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A006645
Self-convolution of Pell numbers (A000129).
16
0, 0, 1, 4, 14, 44, 131, 376, 1052, 2888, 7813, 20892, 55338, 145428, 379655, 985520, 2545720, 6547792, 16777993, 42847988, 109099078, 277040572, 701794187, 1773851304, 4474555476, 11266301976, 28318897549, 71070913036, 178106093666, 445740656420, 1114147888655
OFFSET
0,4
REFERENCES
R. P. Grimaldi, Ternary strings with no consecutive 0's and no consecutive 1's, Congressus Numerantium, 205 (2011), 129-149. (The sequences w_n and z_n)
LINKS
Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.
Sergio Falcon, Half self-convolution of the k-Fibonacci sequence, Notes on Number Theory and Discrete Mathematics (2020) Vol. 26, No. 3, 96-106.
Rigoberto Flórez, Robinson Higuita, and Alexander Ramírez, The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials, arXiv:1808.01264 [math.NT], 2018.
Milan Janjic, Hessenberg Matrices and Integer Sequences , J. Int. Seq. 13 (2010) # 10.7.8, section 3.
FORMULA
a(n) = Sum_{k=0..n} b(k)*b(n-k) with b(k) := A000129(k).
a(n) = Sum_{k=0..floor((n-2)/2)} 2^(n-2)*(n-k-1)*binomial(n-2-k, k)*(1/4)^k, n >= 2.
From Wolfdieter Lang, Apr 11 2000: (Start)
a(n) = ((n-1)*P(n) + n*P(n-1))/4, P(n)=A000129(n).
G.f.: (x/(1 - 2*x - x^2))^2. (End)
a(n) = F'(n, 2), the derivative of the n-th Fibonacci polynomial evaluated at x=2. - T. D. Noe, Jan 19 2006
EXAMPLE
G.f. = x^2 + 4*x^3 + 14*x^4 + 44*x^5 + 131*x^6 + 376*x^7 + 1052*x^8 + ...
MAPLE
a:= n-> (Matrix(4, (i, j)-> if i=j-1 then 1 elif j=1 then [4, -2, -4, -1][i] else 0 fi)^n) [1, 3]: seq(a(n), n=0..40); # Alois P. Heinz, Oct 28 2008
MATHEMATICA
pell[n_] := Simplify[ ((1+Sqrt[2])^n - (1-Sqrt[2])^n)/(2*Sqrt[2])]; a[n_] := First[ ListConvolve[ pp = Array[pell, n+1, 0], pp]]; Table[a[n], {n, 0, 28}] (* Jean-François Alcover, Oct 21 2011 *)
Table[(n Fibonacci[n - 1, 2] + (n - 1) Fibonacci[n, 2])/4, {n, 0, 30}] (* Vladimir Reshetnikov, May 08 2016 *)
PROG
(Sage) taylor( mul(x/(1 - 2*x - x^2) for i in range(1, 3)), x, 0, 28) # Zerinvary Lajos, Jun 03 2009
CROSSREFS
a(n)= A054456(n-1, 1), n>=1 (second column of triangle), A054457.
Sequence in context: A007466 A062109 A118042 * A094309 A000300 A005323
KEYWORD
nonn,easy
EXTENSIONS
Sum formulas and cross-references added by Wolfdieter Lang, Aug 07 2002
STATUS
approved