A078370 a(n) = 4*(n+1)*n + 5.

5, 13, 29, 53, 85, 125, 173, 229, 293, 365, 445, 533, 629, 733, 845, 965, 1093, 1229, 1373, 1525, 1685, 1853, 2029, 2213, 2405, 2605, 2813, 3029, 3253, 3485, 3725, 3973, 4229, 4493, 4765, 5045, 5333, 5629, 5933, 6245, 6565, 6893, 7229, 7573, 7925, 8285

Offset

0,1

Comments

This is the generic form of D in the (nontrivially) solvable Pell equation x^2 - D*y^2 = -4. See A078356, A078357.

1/5 + 1/13 + 1/29 + ... = (Pi/8)*tanh Pi [Jolley]. - Gary W. Adamson, Dec 21 2006

Appears in A054413 and A086902 in relation to sequences related to the numerators and denominators of continued fractions convergents to sqrt((2*n+1)^2 + 4), n = 1, 2, 3, ... . - Johannes W. Meijer, Jun 12 2010

(2*n + 1 + sqrt(a(n)))/2 = [2*n + 1; 2*n + 1, 2*n + 1, ...], n >= 0, with the regular continued fraction with period length 1. This is the odd case. See A087475 for the general case with the Schroeder reference and comments. For the even case see A002522.

Primes in the sequence are in A005473. - Russ Cox, Aug 26 2019

The continued fraction expansion of sqrt(a(n)) is [2n+1; {n, 1, 1, n, 4n+2}]. For n=0, this collapses to [2; {4}]. - Magus K. Chu, Aug 27 2022

Discriminant of the binary quadratic forms y^2 - x*y - A002061(n+1)*x^2. - Klaus Purath, Nov 10 2022

References

L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 176.

Links

Indranil Ghosh, Table of n, a(n) for n = 0..1000

Leo Tavares, Square illustration

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

a(n) = (2n + 1)^2 + 4.

a(n) = 4*(n+1)*n + 5 = 8*binomial(n+1, 2) + 5, hence subsequence of A004770 (5 (mod 8) numbers). [Typo fixed by Zak Seidov, Feb 26 2012]

G.f.: (5 - 2*x + 5*x^2)/(1 - x)^3.

a(n) = 8*n + a(n-1), with a(0) = 5. - Vincenzo Librandi, Aug 08 2010

a(n) = A016754(n) + 4. - Leo Tavares, Feb 22 2023

Mathematica

Table[4 n (n + 1) + 5, {n, 0, 45}] (* or *)
Table[8 Binomial[n + 1, 2] + 5, {n, 0, 45}] (* or *)
CoefficientList[Series[(5 - 2 x + 5 x^2)/(1 - x)^3, {x, 0, 45}], x] (* Michael De Vlieger, Jan 04 2017 *)

Prog

(PARI) a(n)=4*n^2+4*n+5 \\ Charles R Greathouse IV, Sep 24 2015
(Python) a= lambda n: 4*n**2+4*n+5 # Indranil Ghosh, Jan 04 2017
(Scala) (1 to 99 by 2).map(n => n * n + 4) // Alonso del Arte, May 29 2019
(Magma) [4*n^2+4*n+5 : n in [0..80]]; // Wesley Ivan Hurt, Aug 29 2022

Crossrefs

Subsequence of A077426 (D values (not a square) for which Pell x^2 - D*y^2 = -4 is solvable in positive integers).

Cf. A005473.

Cf. A016754.

Sequence in context: A220500 A130230 A106931 * A308464 A247903 A350687

Adjacent sequences: A078367 A078368 A078369 * A078371 A078372 A078373

Keyword

nonn,easy

Author

Wolfdieter Lang, Nov 29 2002

Extensions

More terms from Max Alekseyev, Mar 03 2010

Status

approved