A026937 a(n) = Sum_{k=0..n} (k+1)*T(n, n-k), where T is given by A008288.

1, 3, 10, 30, 87, 245, 676, 1836, 4925, 13079, 34446, 90090, 234227, 605865, 1560200, 4002072, 10230201, 26069995, 66251090, 167941494, 424753615, 1072057117, 2700704172, 6791746500, 17052595573, 42752015487, 107035180630, 267634562754, 668407232235, 1667467065425

Offset

0,2

Links

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

Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 23.

Index entries for linear recurrences with constant coefficients, signature (4,-2,-4,-1).

Formula

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

a(n) = Sum_{k=0..n+1} A000129(k)*A001333(n+1-k). - Graeme McRae, Aug 03 2006 and Michel Marcus, Aug 01 2023

a(n) = A006645(n+2) - A006645(n+1). - R. J. Mathar, Jan 27 2011

a(n) = 4*a(n-1) - 2*a(n-2) - 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 20 2012

a(n) = ((n+2)/2)*A000129(n+1). - G. C. Greubel, May 25 2021

a(n) = ((n+2)/8)*((sqrt(2) + 2)*(1 + sqrt(2))^n - (sqrt(2) - 2)*(1 - sqrt(2))^n). - Peter Luschny, Jul 31 2023

Maple

with (combinat):seq(add(fibonacci(n, 2), k=0..n)/2, n=1..27); # Zerinvary Lajos, May 25 2008

Mathematica

CoefficientList[Series[(1-x)/(1-2x-x^2)^2, {x, 0, 40}], x]  (* Harvey P. Dale, Mar 22 2011 *)
LinearRecurrence[{4, -2, -4, -1}, {1, 3, 10, 30}, 40] (* Vincenzo Librandi, Jun 20 2012 *)
Table[(1/2)*(n+2)*Fibonacci[n+1, 2], {n, 0, 40}] (* G. C. Greubel, May 25 2021 *)

Prog

(Magma) I:=[1, 3, 10, 30]; [n le 4 select I[n] else 4*Self(n-1)-2*Self(n-2)-4*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jun 20 2012
(PARI) my(x='x+O('x^40)); Vec((1-x)/(1-2*x-x^2)^2) \\ Altug Alkan, Sep 20 2018
(PARI) a(n) = my(w=quadgen(8)); (n/8)*((2+w)*(1+w)^n - (w-2)*(1-w)^n); \\ Michel Marcus, Jul 31 2023
(Sage) [(1/2)*(n+2)*lucas_number1(n+1, 2, -1) for n in (0..40)] # G. C. Greubel, May 25 2021

Crossrefs

Cf. A000129, A001333, A006645, A008288, A364553.

Sequence in context: A117869 A092756 A027205 * A320564 A094306 A257596

Adjacent sequences: A026934 A026935 A026936 * A026938 A026939 A026940

Keyword

nonn,easy

Author

Clark Kimberling

Status

approved