A293710 Expansion of x^2/(1 - 4*x - 4*x^2 - x^3).

0, 0, 1, 4, 20, 97, 472, 2296, 11169, 54332, 264300, 1285697, 6254320, 30424368, 148000449, 719953588, 3502240516, 17036776865, 82876023112, 403153440424, 1961154631009, 9540108308844, 46408205199836, 225754408665729, 1098190563771104, 5342188094947168

Offset

0,4

Comments

This sequence is a generalization of the tribonacci sequence wherein the coefficients of the terms on the right hand side of the recurrence relation are terms of (a + b)^2. Thus we have a(n+2) = p^2 a(n+1) + 2*p*m a(n) + m^2 a(n-1), with a(0)=0, a(1)=0, a(2)=1. The further extension is a q-bonacci sequence (qB)n whose recurrence relation has terms on the right hand side with coefficients which are terms of (a + b)^q. For this sequence p = 2 and m = 1: a(n+2) = 4*a(n+1) + 4*a(n) + a(n-1).

References

S. Arolkar, Y. S. Valaulikar, Python Programming Language Codes For Some Properties Of Fibonacci Sequence Extensions, published in Conference Proceedings ISBN: 978-81-930850-2-8, pp. 85-90.

Links

Table of n, a(n) for n=0..25.

S. Arolkar, Y. S. Valaulikar, On an Extension of Fibonacci Sequence, Bulletin of Marathwada Mathematical Society, Aurangabad, Maharashtra, India 17(2)(2016), 1-8.

S. Arolkar, Y. S. Valaulikar, On a B-q bonacci Sequence, International Journal of Advances in Mathematics volume 2017 (1), 1-8, 2017.

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

Formula

G.f.: x^2/(1-x*(2+x)^2).

a(n+2) = 4*a(n+1) + 4*a(n) + a(n-1).

Prog

# (Python)
# also generates the terms a(n), where n < 0. For example a(-1) = 1, a(-2)= -4, ...
def a(n):
    if n == 0:
       return 0
    elif n == 1:
       return 0
    elif n== 2:
       return 1
    elif n < 0:
       return expand(a(n+3)- 4*a(n+2) - 4*a(n+1))
    else:
       return expand(4*a(n-1) + 4*a(n-2) + a(n-3))
m1=input(' Enter the first term to be generated')
m2=input(' Enter the last term to be generated')
for i in range (m1, (m2)+1):
    print a(i)

Crossrefs

Sequence in context: A151254 A232493 A240778 * A098225 A073532 A178874

Adjacent sequences: A293707 A293708 A293709 * A293711 A293712 A293713

Keyword

nonn,easy

Author

S. Arolkar and Y S Valaulikar, Nov 07 2017

Status

approved