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Row sums of Fibonacci triangle shown below.
0

%I #17 Aug 09 2015 01:42:42

%S 1,3,16,123,1453,27060,803383,38256129,2932126904,362464081089,

%T 72358024951979,23344004888219544,12176743686773409053,

%U 10272520597198595537175,14018081932741301581509848

%N Row sums of Fibonacci triangle shown below.

%D Thomas Koshy, "Elementary Number Theory with Applications", p. 143.

%D T. Koshy, Fibonacci and Lucas Numbers with Applications, Wiley-Interscience, 2001, see p. 16.

%F a(n) is asymptotic to (1/2+3/2/sqrt(5))*phi^(n*(n+1)/2) where phi=(1+sqrt(5))/2. - _Benoit Cloitre_, Oct 19 2003

%F a(n) = Sum(i=(n(n-1)/2)+1 to n(n+1)/2) fibonacci(i) - _Sam Alexander_, Oct 19 2003

%F a(n) = F(T(n)+2) - F(T(n-1)+2) where T(n) = n-th triangular number. a(n) = A000045(A000217(n)+2) - A000045(A000217(n-1)+2). - _Jonathan Vos Post_, Dec 17 2006

%e 1

%e 1 2

%e 3 5 8

%e 13 21 34 55

%e 89 144 233 377 610

%e ...

%t Table[Plus@@Fibonacci[Range[((n - 1)^2 + n - 1)/2 + 1, (n^2 + n)/2]], {n, 15}] (* _Alonso del Arte_, Feb 10 2012 *)

%Y Cf. A000045.

%Y Cf. A000217.

%K nonn,easy

%O 1,2

%A _Gary W. Adamson_, Oct 18 2003

%E More terms from _Benoit Cloitre_ and _Ray Chandler_, Oct 19 2003