A065450 Make an infinite chessboard from the squares in the first quadrant; sequence gives number of squares a knight can reach in n moves starting at the origin.

1, 2, 10, 22, 37, 54, 76, 100, 129, 160, 196, 234, 277, 322, 372, 424, 481, 540, 604, 670, 741, 814, 892, 972, 1057, 1144, 1236, 1330, 1429, 1530, 1636, 1744, 1857, 1972, 2092, 2214, 2341, 2470, 2604, 2740, 2881, 3024, 3172, 3322, 3477, 3634, 3796, 3960

Offset

0,2

Comments

The first conjecture is true: Partial sums of A047356 = b(n) = (14*(n*(n+1)) + 2*n + 5 + 3*(-1)^n )/8, since A047356(n)=(14*n+1+3*(-1)^n)/4. And b(n) has g.f. (4*x^2 + 2*x + 1)/(-x^4 + 2*x^3 - 2*x + 1). The difference a(n) - b(n) = 0,2,2,0,0,0,0,0,0..., which has g.f. 2*x^2 + 2*x. Then (4*x^2 + 2*x + 1)/(-x^4 + 2*x^3 - 2*x + 1) + 2*x^2 + 2*x = (-2*x^6 + 2*x^5 + 4*x^4 - 4*x^3 + 2*x^2 + 4*x + 1)/(-x^4 + 2*x^3 - 2*x + 1). - Vim Wenders, Apr 16 2008

Links

Table of n, a(n) for n=0..47.

Formula

Conjectures: G.f.: [1+6x^2+4x^3-4x^4-2x^5+2x^6]/[(1+x)*(1-x)^3]. For n>3, partial sums of A047356. - Ralf Stephan, Mar 06 2004

The second conjecture "For n>3, partial sums of A047356" is also true. From the last possible move, we can either move back to the second last possible move or to b(n)=A047883(n) new squares. So a(n) = a(n-2)+b(n). For n>6, b(n)=7(n-1)+4=A017029(n-1). But a number of the form 7n+4 is naturally the sum of two consecutive terms in A047356 (4=1+3,11=3+8,18=8+10, ...). The conjecture follows. - Vim Wenders, Apr 12 2008

Crossrefs

Cf. A098498.

Sequence in context: A339486 A273993 A225290 * A090288 A331132 A032526

Adjacent sequences: A065447 A065448 A065449 * A065451 A065452 A065453

Keyword

nonn,easy

Author

Bodo Zinser, Nov 18 2001

Extensions

More terms from Don Reble, Nov 28 2001

Status

approved