A017497 -id:A017497 - 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

A380820 a(0) = 0, a(1) = 1, and for n >= 2, a(n) = a(n-1) + a(n-2) if a(n-1) < n, otherwise a(n-1) - n.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 1, 9, 0, 9, 9, 18, 5, 23, 8, 31, 14, 45, 26, 6, 32, 10, 42, 18, 60, 34, 7, 41, 12, 53, 22, 75, 42, 8, 50, 14, 64, 26, 90, 50, 9, 59, 16, 75, 30, 105, 58, 10, 68, 18, 86, 34, 120, 66, 11, 77, 20, 97, 38, 135, 74, 12, 86, 22, 108, 42, 150

Offset: 0

Ya-Ping Lu, Feb 04 2025

Sequence starts with the first 7 Fibonacci numbers. For n >= 12, a(n) takes the values of (8*n+30)/7, (n+22)/7, (9*n+35)/7, (2*n+26)/7, (11*n+41)/7, (4*n+30)/7, and (15*n+45)/7 sequentially for n = 5, 6, 0, 1, 2, 3, 4 mod 7 (see plot in Links), which correspond to A017089 (n>=2), A000027 (n>=5), A017221 (n>=2), A005843 (n>=4), A017497 (n>=2), A016825 (n>=3), and A008597 (n>=3), respectively.

Terms for n >= 16 are the same as A322558(n) for n >= 17.

Ya-Ping Lu, Plot of A380820
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,2,0,0,0,0,0,0,-1).

Cf. A000027, A000045, A005843, A008597, A016825, A017089, A017221, A017497, A322558.

s={0,1};Do[AppendTo[s,If[s[[-1]]James C. McMahon, Feb 14 2025 *)

def A380820(n): R = [0, 1, 1, 2, 3, 5, 8, 1, 9, 0, 9, 9]; X = [9, 2, 11, 4, 15, 8, 1]; Y = [35, 26, 41, 30, 45, 30, 22]; return R[n] if n < 12 else (X[n%7]*n + Y[n%7])//7

a(n) = A322558(n+1) for n >= 16.

A017507 a(n) = (11*n + 9)^11.

Original entry on oeis.org

31381059609, 204800000000000, 25408476896404831, 717368321110468608, 9269035929372191597, 73786976294838206464, 422351360321044921875, 1903193578437064103936, 7153014030880804126753, 23316389970546096340992, 67766737102405685929319, 179216039403700000000000

Offset: 0

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Powers of the form (11*n+9)^m: A017497 (m=1), A017498 (m=2), A017499 (m=3), A017500 (m=4), A017501 (m=5), A017502 (m=6), A017503 (m=7), A017504 (m=8), A017505 (m=9), A017506 (m=10), this sequence (m=11), A017508 (m=12).

List([0..20], n-> (11*n+9)^11); # G. C. Greubel, Oct 29 2019

[(11*n+9)^11: n in [0..20]]; // G. C. Greubel, Oct 29 2019

seq((11*n+9)^11, n=0..20); # G. C. Greubel, Oct 29 2019

(11*Range[20] -2)^11 (* G. C. Greubel, Oct 29 2019 *)

vector(21, n, (11*n-2)^11) \\ G. C. Greubel, Oct 29 2019

[(11*n+9)^11 for n in (0..20)] # G. C. Greubel, Oct 29 2019

From G. C. Greubel, Oct 29 2019: (Start)
G.f.: (31381059609 + 204423427284692*x + 22952948046339025*x^2 + 425976494520496656*x^3 +2292535084833793602*x^4 +4416340564654562280*x^5 +3258008937067466010*x^6 +892801175894641200*x^7 +78407266240574469*x^8 +1500175218575748*x^9 +1792160369461*x^10 +2048*x^11)/(1-x)^12
E.g.f.: (31381059609 + 204768618940391*x + 12499454138732220*x^2 + 106959543173365751*x^3 +272966430737075240*x^4 +286353492156140222*x^5 + 145413296465578218*x^6 +38972231675790897*x^7 +5719308232166595*x^8 + 455590863116565*x^9 +18259946919104*x^10 +285311670611*x^11)*exp(x). (End)

A299647 Positive solutions to x^2 == -2 (mod 11).

Original entry on oeis.org

3, 8, 14, 19, 25, 30, 36, 41, 47, 52, 58, 63, 69, 74, 80, 85, 91, 96, 102, 107, 113, 118, 124, 129, 135, 140, 146, 151, 157, 162, 168, 173, 179, 184, 190, 195, 201, 206, 212, 217, 223, 228, 234, 239, 245, 250, 256, 261, 267, 272, 278, 283, 289, 294, 300, 305, 311, 316

Offset: 1

Bruno Berselli, Mar 06 2018

Positive numbers congruent to {3, 8} mod 11.

Equivalently, interleaving of A017425 and A017485.

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

Subsequence of A106252, A279000.
Cf. A017425, A017485.
Cf. A017497: positive solutions to x == -2 (mod 11).
Cf. A017437: positive solutions to x^3 == -2 (mod 11).
Nonnegative solutions to x^2 == -2 (mod j): A005843 (j=2), A001651 (j=3), A047235 (j=6), A156638 (j=9), this sequence (j=11).

List([1..60], n -> 5*n-2+(2*n-(-1)^n-3)/4);

[(11(2n-1)-(-1)^n)>>2 for n in 1:60] # Peter Luschny, Mar 07 2018

[5*n-2+(2*n-(-1)^n-3)/4: n in [1..60]];

Table[5 n - 2 + (2 n - (-1)^n - 3)/4, {n, 1, 60}]
CoefficientList[ Series[(3 + 5x + 3x^2)/((x - 1)^2 (x + 1)), {x, 0, 57}], x] (* or *)
LinearRecurrence[{1, 1, -1}, {3, 8, 14}, 58] (* Robert G. Wilson v, Mar 08 2018 *)

makelist(5*n-2+(2*n-(-1)^n-3)/4, n, 1, 60);

vector(60, n, nn; 5*n-2+(2*n-(-1)^n-3)/4)

[5*n-2+(2*n-(-1)**n-3)/4 for n in range(1, 60)]

[5*n-2+(2*n-(-1)^n-3)/4 for n in (1..60)]

O.g.f.: x*(3 + 5*x + 3*x^2)/((1 + x)*(1 - x)^2).
E.g.f.: (-1 + 12*exp(x) - 11*exp(2*x) + 22*x*exp(2*x))*exp(-x)/4.
a(n) = -a(-n+1) = a(n-1) + a(n-2) - a(n-3).
a(n) = 5*n - 2 + (2*n - (-1)^n - 3)/4.
a(n) = 4*n - 1 + floor((n - 1)/2) + floor((3*n - 1)/3).
a(n+k) - a(n) = 11*k/2 + (1 - (-1)^k)*(-1)^n/4.
a(n+k) + a(n) = 11*(2*n + k - 1)/2 - (1 + (-1)^k)*(-1)^n/4.
E.g.f.: 3 + ((22*x - 11)*exp(x) - exp(-x))/4. - David Lovler, Aug 08 2022
Sum_{n>=1} (-1)^(n+1)/a(n) = tan(5*Pi/22)*Pi/11. - Amiram Eldar, Feb 27 2023
From Amiram Eldar, Nov 23 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = cosec(3*Pi/22)/2.
Product_{n>=1} (1 + (-1)^n/a(n)) = sec(5*Pi/22)*sin(2*Pi/11). (End)

cp's OEIS Frontend

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

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A380820 a(0) = 0, a(1) = 1, and for n >= 2, a(n) = a(n-1) + a(n-2) if a(n-1) < n, otherwise a(n-1) - n.

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A017507 a(n) = (11*n + 9)^11.

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A299647 Positive solutions to x^2 == -2 (mod 11).

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