A320661 a(n) = 2^(n+3) - 6*n - 7.

1, 3, 13, 39, 97, 219, 469, 975, 1993, 4035, 8125, 16311, 32689, 65451, 130981, 262047, 524185, 1048467, 2097037, 4194183, 8388481, 16777083, 33554293, 67108719, 134217577, 268435299, 536870749, 1073741655, 2147483473, 4294967115, 8589934405, 17179868991

Offset: 0

Author

Paul Curtz, Nov 14 2018

Keywords

• nonn
• easy

Comments

Companion to A247618 which has the same recurrence.

For this recurrence the main sequence is A000295.

Links

• Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Crossrefs

Cf. A000225, A000295, A010701, A016921, A100402, A129756, A247618.

Programs

GAP

List([0..40], n -> 2^(n+3) -6*n -7); # G. C. Greubel, Nov 15 2018

Magma

[2^(n+3) -6*n -7: n in [0..40]]; // G. C. Greubel, Nov 15 2018

Mathematica

a[n_]:=2^(n+3) - 6*n - 7; Array[a,32,0] (* Amiram Eldar, Nov 14 2018 *)

PARI

vector(40, n, n--; 2^(n+3) -6*n -7) \\ G. C. Greubel, Nov 15 2018

Sage

[2^(n+3) -6*n -7 for n in range(40)] # G. C. Greubel, Nov 15 2018

Formula

a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3).

a(n+1) = a(n-1) + 12*A000225(n). a(-1) = 3.

a(2*n) mod 9 = period 3: repeat [1, 4, 7].

a(2*n+1) mod 9 = 3.

a(n) mod 9 = period 6: repeat [1, 3, 4, 3, 7, 3].

a(n) mod 10 = period 20: repeat [1, 3, 3, 9, 7, 9, 9, 5, 3, 5, 5, 1, 9, 1, 1, 7, 5, 7, 7, 3] = Im(n). Im(n-1) = [3, 1, 3, 3, 9, 7, 9, 9, 5, 3, 5, 5, 1, 9, 1, 1, 7, 5, 7, 7]. Disordered [1, 1, 1, 1, 3, 3, 3, 3, 5, 5, 5, 5, 7, 7, 7, 7, 9, 9, 9, 9].

a(n+1) - a(n) = 2^(n+3) - 6.

From G. C. Greubel, Nov 15 2018: (Start)

G.f.: (1-x+6*x^2)/((1-2*x)*(1-x)^2).

E.g.f.: 8*exp(2*x) - (7 + 6*x)*exp(x). (End)

Extensions

More terms from Amiram Eldar, Nov 14 2018