A001582 -id:A001582 - cp's OEIS frontend

A166989 G.f.: A(x) = 1/(1 - 2x - 7x^2 - 2*x^3 + x^4).

1, 2, 11, 38, 156, 598, 2353, 9166, 35843, 139956, 546792, 2135796, 8343205, 32590610, 127308455, 497301794, 1942600788, 7588340434, 29642181517, 115790645854, 452310642407, 1766851828392, 6901817263824, 26960427965352

Offset: 0

Paul D. Hanna, Oct 26 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,7,2,-1).

Cf. A000204, A000045, A002203, A000129, A001582.

LinearRecurrence[{2, 7, 2, -1}, {1, 2, 11, 38}, 100] (* G. C. Greubel, May 30 2016 *)

{a(n)=polcoeff(1/(1-2*x-7*x^2-2*x^3+x^4+x*O(x^n)),n)}

{a(n)=if(n<0,0,if(n==0,1,2*a(n-1)+7*a(n-2)+2*a(n-3)-a(n-4)))}

G.f.: A(x) = exp( Sum_{n>=1} A000204(n)*A002203(n)*x^n/n ) where A000204 (Lucas numbers) forms the logarithmic derivative of the Fibonacci numbers (A000045) and A002203 forms the logarithmic derivative of the Pell numbers (A000129).
Recurrence: a(n) = 2*a(n-1) + 7*a(n-2) + 2*a(n-3) - a(n-4) where a(k)=0 for k<0 with a(0)=1.
Radius of convergence: r = f*p where f=(sqrt(5)-1)/2, p=sqrt(2)-1:
(f*p-x)*(1/(f*p)-x)*(f/p+x)*(p/f+x) = 1 - 2*x - 7*x^2 - 2*x^3 + x^4.
For n >= 2, a(n) - a(n-2) = Fibonacci(n+1)*Pell(n+1) = A001582(n). - Peter Bala, Aug 30 2015
a(n) = (1/2) * Sum_{i=0..n+2} (-1)^(n-i+1) * Lucas(2*i-n-2) * Pell(i) * Pell(n+2-i). - Vladimir Kruchinin, Jan 10 2025

A200541 Product of Fibonacci and tribonacci numbers: a(n) = A000045(n+1)*A000073(n+2).

Original entry on oeis.org

1, 1, 4, 12, 35, 104, 312, 924, 2754, 8195, 24386, 72576, 215991, 642785, 1912960, 5693016, 16942573, 50421592, 150056090, 446571180, 1329008590, 3955167387, 11770690808, 35029911168, 104250013425, 310251009501, 923315841860, 2747814245904, 8177573467339, 24336691577000

Offset: 0

Paul D. Hanna, Nov 18 2011

nonn

Limit a(n+1)/a(n) = (sqrt(5)+1)/2 * (1+(19+3*sqrt(33))^(1/3)+(19-3*sqrt(33))^(1/3))/3 = 2.9760284849940...

			G.f.: A(x) = 1 + x + 4*x^2 + 12*x^3 + 35*x^4 + 104*x^5 + 312*x^6 + 924*x^7 + 2754*x^8 +...+ A000045(n+1)*A000073(n+2)*x^n +...
where tribonacci numbers (A000073) begin:
[1,1,2,4,7,13,24,44,81,149,274,504,927,1705,3136,5768,10609,...],
and Fibonacci numbers (A000045) begin:
[1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,...].

Paul D. Hanna, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (1,4,5,2,-1,1).

Cf. A001582, A000045, A000073.

Module[{nn=30,fs,ts},fs=Fibonacci[Range[nn]];ts=LinearRecurrence[{1,1,1},{1,1,2},nn];Times@@@Thread[{fs,ts}]] (* or *) LinearRecurrence[ {1,4,5,2,-1,1},{1,1,4,12,35,104},30] (* Harvey P. Dale, Dec 14 2016 *)

{a(n)=polcoeff((1-x^2-x^3)/(1-x-4*x^2-5*x^3-2*x^4+x^5-x^6 +x*O(x^n)),n)}

{A000073(n)=polcoeff(x^2/(1-x-x^2-x^3+x^3*O(x^n)),n)}
{a(n)=fibonacci(n+1)*A000073(n+2)}

G.f.: (1 - x^2 - x^3) / (1 - x - 4*x^2 - 5*x^3 - 2*x^4 + x^5 - x^6).

A344684 Sum of two consecutive products of Fibonacci and Pell numbers: F(n)P(n) + F(n+1)P(n+1).

Original entry on oeis.org

1, 3, 12, 46, 181, 705, 2757, 10765, 42058, 164280, 641739, 2506789, 9792253, 38251227, 149420064, 583676434, 2280003517, 8906330973, 34790619369, 135901886149, 530870766310

Offset: 0

Greg Dresden and Hexuan Wang, Aug 17 2021

a(n) is the numerator of the continued fraction [1,...,1,2,...,2] with n 1's followed by n 2's.

			For n=3, a(3)=46 which is F(3)*P(3) + F(4)*P(4) = 2*5 + 3*12 = 46. Also, the continued fraction [1,1,1,2,2,2] with 3 1's followed by 3 2's has numerator 46.

Index entries for linear recurrences with constant coefficients, signature (2,7,2,-1).

Cf. A000045, A000129, A001582.

Table[Fibonacci[n] Fibonacci[n, 2] + Fibonacci[n + 1] Fibonacci[n + 1, 2], {n, 0, 30}]

P(n) = ([2, 1; 1, 0]^n)[2, 1]; \\ A000129
a(n) = fibonacci(n)*P(n)+ fibonacci(n+1)*P(n+1); \\ Michel Marcus, Aug 18 2021

a(n) = F(n)*P(n) + F(n+1)*P(n+1) for F(n) = A000045(n) the Fibonacci numbers and P(n) = A000129(n) the Pell numbers.
a(n) = 2*a(n-1) + 7*a(n-2) + 2*a(n-3) - a(n-4).
G.f.: (1 + x - x^2 - x^3)/(1 - 2*x - 7*x^2 - 2*x^3 + x^4).
a(n) = A001582(n-1) + A001582(n) for n >= 1.

cp's OEIS Frontend

A166989 G.f.: A(x) = 1/(1 - 2x - 7x^2 - 2*x^3 + x^4).

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A200541 Product of Fibonacci and tribonacci numbers: a(n) = A000045(n+1)*A000073(n+2).

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A344684 Sum of two consecutive products of Fibonacci and Pell numbers: F(n)P(n) + F(n+1)P(n+1).

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cp's OEIS Frontend

A166989 G.f.: A(x) = 1/(1 - 2*x - 7*x^2 - 2*x^3 + x^4).

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A200541 Product of Fibonacci and tribonacci numbers: a(n) = A000045(n+1)*A000073(n+2).

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A344684 Sum of two consecutive products of Fibonacci and Pell numbers: F(n)*P(n) + F(n+1)*P(n+1).

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Keywords

Comments

Examples

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Crossrefs

Programs

Mathematica

PARI

Formula

A166989 G.f.: A(x) = 1/(1 - 2x - 7x^2 - 2*x^3 + x^4).

A344684 Sum of two consecutive products of Fibonacci and Pell numbers: F(n)P(n) + F(n+1)P(n+1).