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

Author

Paul Barry, Aug 29 2004

Keywords

• easy
• nonn

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).

Programs

GAP

Concatenation([1,2], List([2..40], n-> 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

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 *)

PARI

{a(n) = if(n==0,1, if(n==1,2, 2^(n-3)*(3*n+4)))}; \\ 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

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