A053439 Expansion of (1+x+2*x^3)/((1-x)*(1-x^2)^2).

1, 2, 4, 8, 11, 18, 22, 32, 37, 50, 56, 72, 79, 98, 106, 128, 137, 162, 172, 200, 211, 242, 254, 288, 301, 338, 352, 392, 407, 450, 466, 512, 529, 578, 596, 648, 667, 722, 742, 800, 821, 882, 904, 968, 991, 1058, 1082, 1152, 1177, 1250, 1276

Offset: 0

Author

N. J. A. Sloane, Jan 12 2000

Keywords

• nonn
• easy

Comments

a(n) gives the number of vertices encountered along the shortest walk that encounters every edge at least once on the complete graph with n + 1 vertices. - Peter Kagey, Nov 17 2016

Examples

			G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 11*x^4 + 18*x^5 + 22*x^6 + 32*x^7 + 37*x^8 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Cf. A128223.

Programs

Magma

I:=[1,2,4,8,11]; [n le 5 select I[n] else Self(n-1) +2*Self(n-2) -2*Self(n-3) -Self(n-4) +self(n-5): n in [1..30]]; // G. C. Greubel, May 26 2018

Mathematica

CoefficientList[Series[(1+x+2x^3)/((1-x)(1-x^2)^2),{x,0,50}],x] (* or *)
LinearRecurrence[{1,2,-2,-1,1},{1,2,4,8,11},50] (* Harvey P. Dale, Apr 26 2011 *)

PARI

x='x+O('x^30); Vec((1+x+2*x^3)/((1-x)*(1-x^2)^2)) \\ G. C. Greubel, May 26 2018

Formula

Even: a(2*n)= 2* n^2 +n +1, odd: a(2*n-1)= 2* n^2. - Frank Ellermann, Feb 11 2002

a(n) = Sum_{k=0..n} binomial(n, k mod 2). - Paul Barry, Jul 24 2003

a(n) = A128223(n) + 1. - Peter Kagey, Nov 17 2016

E.g.f.: (1 + x)*((2 + x)*cosh(x) + (1 + x)*sinh(x))/2. - Ilya Gutkovskiy, Nov 17 2016