A093348 -id:A093348 - cp's OEIS frontend

A094116 Partial sums of A093348.

0, 1, 1, 2, 2, 7, 11, 16, 20, 25, 25, 26, 26, 27, 27, 32, 36, 41, 45, 50, 50, 51, 51, 52, 52, 77, 101, 126, 150, 175, 195, 216, 236, 257, 277, 302, 326, 351, 375, 400, 420, 441, 461, 482, 502, 527, 551, 576, 600, 625, 625, 626, 626, 627, 627, 632, 636, 641, 645, 650

Offset: 1

Benoit Cloitre, May 03 2004

nonn

a(n)=sum(k=1,n,-sum(i=1,k-1,(-1)^i*5^valuation(i,5)))

a(5^k)=(25^k-1)/12

A093347 A 3-fractal "castle" starting with 0.

Original entry on oeis.org

0, 1, 0, 3, 2, 3, 0, 1, 0, 9, 8, 9, 6, 7, 6, 9, 8, 9, 0, 1, 0, 3, 2, 3, 0, 1, 0, 27, 26, 27, 24, 25, 24, 27, 26, 27, 18, 19, 18, 21, 20, 21, 18, 19, 18, 27, 26, 27, 24, 25, 24, 27, 26, 27, 0, 1, 0, 3, 2, 3, 0, 1, 0, 9, 8, 9, 6, 7, 6, 9, 8, 9, 0, 1, 0, 3, 2, 3, 0, 1, 0, 81, 80, 81, 78, 79, 78, 81

Offset: 1

Benoit Cloitre, Apr 26 2004

Amiram Eldar, Table of n, a(n) for n = 1..10000
Benoit Cloitre, Graph of a(n) for n=1 up to 9
Benoit Cloitre, Graph of a(n) for n=1 up to 27
Benoit Cloitre, Graph of a(n) for n=1 up to 81
Benoit Cloitre, Graph of a(n) for n=1 up to 243

Cf. A093348, A093349

Cf. A038500.

a[n_] := Sum[(-1)^(i+1) * 3^IntegerExponent[i, 3], {i, 1, n-1}]; Array[a, 100] (* Amiram Eldar, Jun 17 2022 *)

a(n)=if(n<2,0,3^floor(log(n-1)/log(3))-a(n-3^floor(log(n-1)/log(3))))

a(n) = my(s=-1); fromdigits([if(d==1,s=-s) |d<-digits(n-1,3)], 3); \\ Kevin Ryde, Jan 01 2024

a(1) = 0 then a(n) = w(n) - a(n-w(n)) where w(n) = 3^floor(log(n-1)/log(3)).
a(3^n) = 0, a(3^n+1) = 3^n, a(3^n+2) = 3^n-1, a(3^n+3) = 3^n, etc.
a(n) = Sum_{i=1..n-1} (-1)^(i-1)*3^valuation(i, 3).

A105774 A "fractal" transform of the Fibonacci numbers: a(1)=1; then if F(n) < k <= F(n+1), a(k) = F(n+1) - a(k - F(n)) where F(n) = A000045(n).

Original entry on oeis.org

1, 1, 2, 4, 4, 7, 7, 6, 12, 12, 11, 9, 9, 20, 20, 19, 17, 17, 14, 14, 15, 33, 33, 32, 30, 30, 27, 27, 28, 22, 22, 23, 25, 25, 54, 54, 53, 51, 51, 48, 48, 49, 43, 43, 44, 46, 46, 35, 35, 36, 38, 38, 41, 41, 40, 88, 88, 87, 85, 85, 82, 82, 83, 77, 77, 78, 80, 80, 69, 69, 70, 72, 72

Offset: 1

Benoit Cloitre, May 04 2005

nonn

Let tau = (1+sqrt(5))/2; then the missing numbers 3,5,8,10,13,16,18,21,... are given by round(tau^2*k) for k > 0 (A004937).
Indices n such that a(n) = a(n+1) are given by floor(tau^2*k) - 1 for k > 0 (A003622).
Numbers n such that a(n) differs from a(n+1) are given by floor(tau*k+1/tau) for k > 0 (A022342).
Indices n giving isolated terms (a(n) differs from a(n-1) and a(n+1)) are given by floor(tau*floor(tau^2*k)) for k > 0 (A003623).
Remove 0's from the first differences of sorted values; then you get a version of the infinite Fibonacci word (A001468). I.e., sorted values are 1,1,2,4,4,6,7,7,9,9,11,12,12,..., first differences are 0,1,2,0,2,1,0,2,0,2,1,0,2,0,1,...; removing 0's gives 1,2,2,1,2,2,1,2,1,2,2,1,2,1,2,2,1,2,... #{ k : a(k)=k}=infty.

			For 1 = F(2) < k <= F(3) = 2 the rule gives a(2) = 2 - a(1) = 1 ... if 5 = F(5) < k <= F(6) = 8 the rule forces a(6) = 8 - a(6-5) = 8 - a(1) = 7; a(7) = 8 - a(2) = 7; a(8) = 8 - a(3) = 6.

Antti Karttunen, Table of n, a(n) for n = 1..10946
Benoit Cloitre and Jeffrey Shallit, Some Fibonacci-Related Sequences, arXiv:2312.11706 [math.CO], 2023.
Jeffrey Shallit, Using automata to prove theorems about sequences, One FLAT World Seminar, 2024.

Cf. A000045, A001468, A003622, A003623, A004937, A006498, A022342, A072649, A085002, A105669, A105670, A105672, A093347, A093348, A283766.

a(A000045(n)) = A006498(n-1) for n >= 1. - Typo corrected by Antti Karttunen, Mar 17 2017
limsup a(n)/n = tau and liminf a(n)/n = (tau+2)/5 where tau = (1+sqrt(5))/2. - Corrected by Jeffrey Shallit, Dec 17 2023
a(n) mod 2 = A085002(n) - Benoit Cloitre, May 10 2005
a(1) = 1; for n > 1, a(n) = A000045(2+A072649(n-1)) - a(n-A000045(1 + A072649(n-1))). - Antti Karttunen, Mar 17 2017

A105669 A "fractal" transform of the Fibonacci numbers F(n)=A000045(n): a(1)=1, then for n>1 if F(n) < k < F(n+1) we have a(k) = F(n+1)-a(k-F(n)) and when k = F(n+1) we force a(F(n+1)) = F(n+1) + (1+(-1)^n)*F(n).

Original entry on oeis.org

1, 2, 2, 4, 7, 7, 6, 6, 12, 11, 11, 9, 20, 20, 19, 19, 17, 14, 14, 15, 15, 33, 32, 32, 30, 27, 27, 28, 28, 22, 23, 23, 25, 54, 54, 53, 53, 51, 48, 48, 49, 49, 43, 44, 44, 46, 35, 35, 36, 36, 38, 41, 41, 40, 40, 88, 87, 87, 85, 82, 82, 83, 83, 77, 78, 78, 80, 69, 69, 70, 70, 72

Offset: 1

Benoit Cloitre, May 03 2005

nonn

Let b denote the sequence of n such that a(n)=a(n+1), then b(n)=floor(tau^2*n) where tau=(1+sqrt(5))/2.
Missing numbers are the nearest integer to tau^2*n, n>=0 (cf. A004937).
#{k>0 : a(k) = k} = infinity.
This kind of "fractal" transform can be applied to any increasing monotonic sequence giving true fractal properties for sequences = (m^n)_{n>0} with m integer >=2, specially when m is odd (cf. A093347, A093348).

			For 5 = F(5) < k <= F(6) = 8 we get a(6) = 8-a(6-5) = 8-a(1) = 7.
a(7) = 8-a(7-5) = 8-a(2) = 6.
a(8) = 8-a(8-5) = 8-a(3) = 6.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000045, A001622, A004937, A105670, A105672, A093347, A093348.

a[n_] := a[n] = If[n <= 1, 1, Fibonacci[(k = Floor[Log[Sqrt[5]*n]/Log[GoldenRatio]]) + 1] - a[n - Fibonacci[k]]]; Array[a, 100] (* Amiram Eldar, Jun 17 2022 *)

f=(1+sqrt(5))/2; a(n)=if(n<2,1,fibonacci(floor(log(sqrt(5)*n)/log(f))+1)-a(n-fibonacci(floor(log(sqrt(5)*n)/log(f)))))

F(2n) = F(2n+1) - F(n+1)^2 + F(n)*F(n-1) for n>0.
a(F(2n-1)) = F(2n)-1 for n>1.
1/tau < a(n)/n < tau.

A105670 a(1)=1 then bracketing n by powers of 2 as f(t)=2^t for f(t) < n <= f(t+1), a(n) = f(t+1) - a(n-f(t)).

Original entry on oeis.org

1, 1, 3, 3, 7, 7, 5, 5, 15, 15, 13, 13, 9, 9, 11, 11, 31, 31, 29, 29, 25, 25, 27, 27, 17, 17, 19, 19, 23, 23, 21, 21, 63, 63, 61, 61, 57, 57, 59, 59, 49, 49, 51, 51, 55, 55, 53, 53, 33, 33, 35, 35, 39, 39, 37, 37, 47, 47, 45, 45, 41, 41, 43, 43, 127, 127, 125, 125, 121, 121, 123

Offset: 1

Benoit Cloitre, May 03 2005

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A006068, A105669, A105672, A093347, A093348.

A062383 := proc(n)
        ceil(log(n)/log(2)) ;
        2^% ;
end proc:
A105670 := proc(n)
        option remember;
        if n = 1 then
                1;
        else
                fn1 := A062383(n) ;
                fn := fn1/2 ;
                fn1-procname(n-fn) ;
        end if;
end proc:
seq(A105670(n),n=1..80) ; # R. J. Mathar, Nov 06 2011

t[0] = 0; t[1] = 1; t[n_?EvenQ] := t[n] = t[n/2]; t[n_?OddQ] := t[n] = 1 - t[(n-1)/2]; a[1] = 1; a[n_?EvenQ] := a[n] = a[n - 1]; a[n_] := a[n] = 2*a[Ceiling[n/2]] - 1 + 2*t[Ceiling[n/2] - 1]; Table[a[n], {n, 1, 71}] (* Jean-François Alcover, Aug 13 2013 *)

b(n,m)=if(n<2,1,m*m^floor(log(n-1)/log(m))-b(n-m^floor(log(n-1)/log(m)),m))

a(2n-1) = a(2n).
a(n) = 2*a(ceiling(n/2)) -1 + 2*t(ceiling(n/2)-1) where t(n) = A010060(n) is the Thue-Morse sequence.
a(2n-1) = a(2n) = 2*A006068(n-1)+1. - Jeffrey Shallit, Mar 15 2025

Typo in data corrected by Jean-François Alcover, Aug 13 2013

A105672 a(1)=1, then bracketing n with powers of 3 as f(t)=3^t for f(t) < n <= f(t+1), a(n) = f(t+1) - a(n-f(t)).

Original entry on oeis.org

1, 2, 1, 8, 7, 8, 1, 2, 1, 26, 25, 26, 19, 20, 19, 26, 25, 26, 1, 2, 1, 8, 7, 8, 1, 2, 1, 80, 79, 80, 73, 74, 73, 80, 79, 80, 55, 56, 55, 62, 61, 62, 55, 56, 55, 80, 79, 80, 73, 74, 73, 80, 79, 80, 1, 2, 1, 8, 7, 8, 1, 2, 1, 26, 25, 26, 19, 20, 19, 26, 25, 26, 1, 2, 1, 8, 7, 8, 1, 2, 1

Offset: 1

Benoit Cloitre, May 03 2005

nonn

Cf. A105669, A105670, A093347, A093348.

A105672 := proc(n)
        option remember;
        if n = 1 then
                1;
        else
                fn1 := A064235(n) ;
                fn := fn1/3 ;
                fn1-procname(n-fn) ;
        end if;
end proc:
seq(A105672(n),n=1..80) ; # R. J. Mathar, Nov 06 2011

A064235[n_] := 3^Ceiling[Log[3, n]]; a[1] = 1; a[n_] := a[n] = A064235[n] - a[n - A064235[n]/3]; Table[a[n], {n, 1, 81}] (* Jean-François Alcover, Jul 09 2013, after R. J. Mathar *)

b(n,m)=if(n<2,1,m*m^floor(log(n-1)/log(m))-b(n-m^floor(log(n-1)/log(m)),m))

a(n+1) = 1 + Sum_{k=1..n} (-1)^k*(2-3*3^valuation(k, 3)).

a(n) = A064235(n) - a(n - A064235(n)/3). - R. J. Mathar, Nov 06 2011

A093349 A 7-fractal "castle" starting with 0.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 0, 7, 6, 7, 6, 7, 6, 7, 0, 1, 0, 1, 0, 1, 0, 7, 6, 7, 6, 7, 6, 7, 0, 1, 0, 1, 0, 1, 0, 7, 6, 7, 6, 7, 6, 7, 0, 1, 0, 1, 0, 1, 0, 49, 48, 49, 48, 49, 48, 49, 42, 43, 42, 43, 42, 43, 42, 49, 48, 49, 48, 49, 48, 49, 42, 43, 42, 43, 42, 43, 42, 49, 48, 49, 48, 49, 48, 49, 42, 43

Offset: 1

Benoit Cloitre, Apr 26 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A093347, A093348

a[n_] := Sum[(-1)^(i+1) * 7^IntegerExponent[i, 7], {i, 1, n-1}]; Array[a, 100] (* Amiram Eldar, Jun 17 2022 *)

a(n)=if(n<2,0,7^floor(log(n-1)/log(7))-a(n-7^floor(log(n-1)/log(7))))

a(n)=-sum(i=1,n-1,(-1)^i*7^valuation(i,7))

a(1) = 0 then a(n) = w(n) - a(n-w(n)) where w(n) = 7^floor(log(n-1)/log(7)).

a(n) = Sum_{i=1..n-1} (-1)^(i-1)*7^valuation(i, 7).

cp's OEIS Frontend

A094116 Partial sums of A093348.

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A093347 A 3-fractal "castle" starting with 0.

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A105774 A "fractal" transform of the Fibonacci numbers: a(1)=1; then if F(n) < k <= F(n+1), a(k) = F(n+1) - a(k - F(n)) where F(n) = A000045(n).

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A105669 A "fractal" transform of the Fibonacci numbers F(n)=A000045(n): a(1)=1, then for n>1 if F(n) < k < F(n+1) we have a(k) = F(n+1)-a(k-F(n)) and when k = F(n+1) we force a(F(n+1)) = F(n+1) + (1+(-1)^n)*F(n).

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A105670 a(1)=1 then bracketing n by powers of 2 as f(t)=2^t for f(t) < n <= f(t+1), a(n) = f(t+1) - a(n-f(t)).

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A105672 a(1)=1, then bracketing n with powers of 3 as f(t)=3^t for f(t) < n <= f(t+1), a(n) = f(t+1) - a(n-f(t)).

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A093349 A 7-fractal "castle" starting with 0.

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