A110198 -id:A110198 - cp's OEIS frontend

A201631 a(n) is the number of Fibonacci meanders of length m*n and central angle 360/m degrees where m = 2.

1, 3, 6, 13, 30, 70, 167, 405, 992, 2450, 6090, 15214, 38165, 96069, 242530, 613811, 1556856, 3956316, 10070871, 25674210, 65541142, 167517654, 428635032, 1097874434, 2814611701, 7221917871, 18544968768, 47655572191, 122544150258, 315313433594, 811792614547

Offset: 1

Peter Luschny, Jan 15 2012

nonn

Empirically the partial sums of A051291. - Sean A. Irvine, Jul 13 2022

The above conjecture was proved by Baril et al., which also give a formal definition of the Fibonacci meanders and describe a bijection with a certain class of peakless grand Motzkin paths of length n. - Peter Luschny, Mar 16 2023

			a(3) = 6 = card({100001, 100100, 110000, 111001, 111100, 111111}).

Jean-Luc Baril, Sergey Kirgizov, Rémi Maréchal, and Vincent Vajnovszki, Enumeration of Dyck paths with air pockets, arXiv:2202.06893 [cs.DM], 2022-2023.
Peter Luschny, Fibonacci meanders.

Cf. A110236, A110198, A202411, A203611, A051291, A358734.

Cf. A361574 (case m=3).

A201631 := n -> add(A202411(k),k=0..2*n-1): seq(A201631(i),i=1..9);
# Alternative, using the g.f. of Baril et al.:
S := (x^2 - x + 1 - R)/((x - 1)*(x^2 - x - 1 + R)*R):
R := (((x - 3)*x + 1)*(x^2 + x + 1))^(1/2): ser := series(S, x, 33):
seq(coeff(ser, x, n), n = 1..31); # Peter Luschny, Mar 16 2023
# Using a recurrence:
a := proc(n) option remember; if n < 5 then return [0, 1, 3, 6, 13][n + 1] fi;
(n*(2*n - 1)*(2*n - 3)*(n - 5)*a(n - 5) - (n - 4)*(2*n - 1)^2*(3*n - 5)*a(n - 4) + (2*n - 5)*(n - 3)*(2*n^2 - 3*n + 2)*a(n - 3) - (2*n - 3)*(n - 2)*(2*n^2 - 3*n + 5)*a(n - 2) + (3*n - 4)*(2*n - 1)*(2*n - 5)*(n - 1)*a(n - 1))/(n*(2*n - 3)*(2*n - 5)*(n - 1)) end: seq(a(n), n = 1..31); # Peter Luschny, Mar 16 2023

a[n_] := Sum[A202411[k], {k, 0, 2 n - 1}];
Array[a, 31] (* Jean-François Alcover, Jun 29 2019 *)

a(n) = Sum_{k=0..2n-1} A202411(k).

a(n) = [x^n] (x^2 - x + 1 - R)/((x - 1)*(x^2 - x - 1 + R) * R), where R = (((x - 3)*x + 1)*(x^2 + x + 1))^(1/2). (This is Theorem 21 in Baril et al.) - Peter Luschny, Mar 16 2023

A110197 Number triangle of sums of squared binomial coefficients.

Original entry on oeis.org

1, 2, 1, 3, 5, 1, 4, 14, 10, 1, 5, 30, 46, 17, 1, 6, 55, 146, 117, 26, 1, 7, 91, 371, 517, 251, 37, 1, 8, 140, 812, 1742, 1476, 478, 50, 1, 9, 204, 1596, 4878, 6376, 3614, 834, 65, 1, 10, 285, 2892, 11934, 22252, 19490, 7890, 1361, 82, 1, 11, 385, 4917, 26334, 66352, 82994, 51990, 15761, 2107, 101, 1

Offset: 0

Paul Barry, Jul 15 2005

Alternatively, number square T(n,k) = Sum_{i=0..n} binomial(i+k,k)^2 read by antidiagonals.

			Rows start:
  1;
  2,   1;
  3,   5,   1;
  4,  14,  10,   1;
  5,  30,  46,  17,   1;
  6,  55, 146, 117,  26,   1;
  ...

Columns include A000027, A000330, A024166, A086020, A086023, A086025.
Row sums are A006134.
Antidiagonal sums are A110198.
T(2n,n) gives A112029.

T(n,k) = sum(i=0, n-k, binomial(i+k,k)^2);
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print();); \\ Michel Marcus, Dec 03 2016

T(n,k) = Sum_{i=0..n-k} binomial(i+k,k)^2.

G.f.: 1/((1-x)*sqrt(x^2*y^2-2*x^2*y-2*x*y+x^2-2*x+1)). - Vladimir Kruchinin, Mar 20 2025

cp's OEIS Frontend

A201631 a(n) is the number of Fibonacci meanders of length m*n and central angle 360/m degrees where m = 2.

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