A261587 -id:A261587

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A004090

Sum of digits of Fibonacci numbers.

+10
27

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, 29, 21, 23, 17, 31, 30, 34, 37, 35, 27, 35, 44, 43, 24, 31, 46, 41, 33, 29, 35, 37, 54, 55, 46, 29, 48, 41, 53, 58, 48, 52, 73, 44, 54, 53, 62, 61, 51, 67, 73, 59

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

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

It appears that a(n) - n stays negative for n > 5832, which explains why A020995 is finite. - T. D. Noe, Mar 19 2012

LINKS

T. D. Noe, Table of n, a(n) for n = 0..10000

T. D. Noe, Plot of a(n)-n for n = 0..100000

FORMULA

a(n) = Fibonacci(n) - 9*Sum_{k>0} floor(Fibonacci(n)/10^k). - Hieronymus Fischer, Jun 25 2007

a(n) = A007953(A000045(n)). - Reinhard Zumkeller, Nov 17 2014

MATHEMATICA

Table[Plus@@IntegerDigits@(Fibonacci[n]), {n, 0, 90}] (* Vincenzo Librandi, Jun 18 2015 *)

PROG

(PARI) a(n)=sumdigits(fibonacci(n)) \\ Charles R Greathouse IV, Feb 03 2014

(Haskell)

a004090 = a007953 . a000045 -- Reinhard Zumkeller, Nov 17 2014

(Magma) [&+Intseq(Fibonacci(n)): n in [0..80] ]; // Vincenzo Librandi, Jun 18 2015

CROSSREFS

Cf. A000045, A030132, A007953, A246558, A261587, A068500.

KEYWORD

nonn,base,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

A261575

Table of Fibonacci numbers in base-60 representation: row n contains the sexagesimal digits of A000045(n) in reversed order.

+10
6

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 29, 1, 24, 2, 53, 3, 17, 6, 10, 10, 27, 16, 37, 26, 4, 43, 41, 9, 1, 45, 52, 1, 26, 2, 3, 11, 55, 4, 37, 57, 7, 48, 52, 12, 25, 50, 20, 13, 43, 33, 38, 33, 54, 51, 16, 28, 1, 29, 50, 22, 2, 20, 7, 51, 3, 49, 57, 13, 6

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

A261585(n) = length of n-th row;

T(n,0) = A261606(n) = in base 60: last sexagesimal digit of A000045(n);

T(n,A261607(n)-1) = A261607(n) = in base 60: initial sexagesimal digit of A000045(n);

A000045(n) = sum(T(n,k)*60^k : k = 0..A261585(n)-1).

LINKS

Reinhard Zumkeller, Rows n = 0..1000 of triangle, flattened

Eric Weisstein's World of Mathematics, Sexagesimal

Wikipedia, Sexagesimal

EXAMPLE

A000045(42) = 20*60^4 + 40*60^3 + 20*60^2 + 38*60^1 + 16*60^0 = 267914296.

. ----------------------------------------------------------------------

. n | T(n,*) n | T(n,*) n | T(n,*)

. ----+--------- ----+--------------- ----+-------------------------

. 0 | [0] 21 | [26,2,3] 42 | [16,38,20,40,20]

. 1 | [1] 22 | [11,55,4] 43 | [17,7,55,26,33]

. 2 | [1] 23 | [37,57,7] 44 | [33,45,15,7,54]

. 3 | [2] 24 | [48,52,12] 45 | [50,52,10,34,27,1]

. 4 | [3] 25 | [25,50,20] 46 | [23,38,26,41,21,2]

. 5 | [5] 26 | [13,43,33] 47 | [13,31,37,15,49,3]

. 6 | [8] 27 | [38,33,54] 48 | [36,9,4,57,10,6]

. 7 | [13] 28 | [51,16,28,1] 49 | [49,40,41,12,0,10]

. 8 | [21] 29 | [29,50,22,2] 50 | [25,50,45,9,11,16]

. 9 | [34] 30 | [20,7,51,3] 51 | [14,31,27,22,11,26]

. 10 | [55] 31 | [49,57,13,6] 52 | [39,21,13,32,22,42]

. 11 | [29,1] 32 | [9,5,5,10] 53 | [53,52,40,54,33,8,1]

. 12 | [24,2] 33 | [58,2,19,16] 54 | [32,14,54,26,56,50,1]

. 13 | [53,3] 34 | [7,8,24,26] 55 | [25,7,35,21,30,59,2]

. 14 | [17,6] 35 | [5,11,43,42] 56 | [57,21,29,48,26,50,4]

. 15 | [10,10] 36 | [12,19,7,9,1] 57 | [22,29,4,10,57,49,7]

. 16 | [27,16] 37 | [17,30,50,51,1] 58 | [19,51,33,58,23,40,12]

. 17 | [37,26] 38 | [29,49,57,0,3] 59 | [41,20,38,8,21,30,20]

. 18 | [4,43] 39 | [46,19,48,52,4] 60 | [0,12,12,7,45,10,33]

. 19 | [41,9,1] 40 | [15,9,46,53,7] 61 | [41,32,50,15,6,41,53]

. 20 | [45,52,1] 41 | [1,29,34,46,12] 62 | [41,44,2,23,51,51,26,1]

MATHEMATICA

Reverse[IntegerDigits[Fibonacci[Range[0, 50]], 60], 2] (* Paolo Xausa, Feb 19 2024 *)

PROG

(Haskell)

a261575 n k = a261575_tabf !! n !! k

a261575_row n = a261575_tabf !! n

a261575_tabf = [0] : [1] :

zipWith (add 0) (tail a261575_tabf) a261575_tabf where

add c (a:as) (b:bs) = y : add c' as bs where (c', y) = divMod (a+b+c) 60

add c (a:as) [] = y : add c' as [] where (c', y) = divMod (a+c) 60

add 1 _ _ = [1]

add _ _ _ = []

CROSSREFS

Cf. A000045, A261585 (row lengths), A261587 (row sums), A261598 (row products), A261606 (left edge), A261607 (right edge).

KEYWORD

nonn,tabf,base

AUTHOR

Reinhard Zumkeller, Sep 09 2015

STATUS

approved

A261598

Product of sexagesimal digits of Fibonacci numbers in base-60 representation.

+10
4

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 29, 48, 159, 102, 100, 432, 962, 172, 369, 2340, 156, 2420, 14763, 29952, 25000, 18447, 67716, 22848, 63800, 21420, 217854, 2250, 35264, 34944, 99330, 14364, 1300500, 0, 8726016, 2303910, 544272, 9728000, 5615610, 8419950

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

a(n) is the product of the terms in the n-th row of table A261575.

Conjecture: a(n) = 0 for n > 3329 (empirically checked up to 36000).

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Eric Weisstein's World of Mathematics, Sexagesimal

Wikipedia, Sexagesimal

MAPLE

a:= n-> mul(i, i=convert((<<0|1>, <1|1>>^n)[1, 2], base, 60)):

seq(a(n), n=0..44); # Alois P. Heinz, Jan 22 2022

MATHEMATICA

Apply[Times, IntegerDigits[Fibonacci[Range[0, 50]], 60], {1}] (* Paolo Xausa, Feb 19 2024 *)

PROG

(Haskell)

a261598 = product . a261575_row

(PARI) a(n) = if (n, vecprod(digits(fibonacci(n), 60)), 0); \\ Michel Marcus, Jan 22 2022

CROSSREFS

Cf. A261575, A000045, A246558, A261587.

KEYWORD

nonn,look,base

AUTHOR

Reinhard Zumkeller, Sep 09 2015

STATUS

approved

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