A297740 The number of distinct positions on an infinite chessboard reachable by the (2,3)-leaper in <= n moves.

1, 9, 41, 129, 321, 625, 997, 1413, 1885, 2425, 3033, 3709, 4453, 5265, 6145, 7093, 8109, 9193, 10345, 11565, 12853, 14209, 15633, 17125, 18685, 20313, 22009, 23773, 25605, 27505, 29473, 31509, 33613, 35785, 38025, 40333, 42709, 45153, 47665, 50245, 52893

Offset

0,2

Links

Colin Barker, Table of n, a(n) for n = 0..1000

R. J. Mathar, Fairy chess leaper minimum moves on the infinite chessboard, (2018).

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

a(n) = 34*n^2 + 30*n + 9 for n >= 6.

From Colin Barker, Jan 05 2018: (Start)

G.f.: (1 + x)*(1 + 5*x + 12*x^2 + 20*x^3 + 28*x^4 - 20*x^5 - 24*x^6 + 12*x^8) / (1 - x)^3.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>9. (End)

Mathematica

LinearRecurrence[{3, -3, 1}, {1, 9, 41, 129, 321, 625, 997, 1413, 1885, 2425}, 50] (* Paolo Xausa, Mar 17 2024 *)

Prog

(PARI) Vec((1 + x)*(1 + 5*x + 12*x^2 + 20*x^3 + 28*x^4 - 20*x^5 - 24*x^6 + 12*x^8) / (1 - x)^3 + O(x^40)) \\ Colin Barker, Jan 07 2018

Crossrefs

Cf. A018836 (1,2)-leaper or (1,3)-leaper, A297741 (3,4)-leaper.

Partial sums of A018839.

Cf. A253974, A254129, A254459.

Sequence in context: A373517 A362293 A274323 * A297741 A001846 A271663

Adjacent sequences: A297737 A297738 A297739 * A297741 A297742 A297743

Keyword

nonn,easy

Author

R. J. Mathar, Jan 05 2018

Status

approved