A093968 Inverse binomial transform of n*Pell(n).

0, 1, 2, 6, 8, 20, 24, 56, 64, 144, 160, 352, 384, 832, 896, 1920, 2048, 4352, 4608, 9728, 10240, 21504, 22528, 47104, 49152, 102400, 106496, 221184, 229376, 475136, 491520, 1015808, 1048576, 2162688, 2228224, 4587520, 4718592, 9699328, 9961472, 20447232, 20971520

Offset

0,3

Comments

Binomial transform is A093967.

Binomial transform of (-1)^(n+1)(n*Pell(n-2)) (see A093969).

S-D transform of A001477 (cf. A051159). - Philippe Deléham, Aug 01 2006

a(n) is also the number of projective permutations of vertices of regular n-gons. A permutation of n vertices (AFB...CD) is considered 'projective' if there exists a line so that all the vertices can be projected onto it and the resulted points can be read in the same order: A'F'B'...C'D'. - Anton Zakharov, Jul 25 2016

Links

Table of n, a(n) for n=0..40.

Index entries for linear recurrences with constant coefficients, signature (0,4,0,-4).

Formula

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

a(n) = 2^((n-4)/2)n((1+sqrt(2)) + (1-sqrt(2))(-1)^n).

a(2n) = A036289(n). a(2n+1) = A014480(n). - R. J. Mathar, Jun 02 2011

G.f.: x*G(0)/(1-x) where G(k) = 1 + x/(k+1 - 2*x*(k+1)*(k+2)/(2*x*(k+2) + 1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 01 2013

a(n) = n*2^floor((n-1)/2). - Anton Zakharov, Jul 25 2016

E.g.f.: x*(sqrt(2)*sinh(sqrt(2)*x) + 2*cosh(sqrt(2)*x))/2. - Ilya Gutkovskiy, Jul 25 2016

Sum_{n>=1} 1/a(n) = log(2) + sqrt(2)*log(1+sqrt(2)). - Amiram Eldar, Feb 13 2023

Example

a(3) = 6, as there are only 6 projective permutations of vertices in a triangle ABC: ABC,CBA,ACB,BCA,CAB,BAC and it is equal to the number of simple permutations of three elements.
a(4) = 8, as there are only 8 permutations of vertices in a square, satisfying the projective criterion: ADBC,DACB,DCAB,CDBA,СBDA,BCAD,BACD,ABDC. ADCB is not allowed, cause there is no way to draw a line so that the projections A'B'C'D' of the original points form a line segment B'C' lying inside A'D' on this line. - Anton Zakharov, Jul 25 2016

Mathematica

a[n_] := n*2^Floor[(n - 1)/2]; Array[a, 40, 0] (* Amiram Eldar, Feb 13 2023 *)

Crossrefs

Cf. A000129, A132314.

Sequence in context: A275826 A371238 A143481 * A064713 A162213 A100358

Adjacent sequences: A093965 A093966 A093967 * A093969 A093970 A093971

Keyword

easy,nonn

Author

Paul Barry, Apr 21 2004

Status

approved