A238538 A fourth-order linear divisibility sequence: a(n) = (2^n + 1)*(2^(3*n) - 1)/ ( (2 + 1)*(2^3 - 1) ).

1, 15, 219, 3315, 51491, 811395, 12882499, 205321155, 3278747331, 52408827075, 838132189379, 13406842675395, 214483303960771, 3431523432591555, 54902699475185859, 878429788032676035, 14054769379960303811, 224875452250864496835, 3598000373385828511939

Offset: 1

Author

Peter Bala, Feb 28 2014

Keywords

• nonn
• easy

Comments

This is a fourth-order linear divisibility sequence, that is, the sequence satisfies a linear recurrence of order 4 and if n | m then a(n) | a(m). This is a consequence of the following more general result: The polynomials P(n,x,y) := (x^n + y^n)*(x^(3*n) - y^(3*n)) form a fourth-order linear divisibility sequence in the polynomial ring Z[x,y]. See the Bala link.

Hence, for a fixed integers M and N, the normalized sequence (M^n + N^n)*(M^(3*n) - N^(3*n))/ ( (M + N)*(M^3 - N^3) ) for n = 1,2,3,... is a linear divisibility sequence of order 4. It has the rational o.g.f. x*(1 - 2*M*N*(M^2 - M*N + N^2)*x + (M*N)^4*x^2)/( (1 - M^4*x)*(1 - M^3*N*x)*(1 - M*N^3*x)*(1 - N^4*x) ). This is the case M = 2, N = 1. For other cases see A238539(M = 2, N = -1), A238540(M = 3, N = 1) and A238541(M = 3, N = 2). See also A238536, A238537 and A215466.

Note, these sequences do not belong to the family of linear divisibility sequences of the fourth order studied by Williams and Guy, which have o.g.f.s of the form x*(1 - q*x^2)/Q(x), Q(x) a quartic polynomial and q an integer parameter.

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..831
• Peter Bala, A family of linear divisibility sequences of order four
• E. L. Roettger and H. C. Williams, Appearance of Primes in Fourth-Order Odd Divisibility Sequences, J. Int. Seq., Vol. 24 (2021), Article 21.7.5.
• Wikipedia, Divisibility sequence
• Wikipedia, Lucas Sequence
• H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
• H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume
• Index entries for linear recurrences with constant coefficients, signature (27,-202,432,-256).

Crossrefs

Cf. A215466, A236536, A236537, A238539, A238540, A238541.

Programs

Maple

seq(1/21*(2^n + 1)*(2^(3*n) - 1), n = 1..20);

Mathematica

LinearRecurrence[{27,-202,432,-256},{1,15,219,3315},20] (* Harvey P. Dale, Jul 04 2019 *)

Formula

a(n) = (1/21)*(2^n + 1)*(2^(3*n) - 1) = A000051(n)*A024088(n)/21.

a(n) = (1/21)*(4^n - 1)*(8^n - 1)/(2^n - 1).

O.g.f.: x*(1 - 12*x + 16*x^2)/((1 - x)*(1 - 2*x)*(1 - 8*x)*(1 - 16*x)).

Recurrence equation: a(n) = 27*a(n-1) - 202*a(n-2) + 432*a(n-4) - 256*a(n-4).