

A078370


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


46



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.


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


FORMULA

a(n) = 4*(n+1)*n+5 = 8*binomial(n+1, 2)+5, hence subsequence of A004770 (5 (mod 8) numbers).
G.f.: (52*x+5*x^2)/(1x)^3.
a(n) = 8*n + a(n1) (with a(0)=5).  Vincenzo Librandi, Aug 08 2010


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


CROSSREFS

Subsequence of A077426 (D values (not a square) for which Pell x^2  D*y^2 = 4 is solvable in positive integers).
Sequence in context: A220500 A130230 A106931 * A247903 A240130 A005473
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
Typo in first formula fixed by Zak Seidov, Feb 26 2012


STATUS

approved



