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A293710
Expansion of x^2/(1 - 4*x - 4*x^2 - x^3).
0
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 and 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
S. Arolkar and Y. S. Valaulikar, On an Extension of Fibonacci Sequence, Bulletin of Marathwada Mathematical Society, Aurangabad, Maharashtra, India 17(2)(2016), 1-8.
S. Arolkar and Y. S. Valaulikar, On a B-q bonacci Sequence, International Journal of Advances in Mathematics volume 2017 (1), 1-8, 2017.
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)
from sympy import expand
# 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 = 0
m2 = 25
for i in range (m1, (m2)+1):
print(a(i), end=', ')
CROSSREFS
Sequence in context: A151254 A232493 A240778 * A098225 A073532 A178874
KEYWORD
nonn,easy
AUTHOR
S. Arolkar and Y S Valaulikar, Nov 07 2017
STATUS
approved