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

1,1

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Index entries for linear recurrences with constant coefficients, signature (9,-23,15).

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

Example

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

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 *)

Prog

(PARI) a(n)=(5^n+2*3^n+1)/4; \\ Michel Marcus, Aug 22 2013
(Magma) [(5^n+2*3^n+1)/4: n in [1..30]]; // Vincenzo Librandi, Dec 31 2013

Crossrefs

Cf. A063376, A122983.

Sequence in context: A049160 A191243 A217888 * A049177 A217889 A217890

Adjacent sequences: A146083 A146084 A146085 * A146087 A146088 A146089

Keyword

base,easy,nonn

Author

Jake Foster, Oct 27 2008

Extensions

More terms from Colin Barker, Dec 31 2013

Status

approved