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A004090 Sum of digits of Fibonacci numbers.

%I A004090 #44 Jul 10 2025 14:00:58
%S A004090 0,1,1,2,3,5,8,4,3,7,10,17,9,8,17,7,24,22,19,14,24,20,17,28,27,19,19,
%T A004090 29,21,23,17,31,30,34,37,35,27,35,44,43,24,31,46,41,33,29,35,37,54,55,
%U A004090 46,29,48,41,53,58,48,52,73,44,54,53,62,61,51,67,73,59
%N A004090 Sum of digits of Fibonacci numbers.
%C A004090 a(n) and Fibonacci(n) are congruent modulo 9 which implies that (a(n) mod 9) is equal to (Fibonacci(n) mod 9) A007887(n). Thus (a(n) mod 9) is periodic with Pisano period A001175(9) = 24. - _Hieronymus Fischer_, Jun 25 2007
%C A004090 It appears that a(n) - n stays negative for n > 5832, which explains why A020995 is finite. - _T. D. Noe_, Mar 19 2012
%H A004090 T. D. Noe, <a href="/A004090/b004090.txt">Table of n, a(n) for n = 0..10000</a>
%H A004090 T. D. Noe, <a href="/A004090/a004090_1.gif">Plot of a(n)-n for n = 0..100000</a>
%F A004090 a(n) = Fibonacci(n) - 9*Sum_{k>0} floor(Fibonacci(n)/10^k). - _Hieronymus Fischer_, Jun 25 2007
%F A004090 a(n) = A007953(A000045(n)). - _Reinhard Zumkeller_, Nov 17 2014
%F A004090 A010888(a(n)) = A030132(n) == a(n) (mod 9). - _M. F. Hasler_, Jul 07 2025
%t A004090 Table[Plus@@IntegerDigits@(Fibonacci[n]), {n, 0, 90}] (* _Vincenzo Librandi_, Jun 18 2015 *)
%o A004090 (PARI) a(n)=sumdigits(fibonacci(n)) \\ _Charles R Greathouse IV_, Feb 03 2014
%o A004090 (Haskell)
%o A004090 a004090 = a007953 . a000045 -- _Reinhard Zumkeller_, Nov 17 2014
%o A004090 (Magma) [&+Intseq(Fibonacci(n)): n in [0..80] ]; // _Vincenzo Librandi_, Jun 18 2015
%o A004090 (Python) A004090 = lambda n: sum(int(d)for d in str(A000045(n))) # _M. F. Hasler_, Jul 07 2025
%Y A004090 Cf. A000045 (Fibonacci), A007953 (digit sum), A030132 (digital root of A45), A010888 (digital root), A246558, A261587, A068500.
%K A004090 nonn,base,easy
%O A004090 0,4
%A A004090 _N. J. A. Sloane_