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URL: https://oeis.org/A152107

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A152107
a(n) = ((6+sqrt(5))^n+(6-sqrt(5))^n)/2.
1
1, 6, 41, 306, 2401, 19326, 157481, 1290666, 10606081, 87262326, 718359401, 5915180706, 48713027041, 401185722606, 3304124833001, 27212740595226, 224125017319681, 1845905249384166, 15202987455699881, 125212786737489426
OFFSET
0,2
LINKS
Yassine Otmani, The 2-Pascal Triangle and a Related Riordan Array, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 12.
FORMULA
From Philippe Deléham, Nov 26 2008: (Start)
a(n) = 12*a(n-1)-31*a(n-2), n>1 ; a(0)=1, a(1)=6 .
G.f.: (1-6*x)/(1-12*x+31*x^2).
a(n) = (Sum_{k=0..n} A098158(n,k)*6^(2*k)*5^(n-k))/6^n. (End)
a(n) = Sum_{k=0..n} A027907(n,2k)*5^k . - J. Conrad, Aug 24 2016
E.g.f.: cosh(sqrt(5)*x)*exp(6*x). - Ilya Gutkovskiy, Aug 24 2016
EXAMPLE
For n=3, (6+sqrt(5))^3 = 216 + 108*sqrt(5) + 18*5 + 5*sqrt(5) = 306 + 113*sqrt(5) and (6-sqrt(5))^3 = 306 - 113*sqrt(5), so a(3) = (306 + 113*sqrt(5) + 306 - 113*sqrt(5))/2 = 306. - Michael B. Porter, Aug 25 2016
MAPLE
f:= gfun:-rectoproc({a(n)=12*a(n-1)-31*a(n-2), a(0)=1, a(1)=6}, a(n), remember):
map(f, [$0..50]); # Robert Israel, Aug 25 2016
MATHEMATICA
CoefficientList[Series[(1 - 6 x)/(1 - 12 x + 31 x^2), {x, 0, 19}], x] (* Michael De Vlieger, Aug 25 2016 *)
LinearRecurrence[{12, -31}, {1, 6}, 30] (* Harvey P. Dale, Aug 28 2024 *)
PROG
(Magma) Z<x>:= PolynomialRing(Integers()); N<r5>:=NumberField(x^2-5); S:=[ ((6+r5)^n+(6-r5)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Nov 26 2008
CROSSREFS
Sequence in context: A000402 A186654 A390714 * A143023 A078009 A127848
KEYWORD
nonn
AUTHOR
Al Hakanson (hawkuu(AT)gmail.com), Nov 24 2008
EXTENSIONS
Extended beyond a(6) by Klaus Brockhaus, Nov 26 2008
Typo in name corrected by J. Conrad, Aug 24 2016
STATUS
approved