A216676 -id:A216676 - cp's OEIS frontend

A216699 Digital root of cubes of Fibonacci numbers.

0, 1, 1, 8, 9, 8, 8, 1, 9, 1, 1, 8, 9, 8, 8, 1, 9, 1, 1, 8, 9, 8, 8, 1, 9, 1, 1, 8, 9, 8, 8, 1, 9, 1, 1, 8, 9, 8, 8, 1, 9, 1, 1, 8, 9, 8, 8, 1, 9, 1, 1, 8, 9, 8, 8, 1, 9, 1, 1, 8, 9, 8, 8, 1, 9, 1, 1, 8, 9, 8, 8, 1, 9, 1, 1, 8, 9, 8, 8, 1, 9, 1, 1, 8, 9, 8, 8

Offset: 0

Ravi Bhandari, Sep 15 2012

This sequence repeats after every 8 terms, hence this is periodic with period 8.

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

Cf. A004090, A056570, A216676.

Table[NestWhile[Total[IntegerDigits[#]] &, Fibonacci[n]^3, # > 9 &], {n, 0, 86}] (* T. D. Noe, Oct 15 2012 *)
Join[{0},LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 1},{1, 1, 8, 9, 8, 8, 1, 9},86]] (* Ray Chandler, Aug 25 2015 *)

A216754 Digital root of fourth power of Fibonacci numbers.

Original entry on oeis.org

0, 1, 1, 7, 9, 4, 1, 4, 9, 7, 1, 1, 9, 1, 1, 7, 9, 4, 1, 4, 9, 7, 1, 1, 9, 1, 1, 7, 9, 4, 1, 4, 9, 7, 1, 1, 9, 1, 1, 7, 9, 4, 1, 4, 9, 7, 1, 1, 9, 1, 1, 7, 9, 4, 1, 4, 9, 7, 1, 1, 9, 1, 1, 7, 9, 4, 1, 4, 9, 7, 1, 1, 9, 1, 1, 7, 9, 4, 1, 4, 9, 7, 1, 1, 9, 1, 1, 7, 9, 4, 1, 4, 9, 7, 1, 1, 9, 1, 1, 7, 9, 4, 1, 4, 9, 7, 1, 1, 9

Offset: 0

Ravi Bhandari, Sep 15 2012

This sequence is periodic with period 12. Also, the first (2n - 1) terms are symmetric about n-th term, where n = 6k, k = 1, 2, 3, ...

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

Cf. A004090, A216676, A216699.

Table[NestWhile[Total[IntegerDigits[#]] &, Fibonacci[n]^4, # > 9 &], {n, 0, 86}] (* T. D. Noe, Oct 15 2012 *)
Join[{0},LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1},{1, 1, 7, 9, 4, 1, 4, 9, 7, 1, 1, 9},108]] (* Ray Chandler, Aug 27 2015 *)

Extended by Ray Chandler, Aug 27 2015

A216755 Digital root of the fifth power of Fibonacci(n).

Original entry on oeis.org

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

Offset: 1

Ravi Bhandari, Sep 15 2012

This sequence is periodic with period 24, i.e. gcd(period of digital roots of squares of Fibonacci, period of digital roots of cubes of Fibonacci)

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

Cf. A030132, A216676, A216699

(* First run program for A211821 to define digitalRoot *) Table[digitalRoot[Fibonacci[n]^5], {n, 90}] (* Alonso del Arte, Sep 15 2012 *)
LinearRecurrence[{0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1},{1, 1, 5, 9, 2, 8, 7, 9, 4, 1, 8, 9, 8, 8, 4, 9},100] (* Ray Chandler, Aug 27 2015 *)

a(n) = A010888(A056572(n)).
a(n) = a(n-4) - a(n-12) + a(n-16). - R. J. Mathar, Sep 15 2012
G.f. x*( -1-x-5*x^2-9*x^3-x^4-7*x^5-2*x^6-2*x^8+7*x^9-x^10-5*x^12-8*x^13-x^14-9*x^15 ) / ( (x-1) *(1+x) *(x^2+1) *(x^4+1) *(x^8-x^4+1) ). - R. J. Mathar, Sep 15 2012

A235944 Digital roots of squares of Lucas numbers.

Original entry on oeis.org

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

Offset: 0

Colin Barker, Jan 17 2014

The sequence is periodic with period 12.

			a(5)=4 because A000032[5]=11 and the digital root of 11*11 = 121 is 4.

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

Cf. A000032, A001254, A216676.

LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1},{4, 1, 9, 7, 4, 4, 9, 4, 4, 7, 9, 1},108] (* Ray Chandler, Aug 27 2015 *)
PadRight[{},120,{4,1,9,7,4,4,9,4,4,7,9,1}] (* Harvey P. Dale, Feb 18 2018 *)

Vec(-(x^11+9*x^10+7*x^9+4*x^8+4*x^7+9*x^6+4*x^5+4*x^4+7*x^3+9*x^2+x+4)/(x^12-1) + O(x^100))

a(n) = A010888(A001254(n)).
a(n) = a(n-12).
G.f.: -(x^11 +9*x^10 +7*x^9 +4*x^8 +4*x^7 +9*x^6 +4*x^5 +4*x^4 +7*x^3 +9*x^2 +x +4) / (x^12 -1).

Extended by Ray Chandler, Aug 27 2015

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A216699 Digital root of cubes of Fibonacci numbers.

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A216754 Digital root of fourth power of Fibonacci numbers.

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A216755 Digital root of the fifth power of Fibonacci(n).

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A235944 Digital roots of squares of Lucas numbers.

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