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A026937
a(n) = Sum_{k=0..n} (k+1)*T(n, n-k), where T is given by A008288.
5
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
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
KEYWORD
nonn,easy
STATUS
approved