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
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).
Programs
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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
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Maple
with (combinat):seq(add(fibonacci(n,2),k=0..n)/2,n=1..27); # Zerinvary Lajos, May 25 2008
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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 *) -
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
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Sage
[(1/2)*(n+2)*lucas_number1(n+1,2,-1) for n in (0..40)] # G. C. Greubel, May 25 2021
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) = 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
a(n) = A361732(n+2)/2. - R. J. Mathar, Jun 30 2025