cp's OEIS Frontend

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

%I A078370 #89 Apr 12 2025 12:16:05
%S A078370 5,13,29,53,85,125,173,229,293,365,445,533,629,733,845,965,1093,1229,
%T A078370 1373,1525,1685,1853,2029,2213,2405,2605,2813,3029,3253,3485,3725,
%U A078370 3973,4229,4493,4765,5045,5333,5629,5933,6245,6565,6893,7229,7573,7925,8285,8653,9029
%N A078370 a(n) = 4*(n+1)*n + 5.
%C A078370 This is the generic form of D in the (nontrivially) solvable Pell equation x^2 - D*y^2 = -4. See A078356, A078357.
%C A078370 1/5 + 1/13 + 1/29 + ... = (Pi/8)*tanh Pi [Jolley]. - _Gary W. Adamson_, Dec 21 2006
%C A078370 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 A078370 (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 A078370 Primes in the sequence are in A005473. - _Russ Cox_, Aug 26 2019
%C A078370 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 A078370 Discriminant of the binary quadratic forms y^2  - x*y - A002061(n+1)*x^2. - _Klaus Purath_, Nov 10 2022
%C A078370 From _Klaus Purath_, Apr 08 2025: (Start)
%C A078370 There are no squares in this sequence. The prime factors of these terms are always of the form 4*k + 1.
%C A078370 All a(n) = D satisfy the Pell equation (k*x)^2 - D*(m*y)^2 = -1 for any integer n where m = (D - 3)/2. The values for k and the solutions x, y can be calculated using the following algorithm: k = sqrt(D*m^2 - 1), x(0) = 1, x(1) = 4*D*m^2 - 1, y(0) = 1, y(1) = 4*D*m^2 - 3. The two recurrences are of the form (4*D*m^2 - 2, -1).
%C A078370 It follows from the above that this sequence belongs to A031396. (End)
%D A078370 L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 176.
%H A078370 Indranil Ghosh, <a href="/A078370/b078370.txt">Table of n, a(n) for n = 0..1000</a>
%H A078370 Leo Tavares, <a href="/A078370/a078370.jpg">Square illustration</a>.
%H A078370 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A078370 a(n) = (2*n + 1)^2 + 4.
%F A078370 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 A078370 G.f.: (5 - 2*x + 5*x^2)/(1 - x)^3.
%F A078370 a(n) = 8*n + a(n-1), with a(0) = 5. - _Vincenzo Librandi_, Aug 08 2010
%F A078370 a(n) = A016754(n) + 4. - _Leo Tavares_, Feb 22 2023
%F A078370 From _Elmo R. Oliveira_, Oct 31 2024: (Start)
%F A078370 E.g.f.: (5 + 8*x + 4*x^2)*exp(x).
%F A078370 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
%t A078370 Table[4 n (n + 1) + 5, {n, 0, 45}] (* or *)
%t A078370 Table[8 Binomial[n + 1, 2] + 5, {n, 0, 45}] (* or *)
%t A078370 CoefficientList[Series[(5 - 2 x + 5 x^2)/(1 - x)^3, {x, 0, 45}], x] (* _Michael De Vlieger_, Jan 04 2017 *)
%o A078370 (PARI) a(n)=4*n^2+4*n+5 \\ _Charles R Greathouse IV_, Sep 24 2015
%o A078370 (Python) a= lambda n: 4*n**2+4*n+5 # _Indranil Ghosh_, Jan 04 2017
%o A078370 (Scala) (1 to 99 by 2).map(n => n * n + 4) // _Alonso del Arte_, May 29 2019
%o A078370 (Magma) [4*n^2+4*n+5 : n in [0..80]]; // _Wesley Ivan Hurt_, Aug 29 2022
%Y A078370 Subsequence of A077426 (D values (not a square) for which Pell x^2 - D*y^2 = -4 is solvable in positive integers).
%Y A078370 Cf. A002061, A002522, A004770, A005473, A016754.
%Y A078370 Cf. A054413, A078356, A078357, A086902, A087475.
%Y A078370 Subsequence of A031396.
%K A078370 nonn,easy
%O A078370 0,1
%A A078370 _Wolfdieter Lang_, Nov 29 2002
%E A078370 More terms from _Max Alekseyev_, Mar 03 2010