A010376 -id:A010376 - 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

A153080 a(n) = 13*n + 2.

Original entry on oeis.org

2, 15, 28, 41, 54, 67, 80, 93, 106, 119, 132, 145, 158, 171, 184, 197, 210, 223, 236, 249, 262, 275, 288, 301, 314, 327, 340, 353, 366, 379, 392, 405, 418, 431, 444, 457, 470, 483, 496, 509, 522, 535, 548, 561, 574, 587, 600, 613, 626, 639, 652, 665, 678, 691

Offset: 0

Vincenzo Librandi, Feb 10 2009

Any square mod 13 is one of 0, 1, 3, 4, 9, 10 or 12 (A010376) but not 2, 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 term is a cube. - Bruno Berselli, Feb 19 2016

Numbers k such that GCD(2*k^5+1, 3*k^3+2) > 1. This GCD is 13 if k == 2 (mod 13), or 1 otherwise. - Philippe Deléham, Jan 16 2024

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

Cf. A008595, A010376, A141858, A190991.

Cf. A269100. - Bruno Berselli, Feb 22 2016

I:=[2, 15]; [n le 2 select I[n] else 2*Self(n-1)-1*Self(n-2): n in [1..60]]; // Vincenzo Librandi, Feb 25 2012

A153080:=n->13*n+2: seq(A153080(n), n=0..100); # Wesley Ivan Hurt, Oct 05 2017

Range[2, 1000, 13] (* Vladimir Joseph Stephan Orlovsky, May 29 2011 *)
LinearRecurrence[{2,-1},{2,15},30] (* Vincenzo Librandi, Feb 25 2012 *)

G.f.: (2+11*x)/(1-x)^2. - R. J. Mathar, Jan 05 2011
a(n) = 2*a(n-1) - a(n-2). - Vincenzo Librandi, Feb 25 2012
E.g.f.: exp(x)*(2 + 13*x). - Elmo R. Oliveira, Apr 04 2025

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

A035256 Positive integers of the form x^2+3xy-y^2.

Original entry on oeis.org

1, 3, 4, 9, 12, 13, 16, 17, 23, 25, 27, 29, 36, 39, 43, 48, 49, 51, 52, 53, 61, 64, 68, 69, 75, 79, 81, 87, 92, 100, 101, 103, 107, 108, 113, 116, 117, 121, 127, 129, 131, 139, 144, 147, 153, 156, 157, 159, 169, 172, 173, 179, 181, 183, 191, 192, 196, 199, 204, 207, 208, 211, 212, 221, 225, 233, 237, 243, 244, 251

Offset: 1

N. J. A. Sloane

nonn

This is an indefinite quadratic form of discriminant 13.
Also, positive integers of the form x^2+6xy-4y^2 (an indefinite quadratic form of discriminant 52).
Also, indices of the nonzero terms in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m= 13.
From Klaus Purath, May 07 2023: (Start)
Also, positive integers of the form x^2 + (2m+1)xy + (m^2+m-3)y^2, m, x, y integers. This includes the form in the name.
Also, positive integers of the form x^2 + 2mxy + (m^2-13)y^2, m, x, y integers. This includes the form in the comment above.
This sequence contains all squares. The prime factors of the terms except for {2, 5, 7, 11, 19, ...} = A038884 are terms of the sequence. Also the products of terms belong to the sequence. Thus this set of terms is closed under multiplication.
A positive integer N belongs to the sequence if and only if N (modulo 13) is a term of A010376 and, moreover, in the case that prime factors p of N are terms of A038884, they occur only with even exponents. For these prime factors also p (modulo 13) = {2, 5, 6, 7, 8, 11} applies. (End)

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Primes in this sequence = A038883 and A141188.

Cf. A035195.

formQ[n_] := Reduce[a > 0 && b > 0 && n == a^2 + 3 a*b - b^2, {a, b}, Integers] =!= False; Select[Range[100], formQ] (* Wesley Ivan Hurt, Jun 18 2014 *)

m=13; select(x -> x, direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X)), 1) \\ Fixed by Andrey Zabolotskiy, Jul 30 2020

Entry revised by N. J. A. Sloane, Jun 01 2014

A028726 Nonsquares mod 13.

Original entry on oeis.org

2, 5, 6, 7, 8, 11

Offset: 1

N. J. A. Sloane

			Since 2 is not a perfect square and there are no solutions to x^2 = 2 mod 13, 2 is in the sequence.
Although 3 is not a perfect square either, there are solutions to x^2 = 3 mod 13, such as x = 4, x = 9. Hence 3 is not in the sequence.

Cf. A010376.

Select[Range[0, 12], JacobiSymbol[#, 13] == -1 &] (* Alonso del Arte, Dec 13 2019 *)

(0 to 12).diff((1 to 13).map(n => (n * n) % 13)) // Alonso del Arte, Dec 13 2019

cp's OEIS Frontend

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

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A153080 a(n) = 13*n + 2.

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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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Magma

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A035256 Positive integers of the form x^2+3xy-y^2.

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Mathematica

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Extensions

A028726 Nonsquares mod 13.

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