%I #86 Oct 31 2024 16:26:49

%S 5,13,29,53,85,125,173,229,293,365,445,533,629,733,845,965,1093,1229,

%T 1373,1525,1685,1853,2029,2213,2405,2605,2813,3029,3253,3485,3725,

%U 3973,4229,4493,4765,5045,5333,5629,5933,6245,6565,6893,7229,7573,7925,8285,8653,9029

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

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

%C 1/5 + 1/13 + 1/29 + ... = (Pi/8)*tanh Pi [Jolley]. - _Gary W. Adamson_, Dec 21 2006

%C 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

%C (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.

%C Primes in the sequence are in A005473. - _Russ Cox_, Aug 26 2019

%C 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

%C Discriminant of the binary quadratic forms y^2 - x*y - A002061(n+1)*x^2. - _Klaus Purath_, Nov 10 2022

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

%H Indranil Ghosh, <a href="/A078370/b078370.txt">Table of n, a(n) for n = 0..1000</a>

%H Leo Tavares, <a href="/A078370/a078370.jpg">Square illustration</a>.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = (2*n + 1)^2 + 4.

%F 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]

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

%F a(n) = 8*n + a(n-1), with a(0) = 5. - _Vincenzo Librandi_, Aug 08 2010

%F a(n) = A016754(n) + 4. - _Leo Tavares_, Feb 22 2023

%F From _Elmo R. Oliveira_, Oct 31 2024: (Start)

%F E.g.f.: (5 + 8*x + 4*x^2)*exp(x).

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

%t Table[4 n (n + 1) + 5, {n, 0, 45}] (* or *)

%t Table[8 Binomial[n + 1, 2] + 5, {n, 0, 45}] (* or *)

%t CoefficientList[Series[(5 - 2 x + 5 x^2)/(1 - x)^3, {x, 0, 45}], x] (* _Michael De Vlieger_, Jan 04 2017 *)

%o (PARI) a(n)=4*n^2+4*n+5 \\ _Charles R Greathouse IV_, Sep 24 2015

%o (Python) a= lambda n: 4*n**2+4*n+5 # _Indranil Ghosh_, Jan 04 2017

%o (Scala) (1 to 99 by 2).map(n => n * n + 4) // _Alonso del Arte_, May 29 2019

%o (Magma) [4*n^2+4*n+5 : n in [0..80]]; // _Wesley Ivan Hurt_, Aug 29 2022

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

%Y Cf. A002061, A002522, A004770, A005473, A016754.

%Y Cf. A054413, A078356, A078357, A086902, A087475.

%K nonn,easy

%O 0,1

%A _Wolfdieter Lang_, Nov 29 2002

%E More terms from _Max Alekseyev_, Mar 03 2010