A385602 a(n) is the minimum number of squares from which an n-fold totally concave polyomino (n-TCP) can be made.

21, 50, 95, 149, 215, 293, 383, 485, 599, 725, 863, 1013, 1175, 1349, 1535, 1733, 1943, 2165, 2399, 2645, 2903, 3173, 3455, 3749, 4055, 4373, 4703, 5045, 5399, 5765, 6143, 6533, 6935, 7349, 7775, 8213, 8663, 9125, 9599, 10085, 10583, 11093, 11615, 12149, 12695, 13253, 13823, 14405, 14999

Offset

1,1

Comments

a(n) is the solution to the integer non-linear program: min (n + 1)*(x + y) - 1 where x, y are integers such that x>=y, and x*y - (2n + 1)*x - (n + 1)*y - 2n + 1 >= 0.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Gill Barequet, Neal Madras, and Johann Peters, On t-fold Totally-Concave Polyominoes, 37th Can. Conf. Comput. Geometry, Toronto, ON, Canada, Aug. 13-15, 2025, Paper 34.

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

Formula

a(n) = 6*(n+1)^2-1 for n > 2.

From Stefano Spezia, Aug 24 2025: (Start)

G.f.: x*(21 - 13*x + 8*x^2 - 7*x^3 + 3*x^4)/(1 - x)^3.

E.g.f.: exp(x)*(5 + 18*x + 6*x^2) - 5 - 2*x - 3*x^2/2. (End)

a(n) = A140811(n+1) for n>=3. - Alois P. Heinz, Sep 15 2025

Example

The following are minimal n-TCP for 1 <= n <= 4 (0's represent squares):
    000 00           0 00000 0        00000 00 00 0     00 00 00 00 0
    0 000            000 0 000          0 000 0 000    0 00 00 00 0000
    00  00             00 0 0          000 0 000 0     00 00 00 00  0
     000 0           00 0000 0        00 0  00 0000     00 00 00 0 000
    00 00             00 0 000        0 00000 00  0    0 00 00 00000 0
    n = 1            00 000  0         00  0 0 000     00 00 00  0 00
    # squares = 21   0 00 000         0  00 000  00     00 00 0 000 00
                     000 00 00        0000 00 00  0    0 00 00000 00 0
                     n = 2             0 000 0 0000    00 00  0 00 00
                     # squares = 50   000 0 000 0       00 0 000 00 00
                                      0 00 00 00000    0 00000 00 00 0
                                      n = 3            00  0 00 00 00
                                      # squares = 95    0 000 00 00 00
                                                       0000 00 00 00 0
                                                        0 00 00 00 00
                                                       n = 4
                                                       # squares = 149

Mathematica

LinearRecurrence[{3, -3, 1}, {21, 50, 95, 149, 215}, 50] (* Vincenzo Librandi, Oct 21 2025 *)

Prog

(Magma) I:=[21, 50, 95, 149, 215]; [n le 5 select I[n] else 3*Self(n-1)-3*Self(n-2)+ Self(n-3): n in [1..50]]; // Vincenzo Librandi, Oct 21 2025

Crossrefs

Cf. A140811.

Sequence in context: A147281 A130062 A153441 * A357679 A235884 A053178

Adjacent sequences: A385599 A385600 A385601 * A385603 A385604 A385605

Keyword

nonn,easy

Author

Gill Barequet, Neal Madras, and Johann Peters, Aug 02 2025

Status

approved