A147958 a(n) = ((7 + sqrt(2))^n + (7 - sqrt(2))^n)/2.

1, 7, 51, 385, 2993, 23807, 192627, 1577849, 13036417, 108350935, 904201491, 7566326929, 63431106929, 532418131343, 4472591813139, 37592633210825, 316085049734017, 2658336935367463, 22360719757645683, 188108240644768801

Offset: 0

Author

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

Keywords

• nonn

Comments

7th binomial transform of A077957. Binomial transform of A147957. Inverse binomial transform of A147959. - Philippe Deléham, Nov 30 2008

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (14, -47).

Crossrefs

Cf. A077957, A098158, A147957, A147959.

Programs

Magma

Z:= PolynomialRing(Integers()); N:=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

Mathematica

LinearRecurrence[{14, -47}, {1, 7}, 50] (* G. C. Greubel, Aug 17 2018 *)

PARI

x='x+O('x^30); Vec((1-7*x)/(1-14*x+47*x^2)) \\ G. C. Greubel, Aug 17 2018

Formula

From Philippe Deléham, Nov 19 2008: (Start)

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

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

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

E.g.f.: exp(7*x)*cosh(sqrt(2)*x). - Ilya Gutkovskiy, Aug 11 2017

Extensions

Extended beyond a(6) by Klaus Brockhaus, Nov 19 2008