A244630 -id:A244630 - cp's OEIS frontend

A008599 Multiples of 17.

0, 17, 34, 51, 68, 85, 102, 119, 136, 153, 170, 187, 204, 221, 238, 255, 272, 289, 306, 323, 340, 357, 374, 391, 408, 425, 442, 459, 476, 493, 510, 527, 544, 561, 578, 595, 612, 629, 646, 663, 680, 697, 714, 731, 748, 765, 782, 799, 816, 833, 850, 867, 884

Offset: 0

N. J. A. Sloane

Ivan Panchenko, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences.
Michael Penn, Divisibility by 17: The Lesser Known Rule, YouTube video, 2023.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 329.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008597, A008598, A008600, A244630.

Range[0, 1003, 17] (* Vladimir Joseph Stephan Orlovsky, Jun 01 2011 *)

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

for (n <- 0 to 1003 by 17) print(n + ", ") // Alonso del Arte, Jun 17 2018

(floor(a(n)/10) - 5*(a(n) mod 10)) == 0 (mod 17), see A076311. - Reinhard Zumkeller, Oct 06 2002
From Vincenzo Librandi, Dec 24 2010: (Start)
a(n) = 17*n.
a(n) = 2*a(n-1) - a(n-2).
G.f.: 17*x/(x-1)^2. (End)
From Elmo R. Oliveira, Apr 10 2025: (Start)
E.g.f.: 17*x*exp(x).
a(n) = (A008598(n) + A008600(n))/2. (End)

A016850 a(n) = (5*n)^2.

Original entry on oeis.org

0, 25, 100, 225, 400, 625, 900, 1225, 1600, 2025, 2500, 3025, 3600, 4225, 4900, 5625, 6400, 7225, 8100, 9025, 10000, 11025, 12100, 13225, 14400, 15625, 16900, 18225, 19600, 21025, 22500, 24025, 25600, 27225, 28900, 30625, 32400, 34225, 36100, 38025, 40000, 42025

Offset: 0

N. J. A. Sloane

If we define C(n) = (5*n)^2 (n > 0), the sequence is the first "square-sequence" such that for every n there exists p such that C(n) = C(p) + C(p+n). We observe in fact that p = 3*n because 25 = 3^2 + 4^2. The sequence without 0 is linked with the first nontrivial solution (trivial: n^2 = 0^2 + n^2) of the equation X^2 = 2*Y^2 + 2*n^2 where X = 2*k and Y = 2*p + n which is equivalent to k^2 = p^2 + (p+n)^2 for n given. The second such "square-sequence" is (29*n)^2 (n > 0) because 29^2 = 20^2 + 21^2 and with this relation we obtain (29*n)^2 = (20*n)^2 + (20*n+n)^2. - Richard Choulet, Dec 23 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..800
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000290, A033429, A053742 (first differences), A008587, A008607.

Similar sequences listed in A244630.

[(5*n)^2: n in [0..50]]; // Vincenzo Librandi, Apr 26 2011

(5Range[0, 31])^2 (* Alonso del Arte, Oct 08 2017 *)

a(n)=(5*n)^2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 25*n^2 = 25*A000290(n) = 5*A033429(n). - Omar E. Pol, Jul 03 2014
From Amiram Eldar, Jan 25 2021: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/150.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/300.
Product_{n>=1} (1 + 1/a(n)) = sinh(Pi/5)/(Pi/5).
Product_{n>=1} (1 - 1/a(n)) = sin(Pi/5)/(Pi/5) = 5*sqrt((5-sqrt(5))/2)/(2*Pi). (End)
a(n) = Sum_{i=0..n-1} A053742(i). - John Elias, Jun 30 2021
G.f.: 25*x*(1 + x)/(1 - x)^3. - Stefano Spezia, Jul 08 2023
From Elmo R. Oliveira, Nov 30 2024: (Start)
E.g.f.: 25*x*(1 + x)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
a(n) = n*A008607(n) = A000290(A008587(n)) = A008587(n)^2. (End)

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)

A244636 a(n) = 30*n^2.

Original entry on oeis.org

0, 30, 120, 270, 480, 750, 1080, 1470, 1920, 2430, 3000, 3630, 4320, 5070, 5880, 6750, 7680, 8670, 9720, 10830, 12000, 13230, 14520, 15870, 17280, 18750, 20280, 21870, 23520, 25230, 27000, 28830, 30720, 32670, 34680, 36750, 38880, 41070, 43320, 45630, 48000, 50430

Offset: 0

Vincenzo Librandi, Jul 03 2014

Sequence found by reading the line from 0, in the direction 0, 30, ..., in the square spiral whose vertices are the generalized 17-gonal numbers. - Omar E. Pol, Jul 03 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, A033428, A033429, A033581, A033583, A064761, A249674, A330451.

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

A244636:=n->30*n^2: seq(A244636(n), n=0..50); # Wesley Ivan Hurt, Jul 04 2014

Table[30 n^2, {n, 0, 40}]
CoefficientList[Series[30x (1+x)/(1-x)^3,{x,0,50}],x] (* or *) LinearRecurrence[ {3,-3,1},{0,30,120},50] (* Harvey P. Dale, Dec 02 2021 *)

a(n)=30*n^2 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: 30*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) = 30*A000290(n) = 15*A001105(n) = 10*A033428(n) = 6*A033429(n) = 5*A033581(n) = 3*A033583(n) = 2*A064761(n). - Omar E. Pol, Jul 03 2014
From Elmo R. Oliveira, Dec 02 2024: (Start)
E.g.f.: 30*x*(1 + x)*exp(x).
a(n) = n*A249674(n) = A330451(3*n). (End)

A064762 a(n) = 21*n^2.

Original entry on oeis.org

0, 21, 84, 189, 336, 525, 756, 1029, 1344, 1701, 2100, 2541, 3024, 3549, 4116, 4725, 5376, 6069, 6804, 7581, 8400, 9261, 10164, 11109, 12096, 13125, 14196, 15309, 16464, 17661, 18900, 20181, 21504, 22869, 24276, 25725, 27216, 28749

Offset: 0

Roberto E. Martinez II, Oct 18 2001

Number of edges in a complete 7-partite graph of order 7n, K_n,n,n,n,n,n,n.

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

Similar sequences are listed in A244630.
Cf. A000217, A000290, A033428, A033581, A033582, A033583.
Cf. A008603, A195049.

[21*n^2 : n in [0..50]]; // Wesley Ivan Hurt, Jul 04 2014

21 Range[0, 50]^2 (* Wesley Ivan Hurt, Jul 04 2014 *)
LinearRecurrence[{3,-3,1},{0,21,84},40] (* Harvey P. Dale, Jul 29 2019 *)

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

a(n) = 42*n + a(n-1) - 21 for n > 0, a(0)=0. - Vincenzo Librandi, Aug 07 2010
a(n) = 21*A000290(n) = 7*A033428(n) = 3*A033582(n). - Omar E. Pol, Jul 03 2014
a(n) = t(7*n) - 7*t(n), where t(i) = i*(i+k)/2 for any k. Special case (k=1): a(n) = A000217(7*n) - 7*A000217(n). - Bruno Berselli, Aug 31 2017
From Elmo R. Oliveira, Nov 30 2024: (Start)
G.f.: 21*x*(1 + x)/(1-x)^3.
E.g.f.: 21*x*(1 + x)*exp(x).
a(n) = n*A008603(n) = A195049(2*n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (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)

A244631 a(n) = 19*n^2.

Original entry on oeis.org

0, 19, 76, 171, 304, 475, 684, 931, 1216, 1539, 1900, 2299, 2736, 3211, 3724, 4275, 4864, 5491, 6156, 6859, 7600, 8379, 9196, 10051, 10944, 11875, 12844, 13851, 14896, 15979, 17100, 18259, 19456, 20691, 21964, 23275, 24624, 26011, 27436, 28899, 30400, 31939, 33516

Offset: 0

Vincenzo Librandi, Jul 03 2014

First bisection of A195048. - Bruno Berselli, Jul 03 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. A000290, A008601, A195048.

Cf. similar sequences listed in A244630.

Magma
```
[19*n^2: n in [0..40]];
```
Mathematica
```
Table[19 n^2, {n, 0, 40}]
```

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

G.f.: 19*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) = 19*A000290(n). - Omar E. Pol, Jul 03 2014
From Elmo R. Oliveira, Dec 01 2024: (Start)
E.g.f.: 19*x*(1 + x)*exp(x).
a(n) = n*A008601(n) = A195048(2*n). (End)

A244632 a(n) = 23*n^2.

Original entry on oeis.org

0, 23, 92, 207, 368, 575, 828, 1127, 1472, 1863, 2300, 2783, 3312, 3887, 4508, 5175, 5888, 6647, 7452, 8303, 9200, 10143, 11132, 12167, 13248, 14375, 15548, 16767, 18032, 19343, 20700, 22103, 23552, 25047, 26588, 28175, 29808, 31487, 33212, 34983, 36800, 38663

Offset: 0

Vincenzo Librandi, Jul 03 2014

First bisection of A195058. - Bruno Berselli, Jul 03 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. A000290, A008605, A195058.

Cf. similar sequences listed in A244630.

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

Table[23 n^2, {n, 0, 40}]
LinearRecurrence[{3,-3,1},{0,23,92},50] (* Harvey P. Dale, Jul 14 2024 *)

a(n)=23*n^2 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: 23*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) = 23*A000290(n). - Omar E. Pol, Jul 03 2014
From Elmo R. Oliveira, Dec 01 2024: (Start)
E.g.f.: 23*x*(1 + x)*exp(x).
a(n) = n*A008605(n) = A195058(2*n). (End)

A244634 a(n) = 27*n^2.

Original entry on oeis.org

0, 27, 108, 243, 432, 675, 972, 1323, 1728, 2187, 2700, 3267, 3888, 4563, 5292, 6075, 6912, 7803, 8748, 9747, 10800, 11907, 13068, 14283, 15552, 16875, 18252, 19683, 21168, 22707, 24300, 25947, 27648, 29403, 31212, 33075, 34992, 36963, 38988, 41067, 43200, 45387

Offset: 0

Vincenzo Librandi, Jul 03 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, A016766, A033428, A305548.

Magma
```
[27*n^2: n in [0..40]];
```
Mathematica
```
Table[27 n^2, {n, 0, 40}]
```

a(n)=27*n^2 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: 27*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) = 27*A000290(n) = 9*A033428(n) = 3*A016766(n). - Omar E. Pol, Jul 03 2014
From Elmo R. Oliveira, Dec 01 2024: (Start)
E.g.f.: 27*x*(1 + x)*exp(x).
a(n) = n*A305548(n). (End)

A244635 a(n) = 29*n^2.

Original entry on oeis.org

0, 29, 116, 261, 464, 725, 1044, 1421, 1856, 2349, 2900, 3509, 4176, 4901, 5684, 6525, 7424, 8381, 9396, 10469, 11600, 12789, 14036, 15341, 16704, 18125, 19604, 21141, 22736, 24389, 26100, 27869, 29696, 31581, 33524, 35525, 37584, 39701, 41876, 44109, 46400, 48749

Offset: 0

Vincenzo Librandi, Jul 03 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, A195819.

Magma
```
[29*n^2: n in [0..40]];
```
Mathematica
```
Table[29 n^2, {n, 0, 40}]
```

a(n)=29*n^2 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: 29*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) = 29*A000290(n). - Omar E. Pol, Jul 03 2014
From Elmo R. Oliveira, Dec 01 2024: (Start)
E.g.f.: 29*x*(1 + x)*exp(x).
a(n) = n*A195819(n). (End)

cp's OEIS Frontend

A008599 Multiples of 17.

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A016850 a(n) = (5*n)^2.

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

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A244636 a(n) = 30*n^2.

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A064762 a(n) = 21*n^2.

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

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A244631 a(n) = 19*n^2.

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