A261607 -id:A261607 - cp's OEIS frontend

A008963 Initial digit of Fibonacci number F(n).

0, 1, 1, 2, 3, 5, 8, 1, 2, 3, 5, 8, 1, 2, 3, 6, 9, 1, 2, 4, 6, 1, 1, 2, 4, 7, 1, 1, 3, 5, 8, 1, 2, 3, 5, 9, 1, 2, 3, 6, 1, 1, 2, 4, 7, 1, 1, 2, 4, 7, 1, 2, 3, 5, 8, 1, 2, 3, 5, 9, 1, 2, 4, 6, 1, 1, 2, 4, 7, 1, 1, 3, 4, 8, 1, 2, 3, 5, 8, 1, 2, 3, 6, 9, 1, 2, 4, 6, 1, 1, 2, 4, 7, 1, 1, 3, 5, 8, 1

Offset: 0

N. J. A. Sloane

Benford's law applies since the Fibonacci sequence is of exponential growth: P(d)=log_10(1+1/d), in fact among first 5000 values the digit d=1 appears 1505 times, while 5000*P(1) is about 1505.15. - Carmine Suriano, Feb 14 2011

Wlodarski observed and Webb proved that the distribution of terms of this sequence follows Benford's law. - Amiram Eldar, Sep 23 2019

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
William Webb, Distribution of the first digits of Fibonacci numbers, The Fibonacci Quarterly, Vol. 13, No. 4 (1975), pp. 334-336.
Wikipedia, Benford's law.
J. Wlodarski, Fibonacci and Lucas Numbers Tend to Obey Benford's Law, The Fibonacci Quarterly, Vol. 9, No. 1 (1971), pp. 87-88.
Index entries for sequences related to Benford's law.

Cf. A000045, A003893 (final digit).

Cf. A000030, A261607, A213201.

a008963 = a000030 . a000045  -- Reinhard Zumkeller, Sep 09 2015

F:= combinat[fibonacci]:
a:= n-> parse(""||(F(n))[1]):
seq(a(n), n=0..100);  # Alois P. Heinz, Nov 22 2023

Table[IntegerDigits[Fibonacci[n]][[1]], {n, 0, 100}] (* T. D. Noe, Sep 23 2011 *)

vector(10001,n,f=fibonacci(n-1);f\10^(#Str(f)-1))

a(n) = A000030(A000045(n)). - Amiram Eldar, Sep 23 2019
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{d=1..9} d*log(1+1/d)/log(10) = 3.440236... (A213201). - Amiram Eldar, Jan 12 2023
For n>5, a(n) = floor(10^{alpha*n-beta}), where alpha=log_10(phi), beta=log_10(5)/2, {x}=x-floor(x) denotes the fractional part of x, log_10(phi) = A097348, and phi = (1+sqrt(5))/2 = A001622. - Hans J. H. Tuenter, Aug 20 2025

A261575 Table of Fibonacci numbers in base-60 representation: row n contains the sexagesimal digits of A000045(n) in reversed order.

Original entry on oeis.org

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

Offset: 0

Reinhard Zumkeller, Sep 09 2015

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).

			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]

Reinhard Zumkeller, Rows n = 0..1000 of triangle, flattened
Eric Weisstein's World of Mathematics, Sexagesimal
Wikipedia, Sexagesimal

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

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   _ = []

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

A261606 a(n) = Fibonacci(n) mod 60.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 29, 24, 53, 17, 10, 27, 37, 4, 41, 45, 26, 11, 37, 48, 25, 13, 38, 51, 29, 20, 49, 9, 58, 7, 5, 12, 17, 29, 46, 15, 1, 16, 17, 33, 50, 23, 13, 36, 49, 25, 14, 39, 53, 32, 25, 57, 22, 19, 41, 0, 41, 41, 22, 3, 25, 28, 53

Offset: 0

Reinhard Zumkeller, Sep 09 2015

a(n) = A261575(n,0).

Periodic with period length 120. - Ray Chandler, Sep 23 2015

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Sexagesimal
Wikipedia, Sexagesimal

Cf. A000045, A261575, A261607, A003893.

a261606 n = a261606_list !! n
a261606_list = 0 : 1 : map (flip mod 60)
                           (zipWith (+) a261606_list $ tail a261606_list)

Mod[Fibonacci[Range[67]], 60] (* Michael De Vlieger, Jan 22 2022 *)

a(n) = fibonacci(n) % 60; \\ Michel Marcus, Jan 22 2022

a(0) = 0, a(1) = 1 and for n > 1: a(n) = (a(n-1) + a(n-2)) mod 60.

cp's OEIS Frontend

A008963 Initial digit of Fibonacci number F(n).

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A261575 Table of Fibonacci numbers in base-60 representation: row n contains the sexagesimal digits of A000045(n) in reversed order.

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A261606 a(n) = Fibonacci(n) mod 60.

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Programs

Haskell

Mathematica

PARI

Formula