A379210 - OEIS

%I #18 Dec 24 2024 13:48:27

%S 0,3,4,11,19,32,35,52,68,91,96,123,147,180,187,224,256,299,308,355,

%T 395,448,459,516,564,627,640,707,763,836,851,928,992,1075,1092,1179,

%U 1251,1344,1363,1460,1540,1643,1664,1771,1859,1972,1995,2112,2208,2331,2356,2483

%N List of integers of the form (N^2 - 4)/15.

%C Compare with A204221.

%C The sequence terms occur as exponents in the expansion of Sum_{n >= 1} x^(n*(n-1)) * Product_{k >= 2*n} 1 - x^k = 1 - x^3 - x^4 + x^11 + x^19 - x^32 - x^35 + + - - ....

%C |A379212(n)| is the characteristic function of the numbers in this sequence.

%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 a(n) = (1/32)*(30*n^2 - 30*n + 1) + (-1)^(n)*(5/32)*(2*n - 1) - (-1)^(n*(n+1)/2)*(1/16)*(6*n - 3 + (-1)^n).

%F a(n) is quasi-periodic in n: for n >= 0,

%F a(4*n+1) = 15*n^2 - 26*n + 11; a(4*n+2) = 15*n^2 - 16*n + 4;

%F a(4*n+3) = 15*n^2 - 14*n + 3; a(4*n+4) = 15*n^2 - 4*n.

%F a(1-n) = a(n).

%F 15*a(n) + 4 = A379211(n)^2.

%F G.f: x*(3*x^6 + x^5 + 7*x^4 + 8*x^3 + 7*x^2 + x + 3)/((1 + x)^2*(1 - x)^3*(1 + x^2)^2).

%F E.g.f.: ((15*x^2 + 35*x - 2)*cosh(x) + 2*(cos(x) + 3*x*cos(x) + 2*sin(x) - 3*x*sin(x)) + (15*x^2 + 25*x + 3)*sinh(x))/16. - _Stefano Spezia_, Dec 23 2024

%p A379210 := proc(q) local n;

%p for n from 0 to q do

%p  if type(sqrt(15*n+4), integer) then print(n);

%p fi; od; end:

%p A379210(2500);

%t LinearRecurrence[{1,0,0,2,-2,0,0,-1,1},{0, 3, 4, 11, 19, 32, 35, 52, 68},52] (* _James C. McMahon_, Dec 24 2024 *)

%Y Cf. A204220, A204221, A204542, A379211 (square roots of 15*a(n) + 4), A379212.

%K nonn,easy

%O 1,2

%A _Peter Bala_, Dec 18 2024