A007887 -id:A007887 - 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

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

A137290 Fibonacci(n) mod 30.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 13, 21, 4, 25, 29, 24, 23, 17, 10, 27, 7, 4, 11, 15, 26, 11, 7, 18, 25, 13, 8, 21, 29, 20, 19, 9, 28, 7, 5, 12, 17, 29, 16, 15, 1, 16, 17, 3, 20, 23, 13, 6, 19, 25, 14, 9, 23, 2, 25, 27, 22, 19, 11, 0, 11, 11, 22, 3, 25, 28, 23, 21, 14, 5, 19, 24, 13, 7, 20, 27, 17, 14

Offset: 1

Aaron M. Churchill (churchil(AT)math.udel.edu), Mar 15 2008

nonn

Has period 120.

Michel Marcus, Table of n, a(n) for n = 1..130
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, 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, 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, 0, 0, 0, 0, 0, 1).

Cf. A000045, A003893, A007887, A079345, A105471.

Mod[Fibonacci[Range[80]],30] (* Harvey P. Dale, Sep 12 2022 *)

a(n) = fibonacci(n) % 30 \\ Michel Marcus, Jun 12 2013

A160136 Lodumo_9 of Fibonacci numbers.

Original entry on oeis.org

0, 1, 10, 2, 3, 5, 8, 4, 12, 7, 19, 17, 9, 26, 35, 16, 6, 13, 28, 14, 15, 11, 44, 37, 18, 46, 55, 20, 21, 23, 53, 22, 30, 25, 64, 62, 27, 71, 80, 34, 24, 31, 73, 32, 33, 29, 89, 82, 36, 91, 100, 38, 39, 41, 98, 40, 48, 43, 109, 107, 45, 116, 125, 52, 42, 49, 118, 50, 51, 47, 134

Offset: 0

Philippe Deléham, May 02 2009

Permutation of nonnegative integers.
From Michael De Vlieger, Jan 21 2021: (Start)
The plot is governed by A001175(9) = 24 and is bifurcated into two trajectories that repeat a "constellation" of points we label "red" and "blue" so as to match the linked figures. We might group the terms in a(n) into two classes as to their residue r (mod 24). The red terms have n = r (mod 24) for r in {1, 2, 6, 10, 11, 13, 14, 18, 22, 23}, while the blue terms have r in {0, 3, 4, 5, 7, 8, 9, 12, 15, 16, 17, 19, 20, 21}.
There are 10 residues in the red constellation, and 14 residues in the blue constellation.
For red, we have the displacement a(n + 24) - a(n) = 45, thus the slope m_red = 15/8. For blue, we have the displacement a(n + 24) - a(n) = 18, thus the slope m_blue = 3/4.(End)

Michael De Vlieger, Table of n, a(n) for n = 0..9999
Michael De Vlieger, Plot (n, a(n)) for 1 <= n <=144 illustrating bifurcation into two rays color coded red and blue, and the effect of the Pisano number (mod 9) = 24.
Michael De Vlieger, Plot (n, a(n)) for 1 <= n <= 24 with the 2 rays color coded red and blue, with coordinates of points labeled.
OEIS wiki, Lodumo transform
Index entries for sequences that are permutations of the natural numbers
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1).

Cf. A000045, A007887.

Block[{m = 9, s = Fibonacci[Range[120]]}, Nest[Append[#1, Block[{k = 1}, While[Nand[Mod[k, m] == Mod[s[[#2]], m], FreeQ[#1, k]], k++]; k]] & @@ {#, Length@ #} &, {0}, 120]] (* Michael De Vlieger, Jan 21 2021 *)

a(n) = lod_9(A000045(n)).
a(n) = 2*a(n-24) - a(n-48) for n >= 48. - Philippe Deléham, Mar 09 2023
a(n) = a(n-12) + a(n-24) - a(n-36) for n >= 36. - Ray Chandler, Sep 10 2023

Replaced second 18 by 118 - R. J. Mathar, May 03 2009

A281619 Integer values of (A000045^2-1)/9 where A000045(m) is the m-th Fibonacci number.

Original entry on oeis.org

0, 7, 336, 880, 6032, 15792, 741895, 34853280, 91247072, 625416736, 1637362272, 76921173511, 3613657792752, 9460678925136, 64844458022832, 169764995085840, 7975341111241735, 374671267233275712, 980902112224710592, 6723203096097857600, 17601574218852716736, 826899317018844410887

Offset: 1

Michel Marcus, Jan 25 2017

Also the integer values of A080097/9 where A080097(m) = Fibonacci(n+2)^2 - 1.

The indices of the Fibonacci numbers are 1, 2, 6, 10, and 11 mod 12. See the Wulczy link.

Robert Israel, Table of n, a(n) for n = 1..997
G. Wulczy, Unity with Fibonacci, Problem H-247 and solution, Fib. Quarter. p. 89_90, Vol 15, 1, Feb. 1977.

Cf. A000045, A007887, A080097.

seq(seq((combinat:-fibonacci(12*m+j)^2-1)/9,j=[2,6,10,11,13]),m=0..20); # Robert Israel, Mar 05 2017

Select[(#^2-1)/9&/@Fibonacci[Range[100]],IntegerQ] (* Harvey P. Dale, Feb 07 2017 *)

lista(nn)=v = [1, 2, 6, 10, 11, 13, 14, 18, 22, 23]; for (n=1, nn, j = (n % #v) +1; k = n\#v; print1((fibonacci(24*k+v[j])^2-1)/9, ", "););

a(n)=fibonacci(n\5*12+[1, 2, 6, 10, 11][n%5+1])^2\9 \\ Charles R Greathouse IV, Jan 26 2017

G.f.: x^2*(7*x^10+336*x^9+880*x^8+6032*x^7+15792*x^6+16114*x^5+15792*x^4

+6032*x^3+880*x^2+336*x+7)/(-x^15+103683*x^10-103683*x^5+1). - Robert Israel, Mar 05 2017

A362358 Alternating sum of digits of the Fibonacci numbers, with a plus sign for the last digit.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 2, -1, 1, 0, 1, 1, 2, 3, 5, 8, 2, -1, -10, 0, 12, 1, 2, 3, 5, -3, 13, -1, 1, -11, 1, 12, 2, 3, 5, -3, 13, 10, 1, 0, 1, -10, 13, 3, -17, 19, -9, 10, 1, 0, 1, 12, 13, 3, -6, 8, 2, -1, -10, 0, 1, 12, -9, 3, 5, -3, -9, -23, 1, -22, 34, -10, 2

Offset: 0

Wolfdieter Lang, May 26 2023

a(n) mod 11 = F(n) mod 11 = A105955(n). This is the mod 11 rule applied to F(n) = A000045.

			F(17) = 1597, s(n) = 4 - 1 = 3, a(17) = 7 - 9 + 5 - 1 = 2.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000045, A004090, A007887, A055017, A060384, A105955.

a[n_]:=Sum[(-1)^(IntegerLength[Fibonacci[n]]-i) Part[IntegerDigits[Fibonacci[n]],i],{i,IntegerLength[Fibonacci[n]]}]; Array[a,66,0] (* Stefano Spezia, May 27 2023 *)

a(n) = my(d=Vecrev(digits(fibonacci(n)))); sum(k=1, #d, (-1)^(k+1)*d[k]); \\ Michel Marcus, May 28 2023

Let [f_s(n), f_{s(n)-1}, ..., f_0] be the list of digits of F(n) = A000040(n) with s(n) = A060384(n) - 1, then a(n) = Sum_{j=0..s(n)} (-1)^j*f_j.

a(n) = A055017(A000045(n)), for n >= 0.

A110388 a(n) = F(n)*F(n+1) mod 9, where F(n) = n-th Fibonacci number.

Original entry on oeis.org

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

Offset: 1

Amarnath Murthy, Jul 27 2005

			a(5) = 5*8 mod 9 = 4.

Cf. A007887, A001654.

with(combinat): a:=n->fibonacci(n)*fibonacci(n+1) mod 9: seq(a(n),n=1..130); # Emeric Deutsch, Jul 31 2005

More terms from Emeric Deutsch, Jul 31 2005

cp's OEIS Frontend

A004090 Sum of digits of Fibonacci numbers.

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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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A137290 Fibonacci(n) mod 30.

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A160136 Lodumo_9 of Fibonacci numbers.

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A281619 Integer values of (A000045^2-1)/9 where A000045(m) is the m-th Fibonacci number.

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A362358 Alternating sum of digits of the Fibonacci numbers, with a plus sign for the last digit.

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A110388 a(n) = F(n)*F(n+1) mod 9, where F(n) = n-th Fibonacci number.

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