

A078370


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


53



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
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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



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.


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



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



EXTENSIONS



STATUS

approved



