A269100 -id:A269100 - 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

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

A111369 Numbers k such that 13*k + 11 is prime.

Original entry on oeis.org

0, 2, 6, 12, 14, 20, 26, 30, 36, 42, 50, 56, 72, 80, 84, 86, 90, 96, 114, 120, 122, 134, 140, 152, 156, 164, 170, 174, 180, 182, 204, 206, 210, 212, 216, 222, 230, 236, 246, 254, 260, 266, 272, 282, 294, 300, 306, 314, 332, 342, 344, 350, 356, 360, 380, 384, 390

Offset: 1

Parthasarathy Nambi, Nov 08 2005

			k=156 is a term because 13*k + 11 = 2039 is prime.

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A102721, A269100.

[n: n in [0..100000] |IsPrime(13*n+11)] // Vincenzo Librandi, Nov 13 2010

Select[Range[0,400,2],PrimeQ[13#+11]&] (* Harvey P. Dale, Jun 17 2020 *)

is(n)=isprime(13*n+11) \\ Charles R Greathouse IV, Jun 12 2017

A319524 a(n) is the smallest number that belongs simultaneously to the two arithmetic progressions prime(n) + mprime(n+1) and prime(n+1) + mprime(n+2), m >= 1, n >= 1.

Original entry on oeis.org

8, 33, 40, 128, 115, 302, 226, 226, 835, 401, 734, 1718, 1030, 842, 3121, 3475, 1401, 2339, 5108, 1969, 3233, 2486, 6491, 9692, 10298, 5560, 11552, 6211, 4177, 7987, 6022, 18763, 16678, 21893, 8001, 25585, 13523, 9682, 30961, 32035, 7057, 36089, 19105, 39002, 7162, 47041, 50163, 51752

Offset: 1

Alexandra Hercilia Pereira Silva, Sep 22 2018

Construct a table T in which T(m,n) = prime(n) + m*prime(n+1) as shown below. Then a(n) is defined as the smallest number appearing both in column n and column n+1, so a(1)=8, a(2)=33, a(3)=40, etc.
.
   m\n|  1     2     3     4     5     6     7     8  ...
  ----+--------------------------------------------------
    1 |  5   --8    12    18    24    30    36    42  ...
      |
    2 |  8--  13    19    29    37    47    55    65  ...
      |
    3 | 11    18    26    40    50    64    74    88  ...
      |                  /
    4 | 14    23    33  / 51    63    81    93   111  ...
      |            /   /
    5 | 17    28  / 40-   62    76    98   112   134  ...
      |          /
    6 | 20    33-   47    73    89   115   131   157  ...
      |                             /
    7 | 23    38    54    84   102 / 132   150   180  ...
      |                           /
    8 | 26    43    61    95   115   149   169   203  ...
      |
    9 | 29    48    68   106   128   166   188   226  ...
      |                       /                 /
   10 | 32    53    75   117 / 141   183   207 / 249  ...
      |                     /                 /
   11 | 35    58    82   128   154   200   226   272  ...
      |
   12 | 38    63    89   139   167   217   245   295  ...
      |
   13 | 41    68    96   150   180   234   264   318  ...
      |
   14 | 44    73   103   161   193   251   283   341  ...
      |
   15 | 47    78   110   172   206   268   302   364  ...
      |                                   /
   16 | 50    83   117   183   219   285 / 321   387  ...
      |                                 /
   17 | 53    88   124   194   232   302   340   410  ...
      |
  ... |...   ...   ...   ...   ...   ...   ...   ...  ...
Conjectures:
1. There are infinitely many pairs of consecutive equal terms. (Note that the first pair is (a(7), a(8)).)
2. There exists no N such that the sequence is monotonic for n > N.
From Amiram Eldar, Sep 22 2018: (Start)
Theorem 1: The intersection of the two mentioned arithmetic progressions is always nonempty.
Corollary: The sequence is infinite. (End)
Sequences that derive from this:
1. Positions in {s(n)} at which a(n) occurs: (2,6,5,11,8,17,19,...).
2. Positions in {s(n+1)} at which a(n) occurs: (1,4,3,9,6,15,15,...).
3. Differences between these two sequences: (1,2,2,2,2,4,...).

Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 600 terms from Muniru A Asiru)
Fourth International contest of logical problems, Problem 7, the Ludomind Society.
Fifth International contest of logical problems, Problem 6, the Ludomind Society, 2009.
Olivier Gérard, in reply to Zak Seidov, 11 related sequences, SeqFan list, Apr 14 2016.

Cf. A001043, A016789, A016885, A017041, A017473, A269100.

P:=Filtered([1..10000],IsPrime);;
T:=List([1..Length(P)-1],n->List([1..Length(P)-1],m->P[n]+m*P[n+1]));;
a:=List([1..50],k->Minimum(List([1..Length(T)-1],i->Intersection(T[i],T[i+1]))[k])); # Muniru A Asiru, Sep 26 2018

a[n_]:=ChineseRemainder[{Prime[n],Prime[n+1]},{Prime[n+1],Prime[n+2]} ];Array[a,44] (* Amiram Eldar, Sep 22 2018 *)

Table from Jon E. Schoenfield, Sep 23 2018

More terms from Amiram Eldar, Sep 22 2018

A278126 a(n) = 78*n + 66.

Original entry on oeis.org

66, 144, 222, 300, 378, 456, 534, 612, 690, 768, 846, 924, 1002, 1080, 1158, 1236, 1314, 1392, 1470, 1548, 1626, 1704, 1782, 1860, 1938, 2016, 2094, 2172, 2250, 2328, 2406, 2484, 2562, 2640, 2718, 2796, 2874, 2952, 3030, 3108, 3186, 3264, 3342, 3420, 3498, 3576

Offset: 0

Emeric Deutsch, Nov 13 2016

a(n) (n>=1) is the first Zagreb index of the triple-layered naphthalenophane G(n,n,n) having n hexagons in each layer. The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternately, it is the sum of the degree sums d(i)+d(j) over all edges ij of the graph. The pictorial definition of G(p,q,r) can be viewed in the E. Flapan references.

The M-polynomial of the triple layered naphthalenophane G(p,q,r) is M(G(p,q,r),x,y) = 8*x^2*y^2 + 4*(p + q + r + 2)*x^2*y^3 + (p + q + r - 1)*x^3*y^3 (p, q, r>=1).

Erica Flapan, When Topology Meets Chemistry, Cambridge Univ. Press, Cambridge, 2000.

E. Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.
Erica Flapan and Brian Forcum, Intrinsic chirality of triple-layered naphthalenophane and related graphs, J. Math. Chemistry, 24, 1998, 379-388.
I. Gutman and K. C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50, 2004, 83-92.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A269100, A278127.

Maple
```
seq(78*n+66, n = 0..45);
```

78*Range[0,50]+66 (* or *) LinearRecurrence[{2,-1},{66,144},50] (* Harvey P. Dale, Jul 27 2025 *)

G.f.: 6*(11 + 2*x)/(1 - x)^2.

a(n) = 6*A269100(n).

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

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

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A111369 Numbers k such that 13*k + 11 is prime.

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A319524 a(n) is the smallest number that belongs simultaneously to the two arithmetic progressions prime(n) + m*prime(n+1) and prime(n+1) + m*prime(n+2), m >= 1, n >= 1.

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A319524 a(n) is the smallest number that belongs simultaneously to the two arithmetic progressions prime(n) + mprime(n+1) and prime(n+1) + mprime(n+2), m >= 1, n >= 1.