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.
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) = (2*n + 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
From Elmo R. Oliveira, Oct 31 2024: (Start)
E.g.f.: (5 + 8*x + 4*x^2)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
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).
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Nov 29 2002
EXTENSIONS
More terms from Max Alekseyev, Mar 03 2010
STATUS
approved