A247541 a(n) = 7*n^2 + 1.

1, 8, 29, 64, 113, 176, 253, 344, 449, 568, 701, 848, 1009, 1184, 1373, 1576, 1793, 2024, 2269, 2528, 2801, 3088, 3389, 3704, 4033, 4376, 4733, 5104, 5489, 5888, 6301, 6728, 7169, 7624, 8093, 8576, 9073, 9584, 10109, 10648, 11201, 11768, 12349, 12944, 13553

Offset: 0

Author

Karl V. Keller, Jr., Sep 18 2014

Keywords

• nonn
• easy

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A201602 (primes of the form 7n^2 + 1).

Programs

Magma

[7*n^2+1: n in [0..50]]; // Vincenzo Librandi, Sep 19 2014

Mathematica

a247541[n_Integer] := 7 n^2 + 1; a247541 /@ Range[0, 120] (* Michael De Vlieger, Sep 18 2014 *)
CoefficientList[Series[(1 + 5 x + 8 x^2)/(1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Sep 19 2014 *)
LinearRecurrence[{3,-3,1},{1,8,29},50] (* Harvey P. Dale, Jun 09 2015 *)

PARI

vector(100,n,7*(n-1)^2+1) \\ Derek Orr, Sep 18 2014

Python

for n in range (0,500) : print (7*n**2+1)

Formula

G.f.: (1 + 5*x + 8*x^2)/(1 - x)^3. - Vincenzo Librandi, Sep 19 2014

From Amiram Eldar, Jul 15 2020: (Start)

Sum_{n>=0} 1/a(n) = (1 + (Pi/sqrt(7))*coth(Pi/sqrt(7)))/2.

Sum_{n>=0} (-1)^n/a(n) = (1 + (Pi/sqrt(7))*csch(Pi/sqrt(7)))/2. (End)

From Amiram Eldar, Feb 05 2021: (Start)

Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi/sqrt(7))*sinh(sqrt(2/7)*Pi).

Product_{n>=1} (1 - 1/a(n)) = (Pi/sqrt(7))*csch(Pi/sqrt(7)). (End)

E.g.f.: exp(x)*(1 + 7*x + 7*x^2). - Stefano Spezia, Feb 05 2021