A154609 -id:A154609 - 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

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

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

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

A127547 a(n) = 13*n + 4.

Original entry on oeis.org

4, 17, 30, 43, 56, 69, 82, 95, 108, 121, 134, 147, 160, 173, 186, 199, 212, 225, 238, 251, 264, 277, 290, 303, 316, 329, 342, 355, 368, 381, 394, 407, 420, 433, 446, 459, 472, 485, 498, 511, 524, 537, 550, 563, 576, 589, 602, 615, 628, 641, 654, 667, 680, 693, 706, 719

Offset: 0

Robert H Barbour, Apr 01 2007

Superhighway created by 'LQTL Ant' L90R90L45R45 from iteration 4 where the Ant moves in a 'Moore neighborhood' (nine cells), the L indicates a left turn, the R a right turn, and the numerical value is the size of the turn (in degrees) at each iteration.

Ant Farm algorithm available from Robert H Barbour.

P. Sakar, "A Brief History of Cellular Automata," ACM Computing Surveys, vol. 32, pp. 80-107, 2000.

G. C. Greubel, Table of n, a(n) for n = 0..5000
C. Langton, Studying Artificial Life with Cellular Automata, Physica D: Nonlinear Phenomena, vol. 22, pp. 120-149, 1986.
James Propp, Further Ant-ics, Mathematical Intelligencer, 16 pp. 37-42, 1994.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

A subsequence of A092464.
Cf. A126979, A126980.
Sequences of the form 13*n+q: A008595 (q=0), A190991 (q=1), A153080 (q=2), this sequence (q=4), A154609 (q=5), A186113 (q=6), A269044 (q=7), A269100 (q=11).

[13*n+4: n in [0..60]]; // G. C. Greubel, May 31 2024

Range[4,1000,13] (* Vladimir Joseph Stephan Orlovsky, May 31 2011 *)

[13*n+4 for n in range(61)] # G. C. Greubel, May 31 2024

From Elmo R. Oliveira, Mar 21 2024: (Start)
G.f.: (4+9*x)/(1-x)^2.
E.g.f.: (4 + 13*x)*exp(x).
a(n) = 2*a(n-1) - a(n-2) for n >= 2. (End)

Edited by N. J. A. Sloane, May 10 2007

A155086 Numbers k such that k^2 == -1 (mod 13).

Original entry on oeis.org

5, 8, 18, 21, 31, 34, 44, 47, 57, 60, 70, 73, 83, 86, 96, 99, 109, 112, 122, 125, 135, 138, 148, 151, 161, 164, 174, 177, 187, 190, 200, 203, 213, 216, 226, 229, 239, 242, 252, 255, 265, 268, 278, 281, 291, 294, 304, 307, 317, 320, 330, 333, 343, 346, 356, 359

Offset: 1

Vincenzo Librandi, Jan 20 2009

Numbers k such that k == 5 or 8 mod 13. - Charles R Greathouse IV, Dec 28 2011

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

Cf. A002144, A047221 (m=5), A155095 (m=17), A156619 (m=25), A155096 (m=29), A155097 (m=37), A155098 (m=41), A154609 (bisection).

I:=[5, 8, 18]; [n le 3 select I[n] else Self(n-1)+Self(n-2)-Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 26 2012

LinearRecurrence[{1, 1, -1}, {5, 8, 18}, 50] (* Vincenzo Librandi, Feb 26 2012 *)
Select[Range[1000], PowerMod[#, 2, 13] == 12 &] (* Vincenzo Librandi, Apr 24 2014 *)

a(n) = a(n-1)+a(n-2)-a(n-3).
G.f.: x*(5+3*x+5*x^2)/((1+x)*(x-1)^2) .
a(n) = 13*(n-1/2)/2 -7*(-1)^n/4.
a(n) = a(n-2)+13. - M. F. Hasler, Jun 16 2010
Sum_{n>=1} (-1)^(n+1)/a(n) = tan(3*Pi/26)*Pi/13. - Amiram Eldar, Feb 27 2023
From Amiram Eldar, Nov 25 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = cos(Pi/26)*sec(3*Pi/26) = 1/(2*cos(Pi/13)-1).
Product_{n>=1} (1 + (-1)^n/a(n)) = cosec(5*Pi/26)/2. (End)

Algebra simplified by R. J. Mathar, Aug 18 2009

Edited by N. J. A. Sloane, Jun 23 2010

A156640 a(n) = 169n^2 + 140n + 29.

Original entry on oeis.org

29, 338, 985, 1970, 3293, 4954, 6953, 9290, 11965, 14978, 18329, 22018, 26045, 30410, 35113, 40154, 45533, 51250, 57305, 63698, 70429, 77498, 84905, 92650, 100733, 109154, 117913, 127010, 136445, 146218, 156329, 166778

Offset: 0

Vincenzo Librandi, Feb 15 2009

The identity (57122*n^2 +47320*n +9801)^2 - (169*n^2 +140*n +29)*(4394*n +1820)^2 = 1 can be written as A156735(n)^2 - a(n)*A156636(n)^2 = 1.
The continued fraction expansion of sqrt(a(n)) is [13n+5; {2, 1, 1, 2, 26n+10}]. - Magus K. Chu, Sep 15 2022
From Klaus Purath, Apr 06 2025: (Start)
a(n)*13^2-1 is a square, and a(n) is the sum of two squares (see FORMULA). There are no squares in this sequence. The odd prime factors of these terms are always of the form 4*k + 1.
All a(n) = D satisfy the Pell equation (k*x)^2 - D*(13*y)^2 = -1 for any integer n where a(1-n) = A156639(n). The values for k and the solutions x, y can be calculated using the following algorithm: k = sqrt(D*13^2 - 1), x(0) = 1, x(1) = 4*D*13^2 - 1, y(0) = 1, y(1) = 4*D*13^2 - 3. The two recurrences are of the form (4*D*13^2 - 2, -1).
It follows from the above that this sequence and A156639 belong to A031396. (End)

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

Cf. A156636, A156718, A156735.
Cf. A154609 (13n+5).
Subsequence of A031396.

I:=[29, 338, 985]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]];

A156640:= n-> 169*n^2 + 140*n + 29; seq(A156640(n), n=0..50); # G. C. Greubel, Feb 28 2021

LinearRecurrence[{3,-3,1},{29,338,985},50]
CoefficientList[Series[(29 +251x +58x^2)/(1-x)^3, {x, 0, 60}], x] (* Vincenzo Librandi, May 03 2014 *)

a(n)=169*n^2+140*n+29 \\ Charles R Greathouse IV, Dec 23 2011

[169*n^2 + 140*n + 29 for n in (0..50)] # G. C. Greubel, Feb 28 2021

a(n) = 3*a(n-1) -3*a(n-2) +a(n-3) for n>2.
G.f.: (29 + 251*x + 58*x^2)/(1-x)^3. - Vincenzo Librandi, May 03 2014
E.g.f.: (29 +309*x +169*x^2)*exp(x). - G. C. Greubel, Feb 28 2021
From Klaus Purath, Apr 06 2025: (Start)
a(n) = (5*n + 2)^2 + (12*n + 5)^2 for any integer n.
169*a(n) - 1 = (169*n + 70)^2 for any integer n. (End)

Edited by Charles R Greathouse IV, Jul 25 2010

A154610 Numbers n such that 13n + 5 is prime.

Original entry on oeis.org

0, 2, 6, 8, 18, 24, 32, 38, 44, 56, 62, 66, 72, 74, 78, 84, 86, 92, 98, 108, 114, 128, 132, 134, 144, 162, 164, 174, 176, 182, 186, 198, 204, 206, 224, 228, 246, 248, 254, 258, 266, 272, 276, 282, 284, 296, 302, 318, 324, 326, 336, 342, 368, 378, 386, 392, 396

Offset: 1

Vincenzo Librandi, Jan 15 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A102732, A154609.

[n: n in [0..410] | IsPrime(13*n+5)]; // Vincenzo Librandi, Sep 24 2012

lst={};Do[p=13*n+5;If[PrimeQ[p],AppendTo[lst,n]],{n,0,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 01 2009 *)
Select[Range[0, 100], PrimeQ[13#+5]&] (* Vincenzo Librandi, Sep 24 2012 *)

is(n)=isprime(13*n+5) \\ Charles R Greathouse IV, Sep 24 2012

More terms from Vladimir Joseph Stephan Orlovsky, Jul 01 2009

cp's OEIS Frontend

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

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

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

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

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A127547 a(n) = 13*n + 4.

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A155086 Numbers k such that k^2 == -1 (mod 13).

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A156640 a(n) = 169*n^2 + 140*n + 29.

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

A156640 a(n) = 169n^2 + 140n + 29.