A147958 - OEIS

%I #22 Sep 08 2022 08:45:38

%S 1,7,51,385,2993,23807,192627,1577849,13036417,108350935,904201491,

%T 7566326929,63431106929,532418131343,4472591813139,37592633210825,

%U 316085049734017,2658336935367463,22360719757645683,188108240644768801

%N a(n) = ((7 + sqrt(2))^n + (7 - sqrt(2))^n)/2.

%C 7th binomial transform of A077957. Binomial transform of A147957. Inverse binomial transform of A147959. - _Philippe Deléham_, Nov 30 2008

%H G. C. Greubel, <a href="/A147958/b147958.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (14, -47).

%F From _Philippe Deléham_, Nov 19 2008: (Start)

%F a(n) = 14*a(n-1) - 47*a(n-2), n > 1; a(0)=1, a(1)=7.

%F G.f.: (1 - 7*x)/(1 - 14*x + 47*x^2).

%F a(n) = (Sum_{k=0..n} A098158(n,k)*7^(2k)*2^(n-k))/7^n. (End)

%F E.g.f.: exp(7*x)*cosh(sqrt(2)*x). - _Ilya Gutkovskiy_, Aug 11 2017

%t LinearRecurrence[{14, -47}, {1, 7}, 50] (* _G. C. Greubel_, Aug 17 2018 *)

%o (Magma) Z<x>:= PolynomialRing(Integers()); N<r2>:=NumberField(x^2-2); S:=[ ((7+r2)^n+(7-r2)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // _Klaus Brockhaus_, Nov 19 2008

%o (PARI) x='x+O('x^30); Vec((1-7*x)/(1-14*x+47*x^2)) \\ _G. C. Greubel_, Aug 17 2018

%Y Cf. A077957, A098158, A147957, A147959.

%K nonn

%O 0,2

%A Al Hakanson (hawkuu(AT)gmail.com), Nov 17 2008

%E Extended beyond a(6) by _Klaus Brockhaus_, Nov 19 2008