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A152262
a(n) = 14*a(n-1) - 43*a(n-2), n > 1; a(0)=1, a(1)=7.
3
1, 7, 55, 469, 4201, 38647, 360415, 3383989, 31878001, 300780487, 2840172775, 26828857909, 253476581401, 2395031249527, 22630944493135, 213846879174229, 2020725695234401, 19094743928789767, 180435210107977495
OFFSET
0,2
COMMENTS
Binomial transform of A145301. Inverse binomial transform of A152263. - Philippe Deléham, Dec 03 2008
LINKS
H. D. Nguyen and D. Taggart, Mining the OEIS: Ten Experimental Conjectures, 2013. Mentions this sequence. - From N. J. A. Sloane, Mar 16 2014
FORMULA
a(n) = ((7 + sqrt(6))^n + (7 - sqrt(6))^n)/2.
From Philippe Deléham, Dec 03 2008: (Start)
G.f.: (1-7*x)/(1-14*x+43*x^2).
a(n) = Sum_{k=0..n} A098158(n,k)*7^(2k-n)*6^(n-k). (End)
a(n) = Sum_{k=0..n} A027907(n,2k)*6^k. - J. Conrad, Aug 24 2016
E.g.f.: cosh(sqrt(6)*x)*exp(7*x). - Ilya Gutkovskiy, Aug 24 2016
a(n) = m^n*(ChebyshevU(n, 7/m) - (7/m)*ChebyshevU(n-1, 7/m)), with m = sqrt(43). - G. C. Greubel, May 23 2023
MATHEMATICA
LinearRecurrence[{14, -43}, {1, 7}, 30] (* Harvey P. Dale, Apr 26 2015 *)
PROG
(Magma) Z<x>:= PolynomialRing(Integers()); N<r6>:=NumberField(x^2-6); S:=[ ((7+r6)^n+(7-r6)^n)/2: n in [0..18] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Dec 03 2008
(Magma) [n le 2 select 7^(n-1) else 14*Self(n-1) -43*Self(n-2): n in [1..30]]; // G. C. Greubel, May 23 2023
(SageMath)
@CachedFunction
def a(n): # a = A152262
if (n<2): return 7^n
else: return 14*a(n-1) - 43*a(n-2)
[a(n) for n in range(41)] # G. C. Greubel, May 23 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Al Hakanson (hawkuu(AT)gmail.com), Dec 01 2008
EXTENSIONS
Name from Philippe Deléham, Dec 03 2008
STATUS
approved