A127547 -id:A127547 - cp's OEIS frontend

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

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

A154609 a(n) = 13*n + 5.

Original entry on oeis.org

5, 18, 31, 44, 57, 70, 83, 96, 109, 122, 135, 148, 161, 174, 187, 200, 213, 226, 239, 252, 265, 278, 291, 304, 317, 330, 343, 356, 369, 382, 395, 408, 421, 434, 447, 460, 473, 486, 499, 512, 525, 538, 551, 564, 577, 590, 603, 616, 629, 642, 655, 668, 681, 694

Offset: 0

Vincenzo Librandi, Jan 15 2009

Any square mod 13 is one of 0, 1, 3, 4, 9, 10 or 12 (A010376) but not 5, for this reason there are no squares in sequence. - Bruno Berselli, Feb 19 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A010376,

Sequences of the form 13*n+q: A008595 (q=0), A190991 (q=1), A153080 (q=2), A127547 (q=4), this sequence (q=5), A186113 (q=6), A269044 (q=7), A269100 (q=11).

I:=[5, 18]; [n le 2 select I[n] else 2*Self(n-1)-Self(n-2): n in [1..50]]; // Vincenzo Librandi, Feb 26 2012

Range[5, 1000, 13] (* Vladimir Joseph Stephan Orlovsky, May 31 2011 *)
CoefficientList[Series[(8 x + 5)/(1 - x)^2, {x, 0, 60}], x] (* Vincenzo Librandi, Feb 26 2012 *)

a(n)=13*n+5 \\ Charles R Greathouse IV, Dec 28 2011

[13*n+5 for n in range(61)] # G. C. Greubel, May 31 2024

From Vincenzo Librandi, Feb 26 2012: (Start)
G.f.: (5+8*x)/(1-x)^2.
a(n) = 2*a(n-1) - a(n-2). (End)
E.g.f.: (5 + 13*x)*exp(x). - G. C. Greubel, May 31 2024

A269100 a(n) = 13*n + 11.

Original entry on oeis.org

11, 24, 37, 50, 63, 76, 89, 102, 115, 128, 141, 154, 167, 180, 193, 206, 219, 232, 245, 258, 271, 284, 297, 310, 323, 336, 349, 362, 375, 388, 401, 414, 427, 440, 453, 466, 479, 492, 505, 518, 531, 544, 557, 570, 583, 596, 609, 622, 635, 648, 661, 674, 687, 700, 713, 726, 739

Offset: 0

Bruno Berselli, Feb 19 2016

Any square mod 13 is one of 0, 1, 3, 4, 9, 10 or 12 (A010376) but not 11, 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.
Sequences of the type 13*n + k, for k = 0..12, without squares and cubes:
k =  2: A153080,
k =  6: A186113,
k =  7: A269044,
k = 11: this case.
The sum of the sixth powers of any two terms of the sequence is also a term of the sequence. Example: a(3)^6 + a(8)^6 = a(179129674278) = 2328685765625.
The primes of the sequence are listed in A140373.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Subsequence of A094784, A106389.
Cf. A140373.
Similar sequences of the type k*n+k-2: A023443 (k=1), A005843 (k=2), A016777 (k=3), A016825 (k=4), A016885 (k=5), A016957 (k=6), A017041 (k=7), A017137 (k=8), A017245 (k=9), A017365 (k=10), A017497 (k=11), A017641 (k=12).
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), A269044 (q=7), this sequence (q=11).

Magma
```
[13*n+11: n in [0..60]];
```

13 Range[0,60] + 11
Range[11, 800, 13]
Table[13 n + 11, {n, 0, 60}] (* Bruno Berselli, Feb 22 2016 *)
LinearRecurrence[{2,-1},{11,24},60] (* Harvey P. Dale, Jun 14 2023 *)

Maxima
```
makelist(13*n+11, n, 0, 60);
```
PARI
```
vector(60, n, n--; 13*n+11)
```
Python
```
[13*n+11 for n in range(61)]
```
Sage
```
[13*n+11 for n in range(61)]
```

G.f.: (11 + 2*x)/(1 - x)^2.
a(n) = -A153080(-n-1).
Sum_{i = h..h+13*k} a(i) = a(h*(13*k + 1) + k*(169*k + 35)/2).
Sum_{i >= 0} 1/a(i)^2 = .012486605016510955990... = polygamma(1, 11/13)/13^2.
E.g.f.: (11 + 13*x)*exp(x). - G. C. Greubel, May 31 2024

A186113 a(n) = 13*n + 6.

Original entry on oeis.org

6, 19, 32, 45, 58, 71, 84, 97, 110, 123, 136, 149, 162, 175, 188, 201, 214, 227, 240, 253, 266, 279, 292, 305, 318, 331, 344, 357, 370, 383, 396, 409, 422, 435, 448, 461, 474, 487, 500, 513, 526, 539, 552, 565, 578, 591, 604, 617, 630, 643, 656, 669, 682

Offset: 0

Omar E. Pol, Feb 12 2011

These numbers appear in the G. E. Andrews paper, for example: see the abstract, formula (1.7), etc. Also "13n + 6" appears in the Folsom-Ono paper (see links).
Row 6 of triangle A151890 lists the first seven terms of this sequence.
Any square mod 13 is one of 0, 1, 3, 4, 9, 10 or 12 (A010376) but not 6, 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. - Bruno Berselli, Feb 19 2016

G. C. Greubel, Table of n, a(n) for n = 0..5000
G. E. Andrews, The number of smallest parts in the partitions of n, J. Reine Angew. Math. 624 (2008) 133
Amanda Folsom and Ken Ono, The spt-function of Andrews, Proc. Natl. Acad. Sci. USA 105 (51) (2008) 20152-20156.
Ken Ono, Congruences for the Andrews spt-function, Proc. Natl. Acad. Sci. USA 108 (2011) 473-476.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A092269, A151890.
Sequences of the form 13*n+q: A008595 (q=0), A190991 (q=1), A153080 (q=2),
A127547 (q=4), A154609 (q=5), this sequence (q=6), A269044 (q=7), A269100 (q=11).

[13*n+6: n in [0..60]]; // G. C. Greubel, May 31 2024

Range[6, 1000, 13] (* Vladimir Joseph Stephan Orlovsky, May 31 2011 *)
LinearRecurrence[{2,-1},{6,19},60] (* Harvey P. Dale, May 12 2023 *)

[13*n+6 for n in range(61)] # G. C. Greubel, May 31 2024

G.f.: (6+7*x)/(1-x)^2.

E.g.f.: (6 + 13*x)*exp(x). - G. C. Greubel, May 31 2024

A092464 Numbers congruent to 4 or 9 mod 13.

Original entry on oeis.org

4, 9, 17, 22, 30, 35, 43, 48, 56, 61, 69, 74, 82, 87, 95, 100, 108, 113, 121, 126, 134, 139, 147, 152, 160, 165, 173, 178, 186, 191, 199, 204, 212, 217, 225, 230, 238, 243, 251, 256, 264, 269, 277, 282, 290, 295, 303, 308, 316, 321, 329, 334, 342, 347, 355, 360

Offset: 1

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Mar 25 2004

Numbers k such that k^2 is congruent to 3 (modulo 13).

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

A127547 is a subsequence.

Select[Range[400],MemberQ[{4,9},Mod[#,13]]&] (* or *) Select[Range[400], PowerMod[#,2,13]==3&] (* Harvey P. Dale, Mar 05 2012 *)

From R. J. Mathar, Apr 20 2009: (Start)
a(n) = a(n-2) + 13 = a(n-1) + a(n-2) - a(n-3) = 13*n/2 - 13/4 - 3*(-1)^n/4.
G.f.: x*(4+5*x+4*x^2)/((1+x)*(x-1)^2). (End)
a(n) = 13*(n-1) - a(n-1), (with a(1)=4). - Vincenzo Librandi, Nov 17 2010
Sum_{n>=1} (-1)^(n+1)/a(n) = tan(5*Pi/26)*Pi/13. - Amiram Eldar, Feb 27 2023
From Amiram Eldar, Nov 25 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = 2*sin(5*Pi/26).
Product_{n>=1} (1 + (-1)^n/a(n)) = sin(3*Pi/13)*sec(5*Pi/26). (End)

More terms from Ray Chandler, Mar 27 2004
Edited by N. J. A. Sloane, May 10 2007
Incorrect formula deleted by N. J. A. Sloane, Jun 16 2010

cp's OEIS Frontend

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

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A154609 a(n) = 13*n + 5.

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A269100 a(n) = 13*n + 11.

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A186113 a(n) = 13*n + 6.

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A092464 Numbers congruent to 4 or 9 mod 13.

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