A307028 Longest path length in the stacked book graph S_{n+1} square P_n.

1, 5, 10, 17, 24, 33, 42, 53, 64, 77, 90, 105, 120, 137, 154, 173, 192, 213, 234, 257, 280, 305, 330, 357, 384, 413, 442, 473, 504, 537, 570, 605, 640, 677, 714, 753, 792, 833, 874, 917, 960, 1005, 1050, 1097, 1144, 1193, 1242, 1293, 1344, 1397, 1450, 1505, 1560, 1617, 1674

Offset

1,2

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Eric Weisstein's World of Mathematics, Longest Path.

Eric Weisstein's World of Mathematics, Stacked Book Graph.

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Formula

From Andrew Howroyd, Feb 12 2026: (Start)

a(n) = n^2 + 1 - 2*floor((n-2)/2)*floor((n-3)/2) for n > 1.

a(n) = (2*n^2 + 12*n - 13 + (-1)^n)/4 for n > 1.

G.f.: x*(1 + 3*x - x^3 - x^4)/((1 - x)^3*(1 + x)). (End)

Mathematica

A307028[n_] := If[n == 1, 1, (2*n*(n+6) - 13 + (-1)^n)/4]; Array[A307028, 60] (* or *)
LinearRecurrence[{2, 0, -2, 1}, {1, 5, 10, 17, 24}, 60] (* Paolo Xausa, Mar 31 2026 *)

Prog

(PARI) a(n) = if(n==1, 1, n^2 + 1 - if(n%2, (n-3)^2, (n-4)*(n-2))/2) \\ Andrew Howroyd, Feb 12 2026

Crossrefs

Cf. A307461 (number of longest paths).

Sequence in context: A071978 A313988 A340042 * A276382 A105705 A061409

Adjacent sequences: A307025 A307026 A307027 * A307029 A307030 A307031

Keyword

nonn,easy

Author

Eric W. Weisstein, Mar 20 2019

Extensions

a(8) from Eric W. Weisstein, Apr 09 2019

a(9) onward from Andrew Howroyd, Feb 12 2026

Status

approved