A146086 Number of n-digit numbers with each digit odd where the digits 1 and 3 occur an even number of times.

3, 11, 45, 197, 903, 4271, 20625, 100937, 498123, 2470931, 12295605, 61300877, 305972943, 1528270391, 7636568985, 38168496017, 190799433363, 953868026651, 4768952712765, 23843601302357, 119214519727383, 596062138283711, 2980279310358945, 14901302408615897

Offset: 1

Author

Jake Foster, Oct 27 2008

Keywords

• base
• easy
• nonn

Comments

Let M = [3,1,1,0; 1,3,0,1; 1,0,3,1; 0,1,1,3] be a 4 x 4 matrix. Then a(n) = [M^n] (0,1); n = 1,2,3,.... - _Philippe Deléham, Aug 24 2020

With a(0) = 1, binomial transform of the sequence 1,2,6,20,72,272, ... (see A063376). - Philippe Deléham, Aug 24 2020

Examples

			For n=2 the a(2)=11 numbers are 11, 33, 55, 57, 59, 75, 77, 79, 95, 97, 99.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200
• Index entries for linear recurrences with constant coefficients, signature (9,-23,15).

Crossrefs

Cf. A063376, A122983.

Programs

Magma

[(5^n+2*3^n+1)/4: n in [1..30]]; // Vincenzo Librandi, Dec 31 2013

Mathematica

Table[(5^n + 2 3^n + 1)/4, {n, 1, 30}] (* Vincenzo Librandi, Dec 31 2013 *)
LinearRecurrence[{9,-23,15},{3,11,45},30] (* Harvey P. Dale, Dec 15 2014 *)

PARI

a(n)=(5^n+2*3^n+1)/4; \\ Michel Marcus, Aug 22 2013

Formula

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

From Colin Barker, Dec 31 2013: (Start)

a(n) = 9*a(n-1)-23*a(n-2)+15*a(n-3).

G.f.: -x*(15*x^2-16*x+3) / ((x-1)*(3*x-1)*(5*x-1)). (End)

E.g.f.: exp(3*x)*(cosh(x))^2 - 1. - G. C. Greubel, Jan 31 2016

Extensions

More terms from Colin Barker, Dec 31 2013