A127630 -id:A127630 - cp's OEIS frontend

A045621 a(n) = 2^n - binomial(n, floor(n/2)).

0, 1, 2, 5, 10, 22, 44, 93, 186, 386, 772, 1586, 3172, 6476, 12952, 26333, 52666, 106762, 213524, 431910, 863820, 1744436, 3488872, 7036530, 14073060, 28354132, 56708264, 114159428, 228318856, 459312152, 918624304, 1846943453, 3693886906

Offset: 0

David M Bloom, Brooklyn College

nonn

p(n) = a(n)/2^n is the probability that a majority of heads had occurred at some point after n flips of a fair coin. For example, after 3 flips of a coin, the probability is 5/8 that a majority of heads had occurred at some point. (First flip is heads, p=1/2, or sequence THH, p=1/8.) - Brian Galebach, May 14 2001
Hankel transform is (-1)^n*n. - Paul Barry, Jan 11 2007
Hankel transform of a(n+1) is A127630. - Paul Barry, Sep 01 2009
a(n) is the number of n-step walks on the number line that are positive at some point along the walk. - Benjamin Phillabaum, Mar 06 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..247
Kairi Kangro, Mozhgan Pourmoradnasseri, Dirk Oliver Theis, Short note on the number of 1-ascents in dispersed dyck paths, arXiv:1603.01422 [math.CO], 2016.
S. Mason and J. Parsley, A geometric and combinatorial view of weighted voting, arXiv preprint arXiv:1109.1082 [math.CO], 2011.

Cf. A000108, A001405, A000346, A054336, A068551.

List([0..35], n-> 2^n - Binomial(n, Int(n/2)) ); # G. C. Greubel, Jan 13 2020

[2^n - Binomial(n, Floor(n/2)): n in [0..35]]; // Bruno Berselli, Mar 08 2011

seq( 2^n -binomial(n,floor(n/2)), n=0..35); # G. C. Greubel, Jan 13 2020

Table[2^n - Binomial[n, Floor[n/2]], {n, 0, 35}] (* Roger L. Bagula, Aug 26 2006 *)

{a(n)=if(n<0, 0, 2^n -binomial(n, n\2))} /* Michael Somos, Oct 31 2006 */

[2^n -binomial(n,floor(n/2)) for n in (0..35)] # G. C. Greubel, Jan 13 2020

a(n) = 2^n - A001405(n).
a(2*k) = 2*a(2*k-1), a(2*k+1) = 2*a(2*k) + Catalan(k).
a(n+1) = b(0)*b(n)+b(1)*b(n-1)+...+b(n)*b(0), b(k)=C(k, [ k/2 ]).
G.f.: c(x^2)*x/(1-2*x) where c(x) = g.f. for Catalan numbers A000108.
a(n) = A054336(n, 1) (second column of triangle).
E.g.f.: exp(2*x) - I_0(2*x) - I_1(2*x) where I_n(x) is n-th modified Bessel function as a function of x. - Benjamin Phillabaum, Mar 06 2011
a(2*n+1) = A000346(n); a(2*n) = A068551(n). - Emeric Deutsch, Nov 16 2003
a(n) = Sum_{k=0..n-1} binomial(n, floor(k/2)). - Paul Barry, Aug 05 2004
a(n+1) = 2*a(n) + Catalan(n/2)*(1+(-1)^n)/2. - Paul Barry, Aug 05 2004
a(n+1) = Sum_{k=0..floor(n/2)} 2^(n-2*k)*A000108(k). - Paul Barry, Sep 01 2009
(n+1)*a(n) +2*(-n-1)*a(n-1) +4*(-n+2)*a(n-2) +8*(n-2)*a(n-3) = 0. - R. J. Mathar, Dec 02 2012

Edited by N. J. A. Sloane, Oct 08 2006

Adjustments to formulas (correcting offsets) from Michael Somos, Oct 31 2006

A296064 a(1) = 0; thereafter a(n) is the smallest number (in absolute value) not yet in the sequence such that the arithmetic mean of the first n terms a(1), a(2), ..., a(n) is an integer. Preference is given to positive values of a(n).

Original entry on oeis.org

0, 2, 1, -3, 5, -5, 7, -7, 9, -9, 11, -11, 13, -13, 15, -15, 17, -17, 19, -19, 21, -21, 23, -23, 25, -25, 27, -27, 29, -29, 31, -31, 33, -33, 35, -35, 37, -37, 39, -39, 41, -41, 43, -43, 45, -45, 47, -47, 49, -49, 51, -51

Offset: 1

Enrique Navarrete, Dec 04 2017

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (-1,1,1).

Cf. A296065 (partial sums), A127630.

Essentially the same as A296063.

0, 2, 1, -3, seq(seq(s*i,s=[1,-1]),i=5..100,2); # Robert Israel, Dec 26 2017

Nest[Append[#, Block[{k = 1, s = 1}, While[Nand[FreeQ[#, s k], IntegerQ@ Mean[Append[#, s k]]], If[s == 1, s = -1, k++; s = 1]]; s k]] &, {0}, 51] (* Michael De Vlieger, Dec 12 2017 *)

From Robert Israel, Dec 26 2017: (Start)
a(n) = a(n-3)+a(n-2)-a(n-1) for n >= 7.
G.f.: (2+3*x-4*x^2-x^3+2*x^4)*x^2/((1-x)*(x+1)^2). (End)
a(n) = 1/2+(-1)^n*(1/2-n), n>=4. - R. J. Mathar, May 14 2024

A236377 Real part of Sum_{k=0..n} (k + i^k)^2, where i=sqrt(-1).

Original entry on oeis.org

1, 1, 2, 10, 35, 59, 84, 132, 213, 293, 374, 494, 663, 831, 1000, 1224, 1513, 1801, 2090, 2450, 2891, 3331, 3772, 4300, 4925, 5549, 6174, 6902, 7743, 8583, 9424, 10384, 11473, 12561, 13650, 14874, 16243, 17611, 18980, 20500, 22181, 23861, 25542, 27390

Offset: 0

Bruno Berselli, Jan 24 2014

Corresponding imaginary parts: -i^(n*(n+1))*A052928(n+1).

			For n=6, sum_(k=0)^6 (k + i^k)^2 = 84 + 6*i, therefore a(6) = 84.

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-2,-2,4,-4,3,-1).

Cf. A058373: real part of Sum_{k=0..n} (k + i)^2.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-2*x+3*x^2+4*x^3+11*x^4-10*x^5+9*x^6)/((1+x)*(1+x^2)^2*(1-x)^4)));

LinearRecurrence[{3, -4, 4, -2, -2, 4, -4, 3, -1}, {1, 1, 2, 10, 35, 59, 84, 132, 213}, 50]

G.f.: (1 - 2*x + 3*x^2 + 4*x^3 + 11*x^4 - 10*x^5 + 9*x^6)/((1 + x)*(1 + x^2)^2*(1 - x)^4).
a(n) = 3*a(n-1) -4*a(n-2) +4*a(n-3) -2*a(n-4) -2*a(n-5) +4*a(n-6) -4*a(n-7) +3*a(n-8) -a(n-9).
a(n) = A000330(n) + A127630(n) - A000035(n).

cp's OEIS Frontend

A045621 a(n) = 2^n - binomial(n, floor(n/2)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A296064 a(1) = 0; thereafter a(n) is the smallest number (in absolute value) not yet in the sequence such that the arithmetic mean of the first n terms a(1), a(2), ..., a(n) is an integer. Preference is given to positive values of a(n).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A236377 Real part of Sum_{k=0..n} (k + i^k)^2, where i=sqrt(-1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Formula