A106389 -id:A106389 - cp's OEIS frontend

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

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

A106388 Numbers k such that 11k = 6j^2 + 6j + 1.

Original entry on oeis.org

11, 23, 131, 167, 383, 443, 767, 851, 1283, 1391, 1931, 2063, 2711, 2867, 3623, 3803, 4667, 4871, 5843, 6071, 7151, 7403, 8591, 8867, 10163, 10463, 11867, 12191, 13703, 14051, 15671, 16043, 17771, 18167, 20003, 20423, 22367, 22811, 24863, 25331, 27491, 27983

Offset: 1

Pierre CAMI, May 01 2005

j sequence = A106387

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A106387, A106389, A106390.

LinearRecurrence[{1,2,-2,-1,1},{11,23,131,167,383},50] (* Harvey P. Dale, Jul 26 2018 *)

Vec((11+12*x+86*x^2+12*x^3+11*x^4)/(1+x)^2/(1-x)^3+O(x^99)) \\ Charles R Greathouse IV, Dec 28 2011

a(1)=11, a(2)=23; if n odd a(n)=a(n-1)+54*(n-1), if n even a(n)=a(n-1)+12*(n-1).
a(n) = (66*n*(n-1)-21*(2*n-1)*(-1)^n+23)/4.
G.f.:  x*(11+12*x+86*x^2+12*x^3+11*x^4)/((1+x)^2*(1-x)^3).
a(n)-a(n-1)-2*a(n-2)+2*a(n-3)+a(n-4)-a(n-5) = 0 for n>5.

Formulae corrected and added by Bruno Berselli, Nov 16 2010

More terms from Colin Barker, Apr 16 2014

A106387 Numbers j such that 6j^2 + 6j + 1 = 11k.

Original entry on oeis.org

4, 6, 15, 17, 26, 28, 37, 39, 48, 50, 59, 61, 70, 72, 81, 83, 92, 94, 103, 105, 114, 116, 125, 127, 136, 138, 147, 149, 158, 160, 169, 171, 180, 182, 191, 193, 202, 204, 213, 215, 224, 226, 235, 237, 246, 248, 257, 259, 268, 270, 279, 281, 290, 292, 301, 303

Offset: 1

Pierre CAMI, May 01 2005

k sequence = A106388.

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

Cf. A106388, A106389, A106390.

Select[Range[320],Divisible[6#^2+6#+1,11]&] (* Harvey P. Dale, Sep 10 2011 *)

Vec((4+2*x+5*x^2)/(1+x)/(1-x)^2+O(x^99)) \\ Charles R Greathouse IV, Dec 28 2011

j(1)=4, j(2)=6 then j(n)=j(n-2)+11.
a(n) = 11*n - a(n-1) - 12 (with a(1)=4). - Vincenzo Librandi, Nov 13 2010
a(2k-1) = 11k - 7, a(2k) = 11k - 5. - Ralf Stephan, Nov 15 2010
From Bruno Berselli, Nov 16 2010: (Start)
a(n) = (22*n - 7*(-1)^n - 13)/4.
G.f.: x*(4+2*x+5*x^2)/((1+x)*(1-x)^2).
a(n) - a(n-1) - a(n-2) + a(n-3) = 0 for n > 3.
a(n) - a(n-2) = 11 for n > 2.
a(n) - 2*a(n-1) + a(n-2) = -7*(-1)^n for n > 2. (End)

A106390 Numbers k such that 13k = 6j^2 + 6j + 1.

Original entry on oeis.org

1, 61, 97, 277, 349, 649, 757, 1177, 1321, 1861, 2041, 2701, 2917, 3697, 3949, 4849, 5137, 6157, 6481, 7621, 7981, 9241, 9637, 11017, 11449, 12949, 13417, 15037, 15541, 17281, 17821, 19681, 20257, 22237, 22849, 24949, 25597, 27817, 28501, 30841

Offset: 1

Pierre CAMI, May 01 2005

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A106387, A106388, A106389.

For j sequence see A106389.

f[n_] := Block[{k = (6n(n + 1) + 1)/13}, If[ IntegerQ[k], k, 1]]; Union[ Table[ f[n], {n, 270}]] (* Robert G. Wilson v, May 02 2005 *)

Vec(-x*(x^4+60*x^3+34*x^2+60*x+1)/((x-1)^3*(x+1)^2) + O(x^100)) \\ Colin Barker, Apr 16 2014

a(1)=1, a(2)=61; for odd n a(n) = a(n-1)+18*(n-1), for even n a(n) = a(n-1)+60*(n-1).
a(n) = (25-21*(-1)^n+6*(-13+7*(-1)^n)*n+78*n^2)/4. - Colin Barker, Apr 16 2014
G.f.: -x*(x^4+60*x^3+34*x^2+60*x+1) / ((x-1)^3*(x+1)^2). - Colin Barker, Apr 16 2014

More terms from Robert G. Wilson v, May 02 2005

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Maxima

PARI

Python

Sage

Formula

A106388 Numbers k such that 11k = 6j^2 + 6j + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A106387 Numbers j such that 6j^2 + 6j + 1 = 11k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A106390 Numbers k such that 13k = 6j^2 + 6j + 1.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions