A029898 -id:A029898 - cp's OEIS frontend

A210456 Period of the sequence of the digital roots of Fibonacci n-step numbers.

1, 24, 39, 78, 312, 2184, 1092, 240, 273, 26232, 11553, 9840, 177144, 14348904, 21523359, 10315734, 48417720, 16120104, 15706236, 5036466318, 258149112, 1162261464, 141214768239, 421900912158, 8857200, 2184, 2271, 28578504864, 21938847432216, 148698308091840

Offset: 1

Paolo P. Lava, Robert G. Wilson v, Jan 21 2013

More precisely, start with 0,0,...,0,1 (with n-1 0's and a single 1); thereafter the next term is the digital root (A010888) of the sum of the previous n terms. This is a periodic sequence and a(n) is the length of the period.
Theorem: a(n) <= 9^n.
Conjecture: All entries >1 are divisible by 3.
Additional terms are a(242)=177144, a(243)=177879.
More: a(728)=1594320, a(729)=1596513, a(2186)=14348904, a(2187)=14355471, a(6560)=129140160, a(6561)=129159849, a(19682)=1162261464, a(19683)=1162320519. - Hans Havermann, Jan 30 2013, Feb 08 2013
The modulus-9 Pisano periods of Fibonacci numbers, k-th order sequences. - Hans Havermann, Feb 10 2013
Conjecture: a(3^n-1)=3^(2*n+1)-3, a(3^n)=3^(2*n+1)+3^(n+1)+3 - Fred W. Helenius (fredh(AT)ix.netcom.com), posting to MathFun, Feb 21 2013

			Digital roots of Fibonacci numbers (A030132) are 0, 1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1, 9, 1, 1, 2, 3,... Thus the period is 24 (1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1, 9).

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..100
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number
Eric Weisstein's World of Mathematics, Pisano Period

Cf. Fibonacci numbers, k-th order sequences, A000045 (Fibonacci numbers, k=2), A030132 (digital root, k=2), A001175 (Pisano periods, k=2), A000073 (tribonacci numbers, k=3), A222407 (digital roots, k=3), A046738 (Pisano periods, k=3), A029898 (Pitoun's sequence), A187772, A220555.

Cf. also A010888.

A210456:=proc(q,i)
local d,k,n,v;
v:=array(1..q);
for d from 1 to i do
  for n from 1 to d do v[n]:=0; od; v[d+1]:=1;
  for n from d+2 to q do v[n]:=1+((add(v[k],k=n-d-1..n-1)-1) mod 9);
    if add(v[k],k=n-d+1..n)=9*d and v[n-d]=1 then print(n-d); break;
fi; od; od; end:
A210456 (100000000,100);

f[n_] := f[n] = Block[{s = PadLeft[{1}, n], c = 1}, s = t = Nest[g, s, n]; While[t = g[t]; s != t, c++]; c]; g[lst_List] := Rest@Append[lst, 1 + Mod[-1 + Plus @@ lst, 9]]; Do[ Print[{n, f[n] // Timing}], {n, 100}]

a(23) from Hans Havermann, Jan 30 2013
a(24) from Hans Havermann, Feb 18 2013
a(28) from Robert G. Wilson v, Feb 21 2013
a(29)-a(30) from Hiroaki Yamanouchi, May 04 2015

A086355 Fixed point if [nonzero-digit product]-function at initial-value=prime(n) is iterated.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Jul 21 2003

			n=100, prime(100)=541, iteration list={541,20,2}, a(100)=2.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A051801, A051802, A086353-A086361, A029898, A010888, A038194.

prd[x_] := Apply[Times, DeleteCases[IntegerDigits[x], 0]]; Table[FixedPoint[prd, Prime[w]], {w, 1, 128}]

a(n) = A051802(A000040(n)) = fixed-point of A051801(n-th prime).

A133390 Period 18: repeat 1, 4, 7, 2, 2, 5, 4, 1, 1, 8, 5, 2, 7, 7, 4, 5, 8, 8.

Original entry on oeis.org

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

Offset: 0

Paul Curtz, Nov 23 2007

Disordered, three times 1, 2, 4, 5, 7, 8.

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

PadRight[{},18*5,{1,4,7,2,2,5,4,1,1,8,5,2,7,7,4,5,8,8}] (* Harvey P. Dale, Nov 06 2011 *)

for(i=1,9,print1("1, 4, 7, 2, 2, 5, 4, 1, 1, 8, 5, 2, 7, 7, 4, 5, 8, 8, ")) \\ Charles R Greathouse IV, Jun 02 2011

Sum of digits mod 9 of 1, 4, 7, 11, 20, A130625. Digits of 1/7, A020806 or A029898, 1, 1, 2, 4, 8.

A154529 A090040 mod 9.

Original entry on oeis.org

1, 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

Offset: 0

Paul Curtz, Jan 11 2009

For n>2, equal to 2^(n-2) mod 9 [From Michael B. Porter, Feb 02 2010]

Apart from leading terms the same as A146501, A153130 and A029898. [From R. J. Mathar, Apr 13 2010]

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

Join[{1},LinearRecurrence[{1,0,-1,1},{5,1,2,4},101]] (* Ray Chandler, Jul 15 2015 *)

a(n)=a(n-1)-a(n-3)+a(n-4), n>4. G.f.: (6*x^4+2*x^3+4*x+1-4*x^2)/((1-x)*(1+x)*(x^2-x+1)). [From R. J. Mathar, Feb 25 2009]

Edited by N. J. A. Sloane, Jan 12 2009

Extended by Ray Chandler, Jul 15 2015

A227430 Expansion of x^2(1-x)^3/((1-2x)(1-x+x^2)(1-3*x+3x^2)).

Original entry on oeis.org

0, 0, 1, 3, 6, 10, 15, 21, 29, 45, 90, 220, 561, 1365, 3095, 6555, 13110, 25126, 46971, 87381, 164921, 320001, 640002, 1309528, 2707629, 5592405, 11450531, 23166783, 46333566, 91869970, 181348455, 357913941, 708653429, 1410132405, 2820264810, 5662052980

Offset: 0

Paul Curtz, Jul 11 2013

Consider the binomial transform of 0, 0, 0, 0, 0, 1 (period 6) with its differences:
0, 0, 0, 0, 0,  1,  6, 21, 56, 126,...    d(n): after 0, it is A192080.
0, 0, 0, 0, 1,  5, 15, 35, 70, 126,...    e(n)
0, 0, 0, 1, 4, 10, 20, 35, 56,  85,...    f(n)
0, 0, 1, 3, 6, 10, 15, 21, 29,  45,...    a(n)
0, 1, 2, 3, 4,  5,  6,  8, 16,  45,...    b(n)
1, 1, 1, 1, 1,  1,  2,  8, 29,  85,...    c(n)
0, 0, 0, 0, 0,  1,  6, 21, 56, 126,...    d(n).
a(n) + d(n) = A024495(n),
b(n) + e(n) = A131708(n),
c(n) + f(n) = A024493(n).
a(n) - d(n) =  0,  0,  1,  3,  6,  9,   9,   0,...    A057083(n-2)
b(n) - e(n) =  0,  1,  2,  3,  3,  0,  -9, -27,...    A057682(n)
c(n) - f(n) =  1,  1,  1,  0, -3, -9, -18, -27,...    A057681(n)
d(n) - a(n) =  0,  0, -1, -3, -6, -9,  -9,   0,...   -A057083(n-2)
e(n) - b(n) =  0, -1, -2, -3, -3,  0,   9,  27,...   -A057682(n)
f(n) - c(n) = -1, -1, -1,  0,  3,  9,  18,  27,...   -A057681(n).
The first column is A131531(n).
The first two trisections are multiples of 3. Is the third (1, 10, 29,...) mod 9  A029898(n)?

			a(6)=6*10-15*6+20*3-15*1+6*0=15, a(7)=90-150+120-45+6=21.

Seiichi Manyama, Table of n, a(n) for n = 0..3000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6).

Join[{0},LinearRecurrence[{6,-15,20,-15,6},{0,1,3,6,10},40]] (* Harvey P. Dale, Dec 17 2014 *)

{a(n) = sum(k=0, n\6, binomial(n, 6*k+2))} \\ Seiichi Manyama, Mar 23 2019

a(n) = 6*a(n-1) -15*a(n-2) +20*a(n-3) -15*a(n-4) +6*a(n-5) for n>5, a(0)=a(1)=0, a(2)=1, a(3)=3, a(4)=6, a(5)=10.
a(n) = A024495(n) - A192080(n-5) for n>4.
G.f.: -(x^5 - 3*x^4 + 3*x^3 - x^2)/((1-2*x)*(1-x+x^2)*(1-3*x+3*x^2)). - Ralf Stephan, Jul 13 2013
a(n) = Sum_{k=0..floor(n/6)} binomial(n,6*k+2). - Seiichi Manyama, Mar 23 2019

Definition uses the g.f. of Ralf Stephan.

More terms from Harvey P. Dale, Dec 17 2014

A086356 Fixed point if [nonzero-digit product]-function at initial-value=C[2n,n]=central binomial coefficient is iterated.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Jul 21 2003

			n=10, C[20,10]=184756, iteration list={184756,7560,210,2},
a(100)=2.

Cf. A051801, A051802, A086353-A086361, A029898, A010888, A038194, A000984.

prd[x_] := Apply[Times, DeleteCases[IntegerDigits[x], 0]] Table[FixedPoint[prd, Binomial[2*n, n]], {w, 1, 128}]

a(n)=A051802[A000984(n)]=fixed-point of A051801[C(2n, n)]

A086357 Fixed point if [nonzero-digit-product]-function at initial-value=A002110(n)=n-th primorial is iterated.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Jul 21 2003

			n=7, 7th-primorial=510510, iteration list={510510,25,10,1},
a(100)=2.

Cf. A051801, A051802, A086353-A086361, A029898, A010888, A038194, A000984.

prd[x_] := Apply[Times, DeleteCases[IntegerDigits[x], 0]] q[x_] := Apply[Times, Table[Prime[w], {w, 1, x}]] Table[FixedPoint[prd, q[w]], {w, 1, 128}]

a(n)=A051802[A002110(n)]=fixed-point of A051801[A002110(n)]

A086359 Fixed point if [decimal-digit-sum]-function at initial-value=A000984(n)=C[2n,n] is iterated.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Jul 21 2003

			n=10, C[20,10]=184756, iteration list={184756,31,4},a(10)=4.

Cf. A086353-A086361, A007953, A010888, A038194, A029898.

sud[x_] := Apply[Plus, DeleteCases[IntegerDigits[x], 0]] Table[FixedPoint[sud, Binomial[2*w, w]], {w, 1, 128}]

a(n)=A010888[C[2n, n]]=fixed-point of A007953[C[2n, n]]; It equals C[2n, n] modulo(9); at r=0 use 9.

cp's OEIS Frontend

A210456 Period of the sequence of the digital roots of Fibonacci n-step numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A086355 Fixed point if [nonzero-digit product]-function at initial-value=prime(n) is iterated.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A133390 Period 18: repeat 1, 4, 7, 2, 2, 5, 4, 1, 1, 8, 5, 2, 7, 7, 4, 5, 8, 8.

Views

Author

Keywords

Comments

Links

Programs

Mathematica

PARI

Formula

A154529 A090040 mod 9.

Views

Author

Keywords

Comments

Links

Programs

Mathematica

Formula

Extensions

A227430 Expansion of x^2*(1-x)^3/((1-2*x)*(1-x+x^2)*(1-3*x+3x^2)).

Views

Author

Keywords

Comments

Examples

Links

Programs

Mathematica

PARI

Formula

Extensions

A086356 Fixed point if [nonzero-digit product]-function at initial-value=C[2n,n]=central binomial coefficient is iterated.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A086357 Fixed point if [nonzero-digit-product]-function at initial-value=A002110(n)=n-th primorial is iterated.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A086359 Fixed point if [decimal-digit-sum]-function at initial-value=A000984(n)=C[2n,n] is iterated.

Views

Author

Keywords

Examples

Crossrefs

Programs

A227430 Expansion of x^2(1-x)^3/((1-2x)(1-x+x^2)(1-3*x+3x^2)).