A368205 a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3), with a(0)=1, a(1)=3 and a(2)=7.

1, 3, 7, 14, 25, 40, 56, 63, 37, -71, -350, -945, -2064, -3952, -6783, -10381, -13625, -13330, -2359, 33208, 117672, 288959, 598325, 1099385, 1812546, 2640543, 3197152, 2497824, -1541375, -12816925, -37865849, -86422322, -170718343, -301444536, -476474600, -655816385, -713055419, -351058887, 1028750562, 4501424879, 11797832400, 25361896880, 47988600961

Offset

0,2

Comments

Whittaker's Root Series Formula is applied to the polynomial equation -1+2x+3x^2+x^3. The following infinite series involving the Plastic Ratio (rho) is obtained: rho - 1 = 1/2 - 3/(2*7) + 7/(7*21) - 14/(21*65) + 25/(65*200) - 40/(200*616) + 56/(616*1897) - ...

The terms of the sequence appear in the numerators of the infinite sequence (with alternating signs). The denominators of the sequence are formed by multiplying consecutive terms from the sequence A218836.

Links

Table of n, a(n) for n=0..42.

E. T. Whittaker and G. Robinson, The Calculus of Observations, London: Blackie & Son, Ltd. 1924, pp. 120-123.

Formula

a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3).

a(n) = determinant of the n X n Toeplitz Matrix((3,2,-1,0,0,...,0),(3,1,0,0,0,...,0)).

Example

a(0) = 1,
a(1) = 3*a(0) = 3*1 = 3,
a(2) = 3*a(1) - 2*a(0) = 3*3 - 2*1 = 7,
a(3) = 3*a(2) - 2*a(1) - a(0) = 3*7 - 2*3 - 1 = 14.

Maple

a:=proc(n) local c1, c2, c3;
 option remember;
c1:=3; c2:=2; c3:=1;
if n=0 then 1
elif n=1 then 3
elif n=2 then 7
else c1*a(n-1)-c2*a(n-2)-c3*a(n-3); fi;
end; # N. J. A. Sloane, Dec 31 2023
[seq(a(n), n=0..30)];

Crossrefs

Cf. A218836 (denominator), A060006.

Sequence in context: A089187 A333980 A316319 * A179178 A171973 A253895

Adjacent sequences: A368202 A368203 A368204 * A368206 A368207 A368208

Keyword

sign

Author

Raul Prisacariu, Dec 18 2023

Status

approved