A141354 -id:A141354 - cp's OEIS frontend

A141353 a(n) = Catalan(n) + 2^n - 0^n.

1, 3, 6, 13, 30, 74, 196, 557, 1686, 5374, 17820, 60834, 212108, 751092, 2690824, 9727613, 35423206, 129775862, 477900844, 1767787478, 6565168996, 24468364172, 91486757944, 343068002258, 1289920924540, 4861979955884

Offset: 0

Paul Barry, Jun 27 2008

Hankel transform is A141354.

Cf. A000108 (Catalan numbers), A141351.

f[n_] := Binomial[2n, n]/(n + 1) + 2^n - 0^n; f[0] = 1; Array[f, 29, 0] (* or *)
CoefficientList[ Series[1 + 1/2 (-4 + 2/(1 - 2x) + (1 - Sqrt[1 - 4x])/x), {x, 0, 28}], x] (* Robert G. Wilson v, Mar 18 2018 *)

a(n) = binomial(2*n,n)/(n+1) + 2^n - 0^n; \\ Michel Marcus, Mar 18 2018

G.f.: c(x)+2x/(1-2x), where c(x) is the g.f. of A000108. [corrected by Paul Barry, Oct 18 2010]

Conjecture: (n+1)*a(n) + 2*(-4*n+1)*a(n-1) + 4*(5*n-7)*a(n-2) + 8*(-2*n+5)*a(n-3) = 0. - R. J. Mathar, Nov 15 2012

A181106 Largest odd number strictly less than a square.

Original entry on oeis.org

-1, 3, 7, 15, 23, 35, 47, 63, 79, 99, 119, 143, 167, 195, 223, 255, 287, 323, 359, 399, 439, 483, 527, 575, 623, 675, 727, 783, 839, 899, 959, 1023, 1087, 1155, 1223, 1295, 1367, 1443, 1519, 1599, 1679, 1763, 1847, 1935, 2023, 2115, 2207, 2303, 2399, 2499

Offset: 1

Jerzy Kocik (jkocik(AT)siu.edu), Oct 03 2010

The terms are the negatives of A141354 and therefore have the same generating function except the sign.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Equals minus A141354.

Cf. A120413 (Largest even number strictly less than a square).

Table[n^2-2^Mod[n,2],{n,50}] (* Ray Chandler, Dec 05 2011*)
LinearRecurrence[{2,0,-2,1},{-1,3,7,15},50] (* Harvey P. Dale, Nov 16 2019 *)

Vec(x*(x^3-x^2-5*x+1)/((x-1)^3*(x+1)) + O(x^100)) \\ Colin Barker, Jun 27 2015

a(n) = n^2 - 2^(n mod 2) = -A141354(n-1).
From Colin Barker, Jun 27 2015: (Start)
a(n) = n^2 - 1 for n even; a(n) = n^2 - 2 for n odd.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
G.f.: x*(x^3-x^2-5*x+1) / ((x-1)^3*(x+1)).
(End)

Edited by Ray Chandler, Dec 05 2011

cp's OEIS Frontend

A141353 a(n) = Catalan(n) + 2^n - 0^n.

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