cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A004090 Sum of digits of Fibonacci numbers.

Original entry on oeis.org

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
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

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

Crossrefs

Cf. A000045 (Fibonacci), A007953 (digit sum), A030132 (digital root of A45), A010888 (digital root), A246558, A261587, A068500.

Programs

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
A010888(a(n)) = A030132(n) == a(n) (mod 9). - M. F. Hasler, Jul 07 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

Views

Author

Reinhard Zumkeller, Sep 09 2015

Keywords

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).

Examples

			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]

Links

Crossrefs

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

Programs

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

A261598 Product of sexagesimal digits of Fibonacci numbers in base-60 representation.

Original entry on oeis.org

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
Offset: 0

Views

Author

Reinhard Zumkeller, Sep 09 2015

Keywords

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

Crossrefs

Cf. A261575, A000045, A246558, A261587.

Programs

  • Haskell
    a261598 = product . a261575_row
  • 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 *)
  • PARI
    a(n) = if (n, vecprod(digits(fibonacci(n), 60)), 0); \\ Michel Marcus, Jan 22 2022
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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