A098156 Interleave n+1 and 2n+1 and take binomial transform.

1, 2, 5, 13, 32, 76, 176, 400, 896, 1984, 4352, 9472, 20480, 44032, 94208, 200704, 425984, 901120, 1900544, 3997696, 8388608, 17563648, 36700160, 76546048, 159383552, 331350016, 687865856, 1426063360, 2952790016, 6106906624

Offset

0,2

Comments

Binomial transform of A029579.

An elephant sequence, see A175655. For the central square 16 A[5] vectors, with decimal values between 59 and 440, lead to this sequence (without a(1)). For the corner squares these vectors lead to the companion sequence A066373 (with a leading 1 added). - Johannes W. Meijer, Aug 15 2010

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

David Anderson, E. S. Egge, M. Riehl, L. Ryan, R. Steinke, Y. Vaughan, Pattern Avoiding Linear Extensions of Rectangular Posets, arXiv:1605.06825 [math.CO], 2016.

Colin Defant, Proofs of Conjectures about Pattern-Avoiding Linear Extensions, arXiv:1905.02309 [math.CO], 2019.

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

Formula

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

a(n) = (2 * 0^n + Sum_{k=0..n} (-1)^(n-k)*k*binomial(n,k) + 2^(n+1) + 3*n*2^(n-1) )/4.

a(n) = Sum_{j=0..n} Sum_{k=0..n} binomial(n, 2*(k-j)).

a(n) = Sum_{k=0..n} Sum_{j=0..k} C(n, 2*j). - Paul Barry, Jan 13 2005

a(n) = 2^(n-3)*(3*n+4) for n>=2. - Philip B. Zhang, May 25 2016

E.g.f.: (2 + x + (2 + 3*x)*exp(2*x))/4. - Ilya Gutkovskiy, May 31 2016

Mathematica

CoefficientList[Series[(1-2x+x^2+x^3)/(1-2x)^2, {x, 0, 40}], x] (* Vincenzo Librandi, Jul 21 2013 *)
LinearRecurrence[{4, -4}, {1, 2, 5, 13}, 50] (* Harvey P. Dale, Dec 03 2023 *)

Prog

(PARI) {a(n) = if(n==0, 1, if(n==1, 2, 2^(n-3)*(3*n+4)))}; \\ G. C. Greubel, May 08 2019
(Magma) [1, 2] cat [2^(n-3)*(3*n+4): n in [2..40]]; // G. C. Greubel, May 08 2019
(Sage) [1, 2]+[2^(n-3)*(3*n+4) for n in (2..40)] # G. C. Greubel, May 08 2019
(GAP) Concatenation([1, 2], List([2..40], n-> 2^(n-3)*(3*n+4))) # G. C. Greubel, May 08 2019

Crossrefs

Sequence in context: A086758 A179257 A116702 * A267862 A098586 A199812

Adjacent sequences: A098153 A098154 A098155 * A098157 A098158 A098159

Keyword

easy,nonn

Author

Paul Barry, Aug 29 2004

Status

approved