A030132 -id:A030132 - cp's OEIS frontend

A004090 Sum of digits of Fibonacci numbers.

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

N. J. A. Sloane

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

T. D. Noe, Table of n, a(n) for n = 0..10000
T. D. Noe, Plot of a(n)-n for n = 0..100000

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

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

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

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

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

A004090 = lambda n: sum(int(d)for d in str(A000045(n))) # M. F. Hasler, Jul 07 2025

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

A153130 Period 6: repeat [1, 2, 4, 8, 7, 5].

Original entry on oeis.org

1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5

Offset: 0

Paul Curtz, Dec 19 2008

Digital root of 2^n.
A regular version of Pitoun's sequence: a(n) = A029898(n+1).
Also obtained from permutations of A141425, A020806, A070366, A153110, A153990, A154127, A154687, or A154815.
This sequence and its (again period 6) repeated differences produce the table:
    1,  2,  4,  8,  7,  5,  1,  2,  4,  8,  7, ...
    1,  2,  4, -1, -2, -4,  1,  2,  4, -1, -2, ...
    1,  2, -5, -1, -2,  5,  1,  2, -5, -1, -2, ...
    1, -7,  4, -1,  7, -4,  1, -7,  4, -1,  7, ...
   -8, 11, -5,  8,-11,  5, -8, 11, -5,  8,-11, ...
   19,-16, 13,-19, 16,-13, 19,-16, 13,-19, 16, ...
  -35, 29,-32, 35,-29, 32,-35, 29,-32, 35,-29, ...
   64,-61, 67,-64, 61,-67, 64,-61, 67,-64, 61, ...
If each entry of this table is read modulo 9 we obtain the very regular table:
    1,  2,  4,  8,  7,  5,  1,  2,  4,  8,  7, ...
    1,  2,  4,  8,  7,  5,  1,  2,  4,  8,  7, ...
    1,  2,  4,  8,  7,  5,  1,  2,  4,  8,  7, ...
    1,  2,  4,  8,  7,  5,  1,  2,  4,  8,  7, ...
    1,  2,  4,  8,  7,  5,  1,  2,  4,  8,  7, ...
    1,  2,  4,  8,  7,  5,  1,  2,  4,  8,  7, ...
Also the decimal expansion of the constant 125/1001. - R. J. Mathar, Jan 23 2009
Digital root of the powers of any number congruent to 2 mod 9. - Alonso del Arte, Jan 26 2014

Cecil Balmond, Number 9: The Search for the Sigma Code. Munich, New York: Prestel (1998): 203.

Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1).

Cf. A004000, A007953, A010734, A010888, A029898, A030132, A082365, A145389, A189510.
Cf. digital roots of powers of c mod 9: c = 4, A100402; c = 5, A070366; c = 7, A070403; c = 8, A010689.
Cf. A020806, A070366, A141425, A153110, A153990, A154127, A154687, A154815.

&cat [[1, 2, 4, 8, 7, 5]^^30]; // Wesley Ivan Hurt, Jul 05 2016

seq(op([1, 2, 4, 8, 7, 5]), n=0..40); # Wesley Ivan Hurt, Jul 05 2016

Flatten[Table[{1, 2, 4, 8, 7, 5}, {20}]] (* Paul Curtz, Dec 19 2008 *)
Table[Mod[2^n, 9], {n, 0, 99}] (* Alonso del Arte, Jan 26 2014 *)

a(n)=lift(Mod(2,9)^n) \\ Charles R Greathouse IV, Apr 21 2015

a(n) + a(n+3) = 9 = A010734(n).
G.f.: (1+x+2x^2+5x^3)/((1-x)(1+x)(1-x+x^2)). - R. J. Mathar, Jan 23 2009
a(n) = A082365(n) mod 9. - Paul Curtz, Mar 31 2009
a(n) = -1/2*cos(Pi*n) - 3*cos(1/3*Pi*n) - 3^(1/2)*sin(1/3*Pi*n) + 9/2. - Leonid Bedratyuk, May 13 2012
a(n) = A010888(A004000(n+1)). - Ivan N. Ianakiev, Nov 27 2014
From Wesley Ivan Hurt, Apr 20 2015: (Start)
a(n) = a(n-6) for n>5.
a(n) = a(n-1) - a(n-3) + a(n-4) for n>3.
a(n) = (2+3*(n-1 mod 3))*(n mod 2) + (1+3*(-n mod 3))*(n-1 mod 2). (End)
a(n) = 2^n mod 9. - Nikita Sadkov, Oct 06 2018
From Stefano Spezia, Mar 20 2025: (Start)
E.g.f.: 4*cosh(x) - exp(x/2)*(3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2)) + 5*sinh(x).
a(n) = A007953(2*a(n-1)) = A010888(2*a(n-1)). (End)

Edited by R. J. Mathar, Apr 09 2009

A010077 a(n) = sum of digits of a(n-1) + sum of digits of a(n-2); a(0) = 0, a(1) = 1.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 13, 12, 7, 10, 8, 9, 17, 17, 16, 15, 13, 10, 5, 6, 11, 8, 10, 9, 10, 10, 2, 3, 5, 8, 13, 12, 7, 10, 8, 9, 17, 17, 16, 15, 13, 10, 5, 6, 11, 8, 10, 9, 10, 10, 2, 3, 5, 8, 13, 12, 7, 10, 8, 9, 17, 17, 16, 15, 13, 10, 5, 6, 11, 8, 10, 9, 10, 10

Offset: 0

N. J. A. Sloane, Leonid Broukhis

The digital sum analog (in base 10) of the Fibonacci recurrence. - Hieronymus Fischer, Jun 27 2007
a(n) and Fibonacci(n) = A000045(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 the Pisano period A001175(9)=24. - Hieronymus Fischer, Jun 27 2007
a(n) == A004090(n) (mod 9) (A004090(n) = digital sum of Fibonacci(n)). - Hieronymus Fischer, Jun 27 2007
For general bases p > 2, we have the inequality 2 <= a(n) <= 2p-3 (for n > 2). Actually, a(n) <= 17 = A131319(10) for the base p=10. - Hieronymus Fischer, Jun 27 2007

David A. Corneth, Table of n, a(n) for n = 0..9999
Index entries for Colombian or self numbers and related sequences

Cf. A007953, A007612, A065076.
Cf. A000045, A010073, A010074, A010075, A010076.
Cf. A131294, A131295, A131296, A131297, A131318, A131319, A131320.

a[0] = 0; a[1] = 1; a[n_] := a[n] = Apply[ Plus, IntegerDigits[ a[n - 1] ]] + Apply[ Plus, IntegerDigits[ a[n - 2] ]]; Table[ a[n], {n, 0, 100} ]
nxt[{a_,b_}]:={b, Total[IntegerDigits[a]]+Total[IntegerDigits[b]]}; NestList[ nxt,{0,1},80][[All,1]] (* Harvey P. Dale, Apr 15 2018 *)

first(n) = {n = max(n, 2); my(res = vector(n)); res[2] = 1; for(i = 3, n, res[i] = sumdigits(res[i-1]) + sumdigits(res[i-2]) ); res } \\ David A. Corneth, May 26 2021

Periodic from n=3 with period 24. - Franklin T. Adams-Watters, Mar 13 2006
a(n) = A030132(n-4) + A030132(n-3) for n>3. - Reinhard Zumkeller, Jul 04 2007
a(n) = a(n-1) + a(n-2) - 9*(floor(a(n-1)/10) + floor(a(n-2)/10)). - Hieronymus Fischer, Jun 27 2007
a(n) = floor(a(n-1)/10) + floor(a(n-2)/10) + (a(n-1) mod 10) + (a(n-2) mod 10). - Hieronymus Fischer, Jun 27 2007
a(n) = A059995(a(n-1)) + A059995(a(n-2)) + A010879(a(n-1)) + A010879(a(n-2)). - Hieronymus Fischer, Jun 27 2007
a(n) = Fibonacci(n) - 9*Sum_{k=2..n-1} Fibonacci(n-k+1)*floor(a(k)/10) where Fibonacci(n) = A000045(n). - Hieronymus Fischer, Jun 27 2007

A017641 a(n) = 12*n + 10.

Original entry on oeis.org

10, 22, 34, 46, 58, 70, 82, 94, 106, 118, 130, 142, 154, 166, 178, 190, 202, 214, 226, 238, 250, 262, 274, 286, 298, 310, 322, 334, 346, 358, 370, 382, 394, 406, 418, 430, 442, 454, 466, 478, 490, 502, 514, 526, 538, 550, 562, 574, 586, 598, 610, 622, 634

Offset: 0

N. J. A. Sloane

Exponents e such that x^e + x^2 - 1 is reducible.

If Y is a 4-subset of an (2n+1)-set X then, for n>=3, a(n-2) is the number of 3-subsets of X having at least two elements in common with Y. - Milan Janjic, Dec 16 2007

Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008594, A016969, A017533, A017545, A017593, A030132.

Range[10, 1000, 12] (* Vladimir Joseph Stephan Orlovsky, May 29 2011 *)

a(n)=12*n+10 \\ Charles R Greathouse IV, Jul 10 2016

A030132(a(n)) = 9. - Reinhard Zumkeller, Jul 04 2007
G.f.: 2*(5 + x)/(1 - x)^2. - Stefano Spezia, May 09 2021
Sum_{n>=0} (-1)^n/a(n) = Pi/12 - sqrt(3)*log(2 + sqrt(3))/12. - Amiram Eldar, Dec 12 2021
From Elmo R. Oliveira, Apr 04 2025: (Start)
E.g.f.: 2*exp(x)*(5 + 6*x).
a(n) = 2*A016969(n).
a(n) = 2*a(n-1) - a(n-2) for n >= 2. (End)

A030133 a(n+1) is the sum of digits of (a(n) + a(n-1)).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

a(n) = A010888(A000032(n)). - Reinhard Zumkeller, Aug 20 2011
Similar to the digital roots of several Fibonacci sequences, this digital root sequence for Lucas numbers (A000032) has period 24 with digits summing to 117.
Decimal expansion of 23719213606865169775282 / 111111111111111111111111 = 0.[213472922461786527977538] (periodic). - Daniel Forgues, Feb 27 2017

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A030132, A007953, A049341.

a030133 n = a030133_list !! n
a030133_list =
   2 : 1 : map a007953 (zipWith (+) a030133_list $ tail a030133_list)
-- Reinhard Zumkeller, Aug 20 2011

Transpose[NestList[{Last[#],Total[IntegerDigits[Total[#]]]}&, {2,1}, 100]] [[1]] (* Harvey P. Dale, Jul 25 2011 *)

V=[2,1];for(n=1,100,V=concat(V,sumdigits(V[n]+V[n+1])));V \\ Derek Orr, Feb 27 2017

Vec((2 + x + 3*x^2 + 4*x^3 + 7*x^4 + 2*x^5 + 9*x^6 + 2*x^7 + 2*x^8 + 4*x^9 + 6*x^10 + x^11 + 7*x^12 + 8*x^13 + 6*x^14 + 5*x^15 + 2*x^16 + 7*x^17 + 9*x^18 + 7*x^19 + 7*x^20 + 5*x^21 + 3*x^22 + 8*x^23) / (1 - x^24)  + O(x^80)) \\ Colin Barker, Sep 25 2019

a(n+24) = a(n); a(A017593(n)) = 9. - Reinhard Zumkeller, Jul 04 2007

G.f.: (2 + x + 3*x^2 + 4*x^3 + 7*x^4 + 2*x^5 + 9*x^6 + 2*x^7 + 2*x^8 + 4*x^9 + 6*x^10 + x^11 + 7*x^12 + 8*x^13 + 6*x^14 + 5*x^15 + 2*x^16 + 7*x^17 + 9*x^18 + 7*x^19 + 7*x^20 + 5*x^21 + 3*x^22 + 8*x^23) / (1 - x^24). - Colin Barker, Sep 25 2019

A065076 a(0) = 0, a(1) = 1, a(n) = (sum of digits of a(n-1)) + a(n-2).

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 13, 12, 16, 19, 26, 27, 35, 35, 43, 42, 49, 55, 59, 69, 74, 80, 82, 90, 91, 100, 92, 111, 95, 125, 103, 129, 115, 136, 125, 144, 134, 152, 142, 159, 157, 172, 167, 186, 182, 197, 199, 216, 208, 226, 218, 237, 230, 242, 238, 255, 250, 262, 260, 270

Offset: 0

Bodo Zinser, Nov 09 2001

			a(8) = 12 because a(7) = 13, a(6) = 8 and 4 = 1+3 and 12 = 4 + 8.

Harry J. Smith, Table of n, a(n) for n = 0..1000

Cf. A007953 and A007612.

Cf. A030132.

a065076 n = a065076_list !! n
a065076_list = 0 : 1 : zipWith (+)
                a065076_list (map a007953 $ tail a065076_list)
-- Reinhard Zumkeller, Nov 13 2014

a[0] = 0; a[1] = 1; a[n_] := a[n] = Apply[ Plus, IntegerDigits[ a[n - 1] ]] + a[n - 2]; Table[ a[n], {n, 0, 100} ]
Transpose[NestList[{Last[#],Total[IntegerDigits[Last[#]]]+First[#]}&, {0,1}, 60]][[1]] (* Harvey P. Dale, Dec 07 2011 *)

{ my(a,a1,a2); for (n=0, 60, if (n>1, a=sumdigits(a1) + a2; a2=a1; a1=a, if (n, a=a1=1, a=a2=0)); print1(a, ", ") ) } \\ Harry J. Smith, Oct 06 2009

More terms from Larry Reeves (larryr(AT)acm.org) and Robert G. Wilson v, Nov 13 2001

A216676 Digital roots of squares of Fibonacci numbers.

Original entry on oeis.org

1, 1, 4, 9, 7, 1, 7, 9, 4, 1, 1, 9, 1, 1, 4, 9, 7, 1, 7, 9, 4, 1, 1, 9, 1, 1, 4, 9, 7, 1, 7, 9, 4, 1, 1, 9, 1, 1, 4, 9, 7, 1, 7, 9, 4, 1, 1, 9, 1, 1, 4, 9, 7, 1, 7, 9, 4, 1, 1, 9, 1, 1, 4, 9, 7, 1, 7, 9, 4, 1, 1, 9, 1, 1, 4, 9, 7, 1, 7, 9, 4, 1, 1, 9, 1, 1, 4

Offset: 1

Ravi Bhandari, Sep 14 2012

The first 11 terms are symmetric about 6th term. The first 23 terms are symmetric about 12th term. We can generalize this as follows: the first (2n-1) terms are symmetric about n-th term.
The sequence appears to be periodic with period-length 12. - John W. Layman,  Sep 14 2012
The Fibonacci numbers are periodic modulo any integer. The digital roots of the Fibonacci numbers are given by A030132, a sequence with a period length of 24. Squaring gives {1, 1, 4, 9, 7, 1, 7, 9, 4, 1, 1, 9, 1, 1, 4, 9, 7, 1, 7, 9, 4, 1, 1, 9}, which is a sequence of twelve numbers given twice. Therefore, the previous comment is correct. - Alonso del Arte, Sep 25 2012

			a(7) = 7 because F(7) = 13 and 13^2 = 169, with digits adding up to 16, the digital root is therefore 7.

Andrew Howroyd, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (0,1,0,0,0,-1,0,1).

a = {}; For[n = 1, n <= 100, n++, {fn2 = Fibonacci[n]^2; d = IntegerDigits[fn2]; While[Length[d] > 1, d = IntegerDigits[Total[d]]]; AppendTo[a, d[[1]]] }]; a (* John W. Layman,  Sep 14 2012 *)
ReplaceAll[Table[Mod[Fibonacci[n]^2, 9], {n, 72}], {0 -> 9}] (* Alonso del Arte, Sep 23 2012 *)

fibmod(n, m)=((Mod([1, 1; 1, 0], m))^n)[1, 2]
a(n)=lift(fibmod(n,9)^2-1)+1 \\ Charles R Greathouse IV, Jun 20 2017

a(n) = A010888(A007598(n)).

G.f. ( -1-x-3*x^2-8*x^3-3*x^4+8*x^5-9*x^7-x^6 ) / ( (x-1) *(1+x) *(x^2+1) *(x^4-x^2+1) ). - R. J. Mathar, Sep 15 2012

Terms a(25)-a(72) by John W. Layman, Sep 14 2012

Terms a(73) and beyond from Andrew Howroyd, Feb 25 2018

A049341 a(n+1) = sum of digits of a(n) + a(n-1).

Original entry on oeis.org

3, 6, 9, 6, 6, 3, 9, 3, 3, 6, 9, 6, 6, 3, 9, 3, 3, 6, 9, 6, 6, 3, 9, 3, 3, 6, 9, 6, 6, 3, 9, 3, 3, 6, 9, 6, 6, 3, 9, 3, 3, 6, 9, 6, 6, 3, 9, 3, 3, 6, 9, 6, 6, 3, 9, 3, 3, 6, 9, 6, 6, 3, 9, 3, 3, 6, 9, 6, 6, 3, 9, 3, 3, 6, 9, 6, 6, 3, 9, 3, 3, 6, 9, 6, 6, 3, 9, 3, 3, 6, 9, 6, 6, 3, 9, 3, 3, 6, 9, 6, 6, 3, 9, 3, 3, 6, 9, 6, 6, 3, 9, 3

Offset: 0

Damir Olejar

a(n+1) = A007953(a(n) + a(n-1)) for n > 0.

			After 6,9 we get 6+9 = 15 -> 1+5 = 6.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Michael Gilleland, Some Self-Similar Integer Sequences
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 1).

Cf. A030132, A030133, A049342.

a049341 n =  a030132_list !! n
a049341_list =
   3 : 6 : map a007953 (zipWith (+) a049341_list $ tail a049341_list)
-- Reinhard Zumkeller, Aug 20 2011

LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 1},{3, 6, 9, 6, 6, 3, 9, 3},112] (* Ray Chandler, Aug 27 2015 *)

Period 8.

Definition improved by Reinhard Zumkeller, Aug 20 2011

A112661 Sum of digits of sum of previous 3 terms.

Original entry on oeis.org

1, 1, 1, 3, 5, 9, 8, 4, 3, 6, 4, 4, 5, 4, 4, 4, 3, 2, 9, 5, 7, 3, 6, 7, 7, 2, 7, 7, 7, 3, 8, 9, 2, 10, 3, 6, 10, 10, 8, 10, 10, 10, 3, 5, 9, 8, 4, 3, 6, 4, 4, 5, 4, 4, 4, 3, 2, 9, 5, 7, 3, 6, 7, 7, 2, 7, 7, 7, 3, 8, 9, 2, 10, 3, 6, 10, 10, 8, 10, 10, 10, 3, 5, 9, 8, 4, 3, 6, 4, 4, 5, 4, 4, 4

Offset: 0

Jonathan Vos Post and Andrew Carmichael Post (andrewpost(AT)gmail.com), Dec 29 2005

Sum of digits, not iterated (i.e., not digital sum, reducing to a single digit) as we twice get a term of 10 which we do not reduce to 1. This is to tribonacci (A000073) as A030132 is to Fibonacci (A000045). This sequence has a preamble of 3 terms (1, 1, 1), then enters a cycle of length 39 (ending with 10, 10, 10).

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A000073, A004090, A007953, A010888, A030132.

a[0] = a[1] = a[2] = 1; a[n_] := a[n] = Total@ IntegerDigits[a[n-1] + a[n-2] + a[n-3]]; a /@ Range[0, 93] (* Giovanni Resta, Jun 17 2016 *)

a(n+2) = sum of digits of (a(n) + a(n-1) + a(n-2)). a(n+2) = A007953(a(n) + a(n-1) + a(n-2)).

Data and name corrected by Giovanni Resta, Jun 17 2016

A189510 Digital root of n^n.

Original entry on oeis.org

1, 1, 4, 9, 4, 2, 9, 7, 1, 9, 1, 5, 9, 4, 7, 9, 7, 8, 9, 1, 4, 9, 4, 2, 9, 7, 1, 9, 1, 5, 9, 4, 7, 9, 7, 8, 9, 1, 4, 9, 4, 2, 9, 7, 1, 9, 1, 5, 9, 4, 7, 9, 7, 8, 9, 1, 4, 9, 4, 2, 9, 7, 1, 9, 1, 5, 9, 4, 7, 9, 7, 8, 9, 1, 4, 9, 4, 2, 9, 7, 1, 9, 1, 5, 9, 4, 7

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 02 2011

a(n) = A010888(A000312(n)).

For n >= 1, this sequence is periodic with period 18. The sequence repeats [1,4,9,4,2,9,7,1,9,1,5,9,4,7,9,7,8,9]. - Nathaniel Johnston, May 04 2011

Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

Cf. A010888, A030132, A145389.

A189510 := proc(n) return ((n^n-1) mod 9) + 1: end: seq(A189510(n), n=0..80); # Nathaniel Johnston, May 04 2011

digitalRoot[n_Integer?Positive] := FixedPoint[Plus@@IntegerDigits[#]&,n]; Table[If[n==0,0,digitalRoot[n^n]], {n,0,200}]
Join[{1},LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1},{1, 4, 9, 4, 2, 9, 7, 1, 9, 1, 5, 9, 4, 7, 9, 7, 8, 9},86]] (* Ray Chandler, Aug 27 2015 *)
PadRight[{1},100,{9,1,4,9,4,2,9,7,1,9,1,5,9,4,7,9,7,8}] (* Harvey P. Dale, Jul 31 2025 *)

def A189510(n): return (9,1,4,9,4,2,9,7,1,9,1,5,9,4,7,9,7,8)[n%18] if n else 1 # Chai Wah Wu, Feb 09 2023

From Chai Wah Wu, Feb 09 2023: (Start)
a(n) = a(n-18) for n > 18.
G.f.: (-8*x^18 - 8*x^17 - 7*x^16 - 9*x^15 - 7*x^14 - 4*x^13 - 9*x^12 - 5*x^11 - x^10 - 9*x^9 - x^8 - 7*x^7 - 9*x^6 - 2*x^5 - 4*x^4 - 9*x^3 - 4*x^2 - x - 1)/(x^18 - 1). (End)

a(0) corrected by Reinhard Zumkeller, May 03 2011

cp's OEIS Frontend

A004090 Sum of digits of Fibonacci numbers.

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A153130 Period 6: repeat [1, 2, 4, 8, 7, 5].

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A010077 a(n) = sum of digits of a(n-1) + sum of digits of a(n-2); a(0) = 0, a(1) = 1.

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A017641 a(n) = 12*n + 10.

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A030133 a(n+1) is the sum of digits of (a(n) + a(n-1)).

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A065076 a(0) = 0, a(1) = 1, a(n) = (sum of digits of a(n-1)) + a(n-2).

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A216676 Digital roots of squares of Fibonacci numbers.

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