A233656 a(n) = 3*a(n-1) - 2*(n-1), with a(0) = 1.

1, 3, 7, 17, 45, 127, 371, 1101, 3289, 9851, 29535, 88585, 265733, 797175, 2391499, 7174469, 21523377, 64570099, 193710263, 581130753, 1743392221, 5230176623, 15690529827, 47071589437, 141214768265, 423644304747, 1270932914191, 3812798742521, 11438396227509, 34315188682471

Offset: 0

Author

Richard R. Forberg, Dec 14 2013

Keywords

• nonn
• easy

Comments

Inverse binomial transform of this sequence is A090129.

Conjecture: Last digit of a(n) has repeating pattern of 20 digits as follows: {1, 3, 7, 7, 5, 7, 1, 1, 9, 1, 5, 5, 3, 5, 9, 9, 7, 9, 3, 3}, with an equal frequency of the five odd digits.

Examples

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

Links

• Index entries for linear recurrences with constant coefficients, signature (5,-7,3).

Crossrefs

Cf. A007051, A090129.

Programs

Mathematica

LinearRecurrence[{5,-7,3},{1,3,7},30] (* Harvey P. Dale, Feb 13 2015 *)

PARI

Vec((x^2+2*x-1)/((x-1)^2*(3*x-1)) + O(x^100)) \\ Colin Barker, Dec 17 2013

Formula

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

(a(n+1) - a(n))/2 = A007051(n).

From Colin Barker, Dec 17 2013: (Start)

a(n) = (1 + 3^n + 2*n)/2.

a(n) = 5*a(n-1) - 7*a(n-2) + 3*a(n-3).

G.f.: (x^2+2*x-1) / ((x-1)^2*(3*x-1)). (End)

E.g.f.: exp(x)*(1 + exp(2*x) + 2*x)/2. - Stefano Spezia, Mar 20 2022