A017221 -id:A017221 - cp's OEIS frontend

A017173 a(n) = 9*n + 1.

1, 10, 19, 28, 37, 46, 55, 64, 73, 82, 91, 100, 109, 118, 127, 136, 145, 154, 163, 172, 181, 190, 199, 208, 217, 226, 235, 244, 253, 262, 271, 280, 289, 298, 307, 316, 325, 334, 343, 352, 361, 370, 379, 388, 397, 406, 415, 424, 433, 442, 451, 460, 469, 478

Offset: 0

N. J. A. Sloane

Also all the numbers with digital root 1; A010888(a(n)) = 1. - Rick L. Shepherd, Jan 12 2009
A116371(a(n)) = A156144(a(n)); positions where records occur in A156144: A156145(n+1) = A156144(a(n)). - Reinhard Zumkeller, Feb 05 2009
If A=[A147296] 9*n^2+2*n (n>0, 11, 40, 87, ...); Y=[A010701] 3 (3, 3, 3, ...); X=[A017173] 9*n+1 (n>0, 10, 19, 28, ...), we have, for all terms, Pell's equation X^2 - A*Y^2 = 1. Example: 10^2 - 11*3^2 = 1; 19^2 - 40*3^2 = 1; 28^2 - 87*3^2 = 1. - Vincenzo Librandi, Aug 01 2010

Mohammed Yaseen, Table of n, a(n) for n = 0..10000
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A093644 ((9,1) Pascal, column m=1).
Cf. A010701, A010888, A016777, A116371, A147296, A156144, A156145.
Numbers with digital root m: this sequence (m=1), A017185 (m=2), A017197 (m=3), A017209 (m=4), A017221 (m=5), A017233 (m=6), A017245 (m=7), A017257 (m=8), A008591 (m=9).

a017173 = (+ 1) . (* 9)
a017173_list = [1, 10 ..]  -- Reinhard Zumkeller, Feb 04 2014

Range[1, 1000, 9] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)
LinearRecurrence[{2,-1},{1,10},60] (* Harvey P. Dale, Dec 27 2014 *)

forstep(n=1,500,9,print1(n", ")) \\ Charles R Greathouse IV, May 28 2011

[i+1 for i in range(480) if gcd(i,9) == 9] # Zerinvary Lajos, May 20 2009

G.f.: (1 + 8*x)/(1 - x)^2.
a(n) = 2*a(n-1) - a(n-2) with a(0)=1, a(1)=10. - Vincenzo Librandi, Aug 01 2010
E.g.f.: exp(x)*(1 + 9*x). - Stefano Spezia, Apr 20 2023
a(n) = A016777(3*n). - Elmo R. Oliveira, Apr 12 2025

A017233 a(n) = 9*n + 6.

Original entry on oeis.org

6, 15, 24, 33, 42, 51, 60, 69, 78, 87, 96, 105, 114, 123, 132, 141, 150, 159, 168, 177, 186, 195, 204, 213, 222, 231, 240, 249, 258, 267, 276, 285, 294, 303, 312, 321, 330, 339, 348, 357, 366, 375, 384, 393, 402, 411, 420, 429, 438, 447, 456, 465, 474, 483

Offset: 0

David J. Horn and Laura Krebs Gordon (lkg615(AT)verizon.net), 1985

General form: (q*n-1)*q, cf. A017233 (q=3), A098502 (q=4). - Vladimir Joseph Stephan Orlovsky, Feb 16 2009

Numbers whose digital root is 6; that is, A010888(a(n)) = 6. (Ball essentially says that Iamblichus (circa 350) announced that a number equal to the sum of three integers 3*n, 3*n - 1, and 3*n - 2 has 6 as what is now called the number's digital root.) - Rick L. Shepherd, Apr 01 2014

W. W. R. Ball, A Short Account of the History of Mathematics, Sterling Publishing Company, Inc., 2001 (Facsimile Edition) [orig. pub. 1912], pages 110-111.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Tanya Khovanova, Recursive Sequences.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008585, A008591, A010888, A016789, A017209, A017221, A017233, A019557, A098502.

[9*n+6: n in [0..60]]; // Vincenzo Librandi, Jul 24 2011

Range[6, 1000, 9] (* Vladimir Joseph Stephan Orlovsky, May 28 2011 *)
LinearRecurrence[{2,-1},{6,15},60] (* Harvey P. Dale, Feb 01 2014 *)

a(n)=9*n+6 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: 3*(2+x)/(x-1)^2. - R. J. Mathar, Mar 20 2018
Sum_{n>=0} (-1)^n/a(n) = sqrt(3)*Pi/27 - log(2)/9. - Amiram Eldar, Dec 12 2021
E.g.f.: 3*exp(x)*(2 + 3*x). - Stefano Spezia, Dec 07 2024
From Elmo R. Oliveira, Apr 10 2025: (Start)
a(n) = 3*A016789(n) = A019557(n+1)/2.
a(n) = 2*a(n-1) - a(n-2). (End)

A269044 a(n) = 13*n + 7.

Original entry on oeis.org

7, 20, 33, 46, 59, 72, 85, 98, 111, 124, 137, 150, 163, 176, 189, 202, 215, 228, 241, 254, 267, 280, 293, 306, 319, 332, 345, 358, 371, 384, 397, 410, 423, 436, 449, 462, 475, 488, 501, 514, 527, 540, 553, 566, 579, 592, 605, 618, 631, 644, 657, 670, 683, 696, 709, 722, 735

Offset: 0

Bruno Berselli, Feb 18 2016

After 7 (which corresponds to n=0), all terms belong to A090767 because a(n) = 3*n*2*1 + 2*(n*2+2*1+n*1) + (n+2+1).
This sequence is related to A152741 by the recurrence A152741(n+1) = (n+1)*a(n+1) - Sum_{k = 0..n} a(k).
Any square mod 13 is one of 0, 1, 3, 4, 9, 10 or 12 (A010376) but not 7, and for this reason there are no squares in the sequence. Likewise, any cube mod 13 is one of 0, 1, 5, 8 or 12, therefore no a(k) is a cube.
The sum of the squares of any two terms of the sequence is also a term of the sequence, that is: a(h)^2 + a(k)^2 = a(h*(13*h+14) + k*(13*k+14) + 7). Therefore: a(h)^2 + a(k)^2 > a(a( h*(h+1) + k*(k+1) )) for h+k > 0.
The primes of the sequence are listed in A140371.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Bruno Berselli, A description of the recursive method shown in the third comment: website Matem@ticamente (in Italian), 2008.
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A010376, A022271 (partial sums), A088227, A090767, A140371, A152741.
Similar sequences with closed form (2*k-1)*n+k: A001489 (k=0), A000027 (k=1), A016789 (k=2), A016885 (k=3), A017029 (k=4), A017221 (k=5), A017461 (k=6), this sequence (k=7), A164284 (k=8).
Sequences of the form 13*n+q: A008595 (q=0), A190991 (q=1), A153080 (q=2), A127547 (q=4), A154609 (q=5), A186113 (q=6), this sequence (q=7), A269100 (q=11).

Magma
```
[13*n+7: n in [0..60]];
```

13 Range[0, 60] + 7 (* or *) Range[7, 800, 13] (* or *) Table[13 n + 7, {n, 0, 60}]
LinearRecurrence[{2, -1}, {7, 20}, 60] (* Vincenzo Librandi, Feb 19 2016 *)

Maxima
```
makelist(13*n+7, n, 0, 60);
```
PARI
```
vector(60, n, n--; 13*n+7)
```
Sage
```
[13*n+7 for n in (0..60)]
```

G.f.: (7 + 6*x)/(1 - x)^2.
a(n) =  A088227(4*n+3).
a(n) = -A186113(-n-1).
Sum_{i=h..h+13*k} a(i) = a(h*(13*k + 1) + k*(169*k + 27)/2).
Sum_{i>=0} 1/a(i)^2 = 0.0257568950542502716970... = polygamma(1, 7/13)/13^2.
E.g.f.: exp(x)*(7 + 13*x). - Stefano Spezia, Aug 02 2021

A247682 Odd composite numbers congruent to 5 modulo 9.

Original entry on oeis.org

77, 95, 185, 203, 221, 275, 329, 365, 437, 455, 473, 527, 545, 581, 635, 671, 689, 707, 725, 779, 815, 833, 851, 869, 905, 923, 959, 995, 1067, 1085, 1121, 1139, 1157, 1175, 1211, 1247, 1265, 1337, 1355, 1391, 1445, 1463, 1517, 1535, 1589

Offset: 1

Odimar Fabeny, Sep 22 2014

Subsequence of A017221 (9n + 5).

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

Cf. A017221, A247676, A247678, A247679, A247681, A247683.

Select[18Range[100] + 5, Not[PrimeQ[#]] &] (* Alonso del Arte, Sep 25 2014 *)
Select[Range[5,2000,18],CompositeQ] (* Harvey P. Dale, Feb 21 2016 *)

lista(nn) = {forcomposite(n=1, nn, if ((n % 2) && ((n % 9) == 5), print1(n, ", ")); ); } \\ Michel Marcus, Sep 22 2014

A144650 Triangle read by rows where T(m,n) = 2m*n + m + n + 1.

Original entry on oeis.org

5, 8, 13, 11, 18, 25, 14, 23, 32, 41, 17, 28, 39, 50, 61, 20, 33, 46, 59, 72, 85, 23, 38, 53, 68, 83, 98, 113, 26, 43, 60, 77, 94, 111, 128, 145, 29, 48, 67, 86, 105, 124, 143, 162, 181, 32, 53, 74, 95, 116, 137, 158, 179, 200, 221, 35, 58, 81, 104, 127, 150, 173, 196, 219, 242, 265

Offset: 1

Vincenzo Librandi, Jan 13 2009

First column: A016789, second column: A016885, third column: A017029, fourth column: A017221, fifth column: A017461. - Vincenzo Librandi, Nov 21 2012

			Triangle begins:
   5;
   8, 13;
  11, 18, 25;
  14, 23, 32, 41;
  17, 28, 39, 50,  61;
  20, 33, 46, 59,  72,  85;
  23, 38, 53, 68,  83,  98, 113;
  26, 43, 60, 77,  94, 111, 128, 145;
  29, 48, 67, 86, 105, 124, 143, 162, 181;
  32, 53, 74, 95, 116, 137, 158, 179, 200, 221; etc.

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

Cf. A001105, A001844, A003307, A007742, A104275, A142463, A160378.

Columns k: A016789 (k=1), A016885 (k=2), A017029 (k=3), A017221 (k=4), A017461 (k=5).

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

T[n_,k_]:= 2 n*k + n + k + 1; Table[T[n, k], {n, 11}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 21 2012 *)

flatten([[2*n*k+n+k+1 for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Oct 14 2023

Sum_{n=1..m} T(m, n) = m*(2*m+3)*(m+1)/2 = A160378(n+1) (row sums). - R. J. Mathar, Jan 15 2009, Jan 05 2011
From G. C. Greubel, Oct 14 2023: (Start)
T(n, n) = A001844(n).
T(n, n-1) = A001105(n), n >= 2.
T(n, n-2) = A142463(n-1), n >= 3.
T(n, n-3) = (-1)*A147973(n+2), n >= 4.
Sum_{k=1..n} (-1)^k*T(n, k) = (-1)^n*A007742(floor((n+1)/2)).
G.f.: x*y*(5 - 2*x - 2*x*y - 2*x^2*y + x^2*y^2)/((1-x)^2*(1-x*y)^3). (End)

A017222 a(n) = (9*n + 5)^2.

Original entry on oeis.org

25, 196, 529, 1024, 1681, 2500, 3481, 4624, 5929, 7396, 9025, 10816, 12769, 14884, 17161, 19600, 22201, 24964, 27889, 30976, 34225, 37636, 41209, 44944, 48841, 52900, 57121, 61504, 66049, 70756

Offset: 0

N. J. A. Sloane

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

Sequences of the form (m*n+5)^2: A010864 (m=0), A000290 (m=1), A016754 (m=2), A016790 (m=3), A016814 (m=4), A016850 (m=5), A016970 (m=6), A017042 (m=7), A017126 (m=8), this sequence (m=9), A017330 (m=10), A017450 (m=11), A017582 (m=12).

Cf. A017221, A017246.

[(9*n+5)^2: n in [0..35]]; // Vincenzo Librandi, Jul 24 2011

(9Range[0,30]+5)^2 (* or *) LinearRecurrence[{3,-3,1},{25,196,529},30] (* Harvey P. Dale, May 22 2012 *)

a(n)=(9*n+5)^2 \\ Charles R Greathouse IV, Jun 17 2017

[(9*n+5)^2 for n in range(41)] # G. C. Greubel, Dec 29 2022

a(n) = A017221(n)^2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, May 22 2012
G.f.: (25 + 121*x + 16*x^2)/(1-x)^3. - R. J. Mathar, Mar 20 2018
From G. C. Greubel, Dec 29 2022: (Start)
a(2*n+1) = 4*A017246(n).
a(n) = a(n-1) + 9*(18*n + 1).
E.g.f.: (25 + 171*x + 81*x^2)*exp(x). (End)

A017225 a(n) = (9*n + 5)^5.

Original entry on oeis.org

3125, 537824, 6436343, 33554432, 115856201, 312500000, 714924299, 1453933568, 2706784157, 4704270176, 7737809375, 12166529024, 18424351793, 27027081632, 38579489651, 53782400000, 73439775749

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (6, -15, 20, -15, 6, -1).

Cf. A000584 (n^5), A017221 (9*n+5),

[(9*n+5)^5: n in [0..25]]; // Vincenzo Librandi, Jul 24 2011

G.f.: (3125 + 519074*x + 3256274*x^2 + 2941234*x^3 + 365149*x^4 + 1024*x^5)/(x-1)^6. - R. J. Mathar, Mar 20 2018

A082286 a(n) = 18*n + 10.

Original entry on oeis.org

10, 28, 46, 64, 82, 100, 118, 136, 154, 172, 190, 208, 226, 244, 262, 280, 298, 316, 334, 352, 370, 388, 406, 424, 442, 460, 478, 496, 514, 532, 550, 568, 586, 604, 622, 640, 658, 676, 694, 712, 730, 748, 766, 784, 802, 820, 838, 856, 874, 892, 910, 928, 946

Offset: 0

Cino Hilliard, May 10 2003

Solutions to (11^x + 13^x) mod 19 = 17.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Tanya Khovanova, Recursive Sequences
Leo Tavares, Illustration: Triangles
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A006370, A008600, A016945, A161705.

Cf. A000217, A017221, A022267, A060544.

[18*n + 10: n in [0..60]]; // Vincenzo Librandi, May 05 2011

Range[10, 1000, 18] (* Vladimir Joseph Stephan Orlovsky, Jun 01 2011 *)

a(n)=18*n+10 \\ Charles R Greathouse IV, Jul 10 2016

a(n) = A006370(A016945(n)). - Reinhard Zumkeller, Apr 17 2008
a(n) = 2*A017221(n). - Michel Marcus, Feb 15 2014
a(n) = A060544(n+2) - 9*A000217(n-1). - Leo Tavares, Oct 15 2022
From Elmo R. Oliveira, Apr 08 2024: (Start)
G.f.: 2*(5+4*x)/(1-x)^2.
E.g.f.: 2*exp(x)*(5 + 9*x).
a(n) = 2*a(n-1) - a(n-2) for n >= 2.
a(n) = 2*(A022267(n+1) - A022267(n)). (End)

More terms from Reinhard Zumkeller, Apr 17 2008

A154277 a(n) = 81n^2 - 72n + 17.

Original entry on oeis.org

17, 26, 197, 530, 1025, 1682, 2501, 3482, 4625, 5930, 7397, 9026, 10817, 12770, 14885, 17162, 19601, 22202, 24965, 27890, 30977, 34226, 37637, 41210, 44945, 48842, 52901, 57122, 61505, 66050, 70757, 75626, 80657, 85850, 91205, 96722

Offset: 0

Vincenzo Librandi, Jan 06 2009

The identity (81*n^2 + 90*n + 26)^2 - (9*n^2 + 10*n + 3)*(27*n + 15)^2 = 1 can be written as a(n+1)^2 - A154254(n+1)*A154267(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 (3,-3,1).

Cf. A154254, A154267.

I:=[26, 197, 530]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 02 2012

LinearRecurrence[{3, -3, 1}, {26, 197, 530}, 40] (* Vincenzo Librandi, Feb 02 2012 *)
Table[81n^2-72n+17,{n,0,40}] (* Harvey P. Dale, Oct 16 2022 *)

for(n=0, 22, print1(81*n^2 - 72*n + 17", ")); \\ Vincenzo Librandi, Feb 02 2012

G.f.: (17 - 25*x + 170*x^2)/(1-x)^3. - Vincenzo Librandi, Feb 02 2012
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Feb 02 2012
a(n) = A017221(n-1)^2 + 1 with A017221(-1) = -4. - Bruno Berselli, Feb 02 2012
E.g.f.: (17 + 9*x + 81*x^2)*exp(x). - G. C. Greubel, Sep 09 2016

92205 replaced by 91205 - R. J. Mathar, Jan 07 2009

Edited by Charles R Greathouse IV, Aug 09 2010

A017223 a(n) = (9*n+5)^3.

Original entry on oeis.org

125, 2744, 12167, 32768, 68921, 125000, 205379, 314432, 456533, 636056, 857375, 1124864, 1442897, 1815848, 2248091, 2744000, 3307949, 3944312, 4657463, 5451776, 6331625, 7301384, 8365427

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Sequences of the form (9*n+5)^k: A017221 (k=1), A017222 (k=2), this sequence (k=3), A017224 (k=4), A017225 (k=5), A017226 (k=6), A017227 (k=7), A017228 (k=8), A017229 (k=9), A017230 (k=10), A017231 (k=11).

[(9*n+5)^3: n in [0..35]] ; // Vincenzo Librandi, Jul 24 2011

(9*Range[0,50] + 5)^3 (* G. C. Greubel, Jan 06 2023 *)

[(9*n+5)^3 for n in range(51)] # G. C. Greubel, Jan 06 2023

From Chai Wah Wu, Jul 14 2016: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 3.
G.f.: (125 + 2244*x + 1941*x^2 + 64*x^3)/(1 - x)^4. (End)
E.g.f.: (125 + 2619*x + 3402*x^2 + 729*x^3)*exp(x). - G. C. Greubel, Jan 06 2023

cp's OEIS Frontend

A017173 a(n) = 9*n + 1.

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A017233 a(n) = 9*n + 6.

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A269044 a(n) = 13*n + 7.

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A247682 Odd composite numbers congruent to 5 modulo 9.

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A144650 Triangle read by rows where T(m,n) = 2m*n + m + n + 1.

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A017222 a(n) = (9*n + 5)^2.

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A017225 a(n) = (9*n + 5)^5.

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A154277 a(n) = 81n^2 - 72n + 17.