A261064 - OEIS

%I #60 May 14 2024 16:54:57

%S 1,6,26,100,363,1274,4372,14760,49205,162382,531438,1727180,5580127,

%T 17936130,57395624,182948560,581130729,1840247318,5811307330,

%U 18305618100,57531942611,180441092746,564859072956,1765184603000,5507375961373,17157594341214,53379182394902

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

%C Second column of A201730.

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

%H Vincenzo Librandi, <a href="/A261064/b261064.txt">Table of n, a(n) for n = 1..1000</a>

%H Jean-Luc Baril, Pamela E. Harris, and José L. Ramírez, <a href="https://arxiv.org/abs/2405.05357">Flattened Catalan Words</a>, arXiv:2405.05357 [math.CO], 2024. See p. 21.

%H National Math Contest "Atanas Radev", <a href="http://www.mgyambol.com/docs/Broshura_ZMS_2020_v2.pdf">Problems and Solutions</a>, problem 8.4 ("Задача 8.4" in Bulgarian), Jan 2020.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-22,24,-9).

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

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

%F a(n) = Sum_{k = 1..n} A027907(n, 2k - 1)*k . - _J. Conrad_, Aug 30 2016

%F a(n) = Sum_{k = 0..(n - 1)} binomial(n + 1, k + 2)*A001792(k). - _Ivaylo Kortezov_, Jan 21 2020

%F E.g.f.: exp(x)*(exp(2*x)*(1 + 3*x) - x - 1)/4. - _Stefano Spezia_, May 14 2024

%t LinearRecurrence[{8, -22, 24, -9}, {1, 6, 26, 100}, 30] (* _Vincenzo Librandi_, Aug 31 2016 *)

%t Table[(3^n - 1)(n + 1)/4, {n, 0, 39}] (* _Alonso del Arte_, Jan 19 2020 *)

%o (PARI) first(m)=vector(m,i, (3^i-1)*(i+1)/4); /* _Anders Hellström_, Aug 08 2015 */

%o (Magma) [(3^n-1)*(n+1)/4: n in [1..30]]; // _Vincenzo Librandi_, Aug 31 2016

%Y Cf. A001792, A027907, A201730, A212337.

%K nonn,easy

%O 1,2

%A _R. J. Mathar_, Aug 08 2015