A379210 List of integers of the form (N^2 - 4)/15.

0, 3, 4, 11, 19, 32, 35, 52, 68, 91, 96, 123, 147, 180, 187, 224, 256, 299, 308, 355, 395, 448, 459, 516, 564, 627, 640, 707, 763, 836, 851, 928, 992, 1075, 1092, 1179, 1251, 1344, 1363, 1460, 1540, 1643, 1664, 1771, 1859, 1972, 1995, 2112, 2208, 2331, 2356, 2483

Offset: 1

Author

Peter Bala, Dec 18 2024

Keywords

• nonn
• easy

Comments

Compare with A204221.

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 + + - - ....

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

Links

• Index entries for linear recurrences with constant coefficients, signature (1,0,0,2,-2,0,0,-1,1).

Crossrefs

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

Programs

Maple

A379210 := proc(q) local n;
for n from 0 to q do
 if type(sqrt(15*n+4), integer) then print(n);
fi; od; end:
A379210(2500);

Mathematica

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 *)

Formula

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).

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

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

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

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

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

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).

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