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
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Aug 29 2004
STATUS
approved