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

Author

Raul Prisacariu, Dec 18 2023

Keywords

• sign

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.

Examples

			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.
		

Links

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

Crossrefs

Cf. A218836 (denominator), A060006.

Programs

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)];

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)).