A152107 - OEIS

%I #34 Jun 04 2026 00:20:24

%S 1,6,41,306,2401,19326,157481,1290666,10606081,87262326,718359401,

%T 5915180706,48713027041,401185722606,3304124833001,27212740595226,

%U 224125017319681,1845905249384166,15202987455699881,125212786737489426

%N a(n) = ((6+sqrt(5))^n+(6-sqrt(5))^n)/2.

%H Robert Israel, <a href="/A152107/b152107.txt">Table of n, a(n) for n = 0..1090</a>

%H Yassine Otmani, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL28/Otmani/otmani10.html">The 2-Pascal Triangle and a Related Riordan Array</a>, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 12.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (12,-31).

%F From _Philippe Deléham_, Nov 26 2008: (Start)

%F a(n) = 12*a(n-1)-31*a(n-2), n>1 ; a(0)=1, a(1)=6 .

%F G.f.: (1-6*x)/(1-12*x+31*x^2).

%F a(n) = (Sum_{k=0..n} A098158(n,k)*6^(2*k)*5^(n-k))/6^n. (End)

%F a(n) = Sum_{k=0..n} A027907(n,2k)*5^k . - _J. Conrad_, Aug 24 2016

%F E.g.f.: cosh(sqrt(5)*x)*exp(6*x). - _Ilya Gutkovskiy_, Aug 24 2016

%e 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

%p f:= gfun:-rectoproc({a(n)=12*a(n-1)-31*a(n-2),a(0)=1,a(1)=6},a(n),remember):

%p map(f, [$0..50]); # _Robert Israel_, Aug 25 2016

%t CoefficientList[Series[(1 - 6 x)/(1 - 12 x + 31 x^2), {x, 0, 19}], x] (* _Michael De Vlieger_, Aug 25 2016 *)

%t LinearRecurrence[{12,-31},{1,6},30] (* _Harvey P. Dale_, Aug 28 2024 *)

%o (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

%o (PARI) a(n)=([0,1; -31,12]^n*[1;6])[1,1] \\ _Charles R Greathouse IV_, Jun 04 2026

%Y Cf. A098158, A027907.

%K nonn,easy

%O 0,2

%A Al Hakanson (hawkuu(AT)gmail.com), Nov 24 2008

%E Extended beyond a(6) by _Klaus Brockhaus_, Nov 26 2008

%E Typo in name corrected by _J. Conrad_, Aug 24 2016