OFFSET
1,2
COMMENTS
Equivalently, numbers of the form m*(11*m+2), where m = 0,-1,1,-2,2,-3,3,...
Also, integer values of h*(h+2)/11.
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.: x^2*(9+4*x+9*x^2)/((1+x)^2*(1-x)^3).
a(n) = a(-n+1) = (22*n*(n-1)+7*(-1)^n*(2*n-1)-1)/8 + 1 = (1/176)*(22*n+7*(-1)^n-15)*(22*n+7*(-1)^n-7).
Sum_{n>=2} 1/a(n) = 11/4 - cot(2*Pi/11)*Pi/2. - Amiram Eldar, Mar 15 2022
E.g.f.: (x*(11*x - 7)*cosh(x) + (7 + 7*x + 11*x^2)*sinh(x))/4. - Stefano Spezia, Mar 22 2026
MATHEMATICA
Select[Range[0, 7000], IntegerQ[Sqrt[11 # + 1]] &]
CoefficientList[Series[x (9 + 4 x + 9 x^2)/((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)
PROG
(Magma) [n: n in [0..7000] | IsSquare(11*n+1)];
(Magma) I:=[0, 9, 13, 40, 48]; [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(11*k+1); \\ Bruce Nye, Mar 11 2026
(PARI) concat(0, Vec(x^2*(9+4*x+9*x^2)/((1+x)^2*(1-x)^3)+O(x^50))) \\ Bruce Nye, Mar 14 2026
CROSSREFS
Cf. numbers k such that h*k+1 is a square: A005563 (h=1), A046092 (h=2), A001082 (h=3), A002378 (h=4), A036666 (h=5), A062717 (h=6), A132354 (h=7), A000217 (h=8), A132355 (h=9), A132356 (h=10), A152749 (h=12), A219389 (h=13), A219390 (h=14), A204221 (h=15), A074378 (h=16), A219394 (h=17), A219395 (h=18), A219396 (h=19), A219190 (h=20), A219391 (h=21), A219392 (h=22), A219393 (h=23), A001318 (h=24), A219259 (h=25), A217441 (h=26), A219258 (h=27), A219191 (h=28).
KEYWORD
nonn,easy
AUTHOR
Bruno Berselli, Nov 16 2012
STATUS
approved