A219391 - OEIS

%I #66 Dec 26 2024 10:14:59

%S 0,3,8,19,23,40,55,80,88,119,144,183,195,240,275,328,344,403,448,515,

%T 535,608,663,744,768,855,920,1015,1043,1144,1219,1328,1360,1475,1560,

%U 1683,1719,1848,1943,2080,2120,2263,2368,2519,2563,2720,2835,3000,3048,3219

%N Numbers k such that 21*k + 1 is a square.

%C Equivalently, numbers in increasing order of the form m*(21*m + 2) or m*(21*m + 16) + 3, where m = 0, -1, 1, -2, 2, -3, 3, ....

%C Let F(r) = Product_{n >= 1} 1 - q^(28*n-r). The sequence terms are the exponents in the expansion of F(0)*F(3)*F(8)*F(11)*F(14)*F(17)*F(20)*F(25) = 1 - q^3 - q^8 + q^19 + q^23 - q^40 - q^55 + + - - ... (by the quintuple product identity). Cf. A204221. - _Peter Bala_, Dec 25 2024

%H Bruno Berselli, <a href="/A219391/b219391.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,2,-2,0,0,-1,1).

%F G.f.: x^2*(3 + 5*x + 11*x^2 + 4*x^3 + 11*x^4 + 5*x^5 + 3*x^6)/((1 + x)^2*(1 - x)^3*(1 + x^2)^2).

%F a(n) = a(-n+1) = (42*n*(n-1) + 2*i^(n*(n+1))*(6*n + (-1)^n-3) + 7*(-1)^n*(2*n-1) + 11)/32, where i=sqrt(-1).

%F Sum_{n>=2} 1/a(n) = 21/4 - cot(2*Pi/21)*Pi/2 + Pi/(2*sqrt(3)) - tan(Pi/14)*Pi/2. - _Amiram Eldar_, Mar 16 2022

%p A219391:=proc(q)

%p local n;

%p for n from 1 to q do if type(sqrt(21*n+1), integer) then print(n);

%p fi; od; end:

%p A219391(1000); # _Paolo P. Lava_, Feb 19 2013

%t Select[Range[0, 3300], IntegerQ[Sqrt[21 # + 1]] &]

%t CoefficientList[Series[x (3 + 5 x + 11 x^2 + 4 x^3 + 11 x^4 + 5 x^5 + 3 x^6)/((1 + x)^2 (1 - x)^3 (1 + x^2)^2), {x, 0, 50}], x] (* _Vincenzo Librandi_, Aug 18 2013 *)

%t LinearRecurrence[{1,0,0,2,-2,0,0,-1,1},{0,3,8,19,23,40,55,80,88},60] (* _Harvey P. Dale_, Oct 01 2021 *)

%o (Magma) [n: n in [0..3300] | IsSquare(21*n+1)];

%o (Magma) I:=[0,3,8,19,23,40,55,80,88]; [n le 9 select I[n] else Self(n-1)+2*Self(n-4)-2*Self(n-5)-Self(n-8)+Self(n-9): n in [1..50]]; // _Vincenzo Librandi_, Aug 18 2013

%o (Maxima) makelist((42*n*(n-1)+2*%i^(n*(n+1))*(6*n+(-1)^n-3)+7*(-1)^n*(2*n-1)+11)/32, n, 1, 50);

%Y Cf. similar sequences listed in A219257.

%Y Cf. A219721 (square roots of 21*a(n)+1).

%Y Subsequence of A047528.

%K nonn,easy,changed

%O 1,2

%A _Bruno Berselli_, Nov 20 2012