi. (End)
From Elmo R. Oliveira, Nov 30 2024: (Start)
G.f.: 14*x*(1 + x)/(1-x)^3.
E.g.f.: 14*x*(1 + x)*exp(x).
a(n) = n*A008596(n) = A195145(2*n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

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

Original entry on oeis.org

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

A244630 a(n) = 17*n^2.

Original entry on oeis.org

0, 17, 68, 153, 272, 425, 612, 833, 1088, 1377, 1700, 2057, 2448, 2873, 3332, 3825, 4352, 4913, 5508, 6137, 6800, 7497, 8228, 8993, 9792, 10625, 11492, 12393, 13328, 14297, 15300, 16337, 17408, 18513, 19652, 20825, 22032, 23273, 24548, 25857, 27200, 28577, 29988

Offset: 0

Vincenzo Librandi, Jul 03 2014

First bisection of A195047. - Bruno Berselli, Jul 03 2014

Norms of purely imaginary numbers in Z[sqrt(-17)] (for example, 3*sqrt(-17) has norm 153). - Alonso del Arte, Jun 23 2018

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

Cf. A008599, A195047.

Cf. similar sequences of the type k*n^2: A000290 (k = 1), A001105 (k = 2), A033428 (k = 3), A016742 (k = 4), A033429 (k = 5), A033581 (k = 6), A033582 (k = 7), A139098 (k = 8), A016766 (k = 9), A033583 (k = 10), A033584 (k = 11), A135453 (k = 12), A152742 (k = 13), A144555 (k = 14), A064761 (k = 15), A016802 (k = 16), this sequence (k = 17), A195321 (k = 18), A244631 (k = 19), A195322 (k = 20), A064762 (k = 21), A195323 (k = 22), A244632 (k = 23), A195824 (k = 24), A016850 (k = 25), A244633 (k = 26), A244634 (k = 27), A064763 (k = 28), A244635 (k = 29), A244636 (k = 30).

List([0..45],n->17*n^2); # Muniru A Asiru, Jun 29 2018

Magma
```
[17*n^2: n in [0..40]];
```

seq(17*n^2,n=0..45); # Muniru A Asiru, Jun 29 2018

Mathematica
```
Table[17 n^2, {n, 0, 40}]
```

a(n)=17*n^2 \\ Charles R Greathouse IV, Oct 07 2015

for (i <- 0 to 50) yield 17 * (i * i) // Alonso del Arte, Jun 29 2018

G.f.: 17*x*(1 + x)/(1 - x)^3. [corrected by Bruno Berselli, Jul 03 2014]
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
a(n) = 17*A000290(n). - Omar E. Pol, Jul 03 2014
a(n) = a(-n). - Muniru A Asiru, Jun 29 2018
From Elmo R. Oliveira, Dec 02 2024: (Start)
E.g.f.: 17*x*(1 + x)*exp(x).
a(n) = n*A008599(n) = A195047(2*n). (End)

A186030 a(n) = n(13n-3)/2.

Original entry on oeis.org

0, 5, 23, 54, 98, 155, 225, 308, 404, 513, 635, 770, 918, 1079, 1253, 1440, 1640, 1853, 2079, 2318, 2570, 2835, 3113, 3404, 3708, 4025, 4355, 4698, 5054, 5423, 5805, 6200, 6608, 7029, 7463, 7910, 8370, 8843, 9329, 9828, 10340, 10865, 11403

Offset: 0

Bruno Berselli, Feb 11 2011

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

Cf. numbers of the form n*(d*n+10-d)/2 indexed in A140090.
Cf. A154609 (first differences).
Cf. A008585, A152742.

Magma
```
[ n*(13*n-3)/2 : n in [0..42] ];
```

Table[n (13n-3)/2,{n,0,60}]  (* Harvey P. Dale, Feb 24 2011 *)

a(n)=n*(13*n-3)/2 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: x*(5+8*x)/(1-x)^3.
From Bruno Berselli, Sep 05 2011: (Start)
a(n) - a(-n) = -A008585(n).
a(n) + a(-n) = A152742(n). (End)
E.g.f.: (1/2)*(13*x^2 + 10*x)*exp(x). - G. C. Greubel, Jul 17 2017

A016910 a(n) = (6*n)^2.

Original entry on oeis.org

0, 36, 144, 324, 576, 900, 1296, 1764, 2304, 2916, 3600, 4356, 5184, 6084, 7056, 8100, 9216, 10404, 11664, 12996, 14400, 15876, 17424, 19044, 20736, 22500, 24336, 26244, 28224, 30276, 32400, 34596, 36864, 39204, 41616, 44100, 46656, 49284, 51984, 54756, 57600, 60516, 63504, 66564, 69696, 72900

Offset: 0

N. J. A. Sloane

Areas A of two classes of triangles with integer sides (a,b,c) where a = 9k, b=10k and c = 17k, or a = 3k, b = 25k and c = 26k for k=0,1,2,... These areas are given by Heron's formula A = sqrt(s(s-a)(s-b)(s-c)) = (6k)^2, with the semiperimeter s = (a+b+c)/2. This sequence is a subsequence of A188158. - Michel Lagneau, Oct 11 2013

Sequence found by reading the line from 0, in the direction 0, 36, ..., in the square spiral whose vertices are the generalized 20-gonal numbers A218864. - Omar E. Pol, May 13 2018.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A089491, A188158, A243941.

Cf. similar sequences of the type k*n^2: A000290 (k=1), A001105 (k=2), A033428 (k=3), A016742 (k=4), A033429 (k=5), A033581 (k=6), A033582 (k=7), A139098 (k=8), A016766 (k=9), A033583 (k=10), A033584 (k=11), A135453 (k=12), A152742 (k=13), A144555 (k=14), A064761 (k=15), A016802 (k=16), A244630 (k=17), A195321 (k=18), A244631 (k=19), A195322 (k=20), A064762 (k=21), A195323 (k=22), A244632 (k=23), A195824 (k=24), A016850 (k=25), A244633 (k=26), A244634 (k=27), A064763 (k=28), A244635 (k=29), A244636 (k=30).

[(6*n)^2: n in [0..40]]; // Vincenzo Librandi, May 03 2011

(6*Range[0,40])^2 (* or *) LinearRecurrence[{3,-3,1},{0,36,144},40] (* Harvey P. Dale, Dec 25 2017 *)
Table[36 n^2, {n, 0, 45}] (* Omar E. Pol, Jun 07 2018 *)

a(n)=36*n^2 \\ Charles R Greathouse IV, Jun 10 2016

From Ilya Gutkovskiy, Jun 09 2016: (Start)
O.g.f.: 36*x*(1 + x)/(1 - x)^3.
E.g.f.: 36*x*(1 + x)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
Sum_{n>=1} 1/a(n) = Pi^2/216 = A086726. (End)
Product_{n>=1} a(n)/A136017(n) = Pi/3. - Fred Daniel Kline, Jun 09 2016
a(n) = t(9*n) - 9*t(n), where t(i) = i*(i+k)/2 for any k. Special case (k=1): a(n) = A000217(9*n) - 9*A000217(n). - Bruno Berselli, Aug 31 2017
a(n) = 36*A000290(n) = 18*A001105(n) = 12*A033428 = 9*A016742(n) = 6*A033581(n) = 4*A016766(n) = 3*A135453(n) = 2*A195321(n). - Omar E. Pol, Jun 07 2018
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/432. - Amiram Eldar, Jun 27 2020
From Amiram Eldar, Jan 25 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = sinh(Pi/6)/(Pi/6).
Product_{n>=1} (1 - 1/a(n)) = sin(Pi/6)/(Pi/6) = 3/Pi (A089491). (End)

A244633 a(n) = 26*n^2.

Original entry on oeis.org

0, 26, 104, 234, 416, 650, 936, 1274, 1664, 2106, 2600, 3146, 3744, 4394, 5096, 5850, 6656, 7514, 8424, 9386, 10400, 11466, 12584, 13754, 14976, 16250, 17576, 18954, 20384, 21866, 23400, 24986, 26624, 28314, 30056, 31850, 33696, 35594, 37544, 39546, 41600, 43706

Offset: 0

Vincenzo Librandi, Jul 03 2014

Sequence found by reading the line from 0, in the direction 0, 26, ..., in the square spiral whose vertices are the generalized 15-gonal numbers. - Omar E. Pol, Jul 03 2014

Norms of purely imaginary numbers in Z[sqrt(-26)]. - Alonso del Arte, Dec 25 2014

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

Cf. similar sequences listed in A244630.

Cf. A000290, A001105, A005843, A008595, A152742, A252994, A277082.

Magma
```
[26*n^2: n in [0..40]];
```

A244633:=n->26*n^2: seq(A244633(n), n=0..50); # Wesley Ivan Hurt, Jul 04 2014

Table[26 n^2, {n, 0, 40}]
26 Range[0, 50]^2 (* Wesley Ivan Hurt, Jul 04 2014 *)

a(n)=26*n^2 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: 26*x*(1 + x)/(1 - x)^3. [corrected by Bruno Berselli, Jul 03 2014]
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
a(n) = 26*A000290(n) = 13*A001105(n) = 2*A152742(n). - Omar E. Pol, Jul 03 2014
From Elmo R. Oliveira, Dec 02 2024: (Start)
E.g.f.: 26*x*(1 + x)*exp(x).
a(n) = n*A252994(n) = A005843(n)*A008595(n). (End)

A249327 Rectangular array T(n,k) = f(n)*k^2, where f = A005117 (squarefree numbers); n, k >= 1; read by antidiagonals.

Original entry on oeis.org

1, 4, 2, 9, 8, 3, 16, 18, 12, 5, 25, 32, 27, 20, 6, 36, 50, 48, 45, 24, 7, 49, 72, 75, 80, 54, 28, 10, 64, 98, 108, 125, 96, 63, 40, 11, 81, 128, 147, 180, 150, 112, 90, 44, 13, 100, 162, 192, 245, 216, 175, 160, 99, 52, 14, 121, 200, 243, 320, 294, 252, 250

Offset: 1

Clark Kimberling, Oct 26 2014

Every positive integer occurs exactly once.

			Northwest corner:
1   4    9    16   25    36    49
2   8    18   32   50    72    98
3   12   27   48   75    108   147
5   20   45   80   125   180   245
6   24   54   96   150   216   294

Cf. A005117, A000037 (is partitioned by the rows of A249327, excluding the first).

z = 20; f = Select[Range[10000], SquareFreeQ[#] &];
u[n_, k_] := f[[n]]*k^2; t = Table[u[n, k], {n, 1, 20}, {k, 1, 20}];
TableForm[t] (* A249327 array *)
Table[u[k, n - k + 1], {n, 1, 15}, {k, 1, n}] // Flatten (* A249327 sequence *)

T(1,k) = A000290(k), T(2,k) = A001105(k), T(3,k) = A033428(k), T(4,k) = A033429(k), T(5,.) through T(10,.) are A033581, A033582, A033583, A033584, A152742 and A144555 without initial 0. - M. F. Hasler, Oct 31 2014

A303302 a(n) = 34*n^2.

Original entry on oeis.org

0, 34, 136, 306, 544, 850, 1224, 1666, 2176, 2754, 3400, 4114, 4896, 5746, 6664, 7650, 8704, 9826, 11016, 12274, 13600, 14994, 16456, 17986, 19584, 21250, 22984, 24786, 26656, 28594, 30600, 32674, 34816, 37026, 39304, 41650, 44064, 46546, 49096, 51714, 54400, 57154, 59976, 62866, 65824, 68850, 71944

Offset: 0

Omar E. Pol, May 13 2018

Sequence found by reading the line from 0, in the direction 0, 34, ..., in the square spiral whose vertices are the generalized 19-gonal numbers A303813.

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

Cf. A005843, A008599, A303813.

Cf. similar sequences of the type k*n^2: A000290 (k=1), A001105 (k=2), A033428 (k=3), A016742 (k=4), A033429 (k=5), A033581 (k=6), A033582 (k=7), A139098 (k=8), A016766 (k=9), A033583 (k=10), A033584 (k=11), A135453 (k=12), A152742 (k=13), A144555 (k=14), A064761 (k=15), A016802 (k=16), A244630 (k=17), A195321 (k=18), A244631 (k=19), A195322 (k=20), A064762 (k=21), A195323 (k=22), A244632 (k=23), A195824 (k=24), A016850 (k=25), A244633 (k=26), A244634 (k=27), A064763 (k=28), A244635 (k=29), A244636 (k=30), A244082 (k=32), this sequence (k=34), A016910 (k=36), A016982 (k=49), A017066 (k=64), A017162 (k=81), A017270 (k=100), A017390 (k=121), A017522 (k=144).

[34*n^2: n in [0..50]]; // Vincenzo Librandi Jun 07 2018

Table[34 n^2, {n, 0, 40}]
LinearRecurrence[{3,-3,1},{0,34,136},50] (* Harvey P. Dale, Jul 23 2018 *)

PARI
```
a(n) = 34*n^2;
```

concat(0, Vec(34*x*(1 + x) / (1 - x)^3 + O(x^40))) \\ Colin Barker, Jun 12 2018

a(n) = 34*A000290(n) = 17*A001105(n) = 2*A244630(n).
G.f.: 34*x*(1 + x)/(1 - x)^3. - Vincenzo Librandi, Jun 07 2018
From Elmo R. Oliveira, Dec 02 2024: (Start)
E.g.f.: 34*x*(1 + x)*exp(x).
a(n) = A005843(n)*A008599(n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A363436 Array read by ascending antidiagonals: A(n, k) = k*n^2, with k >= 0.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 4, 2, 0, 0, 9, 8, 3, 0, 0, 16, 18, 12, 4, 0, 0, 25, 32, 27, 16, 5, 0, 0, 36, 50, 48, 36, 20, 6, 0, 0, 49, 72, 75, 64, 45, 24, 7, 0, 0, 64, 98, 108, 100, 80, 54, 28, 8, 0, 0, 81, 128, 147, 144, 125, 96, 63, 32, 9, 0, 0, 100, 162, 192, 196, 180, 150, 112, 72, 36, 10, 0

Offset: 0

Stefano Spezia, Jul 08 2023

			The array begins:
  0,  0,  0,   0,   0,   0,   0, ...
  0,  1,  2,   3,   4,   5,   6, ...
  0,  4,  8,  12,  16,  20,  24, ...
  0,  9, 18,  27,  36,  45,  54, ...
  0, 16, 32,  48,  64,  80,  96, ...
  0, 25, 50,  75, 100, 125, 150, ...
  0, 36, 72, 108, 144, 180, 216, ...
  ...

Cf. A000290 (k = 1), A001105 (k = 2), A033428 (k = 3), A016742 (k = 4), A033429 (k = 5), A033581 (k = 6), A033582 (k = 7), A139098 (k = 8), A016766 (k = 9), A033583 (k = 10), A033584 (k = 11), A135453 (k = 12),  A152742 (k = 13), A144555 (k = 14), A064761 (k = 15), A016802 (k = 16), A244630 (k = 17), A195321 (k = 18), A244631 (k = 19), A195322 (k = 20), A064762 (k = 21), A195323 (k = 22), A244632 (k = 23), A195824 (k = 24), A016850 (k = 25), A244633 (k = 26), A244634 (k = 27), A064763 (k = 28), A244635 (k = 29), A244636 (k = 30).
Cf. A001477 (n = 1), A008586 (n = 2), A008591 (n = 3), A008598 (n = 4), A008607 (n = 5), A044102 (n = 6), A152691 (n = 8).
Cf. A000007 (n = 0 or k = 0), A000578 (main diagonal), A002415 (antidiagonal sums), A004247.

A[n_,k_]:=k n^2; Table[A[n-k,k],{n,0,11},{k,0,n}]//Flatten

O.g.f.: x*y*(1 + x)/((1 - x)^3*(1 - y)^2).
E.g.f.: x*y*(1 + x)*exp(x + y).
A(n, k) = n*A004247(n, k).

cp's OEIS Frontend

A144555 a(n) = 14*n^2.

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

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A244630 a(n) = 17*n^2.

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A186030 a(n) = n*(13*n-3)/2.

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A016910 a(n) = (6*n)^2.

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A244633 a(n) = 26*n^2.

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A249327 Rectangular array T(n,k) = f(n)*k^2, where f = A005117 (squarefree numbers); n, k >= 1; read by antidiagonals.

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

A186030 a(n) = n(13n-3)/2.