OFFSET
1,2
COMMENTS
Equivalently, numbers of the form m*(18*m+2), where m = 0,-1,1,-2,2,-3,3,...
Also, integer values of 2*h*(h+1)/9.
The sequence terms are the exponents in the expansion of Product_{n >= 1} (1 - q^(36*n))*(1 - q^(36*n-16))*(1 - q^(36*n-20)) = 1 - q^16 - q^20 + q^68 + q^76 - q^156 - q^168 + + - - .... - Peter Bala, Dec 24 2024
LINKS
Bruno Berselli, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).
FORMULA
G.f.: 4*x^2*(4 + x + 4*x^2)/((1 + x)^2*(1 - x)^3).
a(n) = a(-n+1) = (18*n*(n-1) + 7*(-1)^n*(2*n-1) - 1)/4 + 2.
Sum_{n>=2} 1/a(n) = 9/2 - cot(Pi/9)*Pi/2. - Amiram Eldar, Mar 15 2022
E.g.f.: (x*(9*x - 7)*cosh(x) + (9*x^2 + 7*x + 7)*sinh(x))/2. - Stefano Spezia, Apr 15 2026
MAPLE
A219395:=proc(q)
local n;
for n from 1 to q do if type(sqrt(18*n+1), integer) then print(n);
fi; od; end:
A219395(1000); # Paolo P. Lava, Feb 19 2013
MATHEMATICA
Select[Range[0, 10000], IntegerQ[Sqrt[18 # + 1]] &]
CoefficientList[Series[4 x (4 + x + 4 x^2)/((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)
LinearRecurrence[{1, 2, -2, -1, 1}, {0, 16, 20, 68, 76}, 50] (* Harvey P. Dale, Dec 24 2014 *)
PROG
(Magma) [n: n in [0..10000] | IsSquare(18*n+1)];
(Magma) I:=[0, 16, 20, 68, 76]; [n le 5 select I[n] else Self(n-1)+2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Aug 18 2013
(PARI) isok(k) = issquare(18*k+1); \\ Bruce Nye, Apr 14 2026
(PARI) concat(0, Vec(4*x^2*(4 + x + 4*x^2)/((1 + x)^2*(1 - x)^3)+O(x^50))) \\ Bruce Nye, Apr 14 2026
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Bruno Berselli, Dec 03 2012
STATUS
approved