A261064 a(n) = (3^n-1)*(n+1)/4.

1, 6, 26, 100, 363, 1274, 4372, 14760, 49205, 162382, 531438, 1727180, 5580127, 17936130, 57395624, 182948560, 581130729, 1840247318, 5811307330, 18305618100, 57531942611, 180441092746, 564859072956, 1765184603000, 5507375961373, 17157594341214, 53379182394902

Offset: 1

Author

R. J. Mathar, Aug 08 2015

Keywords

• nonn
• easy

Comments

Second column of A201730.

Number of non-selfintersecting broken lines in a convex (n+1)-gon. (National Math Contest "Atanas Radev" 2020, Bulgaria) - Ivaylo Kortezov, Jan 18 2020

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Jean-Luc Baril, Pamela E. Harris, and José L. Ramírez, Flattened Catalan Words, arXiv:2405.05357 [math.CO], 2024. See p. 21.
• Ivaylo Kortezov, problem 8.4 ("Задача 8.4" in Bulgarian) in National Math Contest "Atanas Radev" 2020.
• Mark Shattuck, Enumeration of consecutive patterns in flattened Catalan words, arXiv:2502.10661 [math.CO], 2025. See pp. 3, 6.
• Index entries for linear recurrences with constant coefficients, signature (8,-22,24,-9).

Crossrefs

Cf. A001792, A027907, A201730, A212337.

Programs

Magma

[(3^n-1)*(n+1)/4: n in [1..30]]; // Vincenzo Librandi, Aug 31 2016

Mathematica

LinearRecurrence[{8, -22, 24, -9}, {1, 6, 26, 100}, 30] (* Vincenzo Librandi, Aug 31 2016 *)
Table[(3^n - 1)(n + 1)/4, {n, 0, 39}] (* Alonso del Arte, Jan 19 2020 *)

PARI

first(m)=vector(m,i, (3^i-1)*(i+1)/4); /* Anders Hellström, Aug 08 2015 */

Formula

G.f.: -x*(-1 + 2*x) / ( (3*x - 1)^2*(x - 1)^2 ).

a(n) = A212337(n - 1) - 2*A212337(n - 2).

a(n) = Sum_{k = 1..n} A027907(n, 2k - 1)*k . - J. Conrad, Aug 30 2016

a(n) = Sum_{k = 0..(n - 1)} binomial(n + 1, k + 2)*A001792(k). - Ivaylo Kortezov, Jan 21 2020

E.g.f.: exp(x)*(exp(2*x)*(1 + 3*x) - x - 1)/4. - Stefano Spezia, May 14 2024