A006645 Self-convolution of Pell numbers (A000129).

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, 2781614954080, 6937095319664, 17282819159456

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

Michael De Vlieger, Table of n, a(n) for n = 0..2605

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, Vol. 26, No. 3 (2020), 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.

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.

Milan Janjić, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010), Article 10.7.8, section 3.

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

Formula

a(n) = A054456(n-1, 1), n>=1.

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

E.g.f.: exp(x) * (2 * x * cosh(sqrt(2)*x) + sqrt(2) * (2*x-1) * sinh(sqrt(2)*x)) / 8. - Amiram Eldar, Feb 25 2026

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

(SageMath) taylor( mul(x/(1 - 2*x - x^2) for i in range(1, 3)), x, 0, 28) # Zerinvary Lajos, Jun 03 2009
(SageMath)
def A000129(n): return lucas_number1(n, 2, -1)
def A006645(n): return ((n-1)*A000129(n) + n*A000129(n-1))//4
print([A006645(n) for n in range(41)]) # G. C. Greubel, Jan 02 2026
(Magma)
I:=[0, 0, 1, 4]; [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..40]]; // G. C. Greubel, Jan 02 2026

Crossrefs

Cf. A000129, A054456 (second column of triangle), A054457.

Sequence in context: A062109 A118042 A381354 * A094309 A000300 A005323

Adjacent sequences: A006642 A006643 A006644 * A006646 A006647 A006648

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

Sum formulas and cross-references added by Wolfdieter Lang, Aug 07 2002

Status

approved