A117943 -id:A117943 - cp's OEIS frontend

A255822 Number of nonzero terms in A117943 up to index 2^n: a(n) = Sum_{k=1..2^n} A117943(k).

0, 1, 1, 3, 5, 9, 19, 41, 84, 163, 311, 615, 1268, 2612, 5228, 10244, 20172, 40496, 82066, 165284, 329388, 654207, 1305709, 2620890, 5262399, 10524162, 20983533, 41878200, 83848212, 168188942, 337036806, 673896773, 1345770393, 2688442850, 5375471835

Offset: 0

M. F. Hasler, Mar 09 2015

nonn

The sequence seems to grow roughly as a(n) ~ c*2^n with c = 0.313...

Hiroaki Yamanouchi, Table of n, a(n) for n = 0..2000

Cf. A117943, A178931.

N=1;s=0;for(n=1,9e9,s+=A117943(n);n

a(25)-a(34) from Hiroaki Yamanouchi, Mar 10 2015

A126616 a(n) = n for n < 10, a(10*n) = a(n), and if the terms a(10), a(20), a(30), ... are deleted, one gets back the original sequence.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Feb 09 2007

A self-generating sequence.

Invented by Eric Angelini. Might also be called a lizard sequence (une suite du lézard) because it grows back from its tail.

J.-P. Delahaye, La suite du lézard et autres inventions, Pour la Science, No. 353, 2007.

Cf. A117943, A178931, A255824 - A255829.

A126616 := proc(n) option remember ; if n < 10 then n ; elif n mod 10 = 0 then A126616(n/10) ; else A126616( n-floor(n/10) ) ; fi ; end: seq(A126616(n),n=1..120) ; # R. J. Mathar, Oct 02 2007

a[n_] := Module[{m = 10, k = n, q}, While[k >= m, q = Quotient[k, m]; If[Mod[k, m] != 0, k -= q, k = q]]; k];
Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Aug 02 2022, after M. F. Hasler *)

a(n,m=10)=while(n>=m,if(n%m,n-=n\m,n\=m));n \\ M. F. Hasler, Mar 07 2015

More terms from R. J. Mathar, Oct 02 2007

Definition rephrased by M. F. Hasler, Mar 09 2015

A255824 a(n) = n for n < 4; a(4n) = a(n); if every 4th term (a(4), a(8), a(12), ...) is deleted, this gives back the original sequence.

Original entry on oeis.org

1, 2, 3, 1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 3, 1, 2, 1, 3, 1, 1, 2, 1, 1, 3, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 2, 1, 2, 1, 2, 3, 1, 1, 1, 1, 3, 1, 2, 1, 2, 2, 1, 2, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 1, 3, 3, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 2, 2, 1, 2, 1, 1, 1, 2, 1, 3, 3

Offset: 1

M. F. Hasler, Mar 07 2015

nonn

A self-generating sequence. This is the m=4 analog of the m=10 variant A126616. Sequence A117943 is the m=3 analog with all terms decreased by 1.

Cf. A117943, A255825, A255826, A255827, A255828, A255829, A126616.

a(n, m=4)=while(n>=m, if(n%m, n-=n\m, n\=m)); n \\ M. F. Hasler, Mar 07 2015

a(n) = a(n/4) if n == 0 (mod 4); a(n) = a(n - floor(n/4)) otherwise.

A255829 a(n) = n for n < 9; a(9n) = a(n); if every 9th term (a(9), a(18), a(27),...) is deleted, this gives back the original sequence.

Original entry on oeis.org

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

Offset: 1

M. F. Hasler, Mar 07 2015

nonn

A self-generating sequence. This is the m=9 analog of the m=10 variant A126616. Sequence A117943 is the m=3 analog with all terms decreased by 1.

Cf. A117943, A255824 - A255828, A126616.

a(n, m=9)=while(n>=m, if(n%m, n-=n\m, n\=m)); n \\ M. F. Hasler, Mar 07 2015

a(n) = a(n/9) if n==0 (mod 9); a(n) = a(n - floor(n/9)) otherwise.

A178931 This sequence S is generated by the following rules: 2 is in S, and if n is in S, then floor[(3n-1)/2] and 3n are in S.

Original entry on oeis.org

2, 6, 8, 11, 16, 18, 23, 24, 26, 33, 34, 35, 38, 48, 49, 50, 52, 54, 56, 69, 71, 72, 73, 74, 77, 78, 80, 83, 99, 102, 103, 105, 106, 107, 109, 110, 114, 115, 116, 119, 124, 144, 147, 148, 150, 152, 154, 156, 157, 158, 160, 162, 163, 164, 168, 170, 172, 173, 178, 185

Offset: 1

Clark Kimberling, Dec 30 2010

nonn

This sequence results from flattening and sorting the tree at A183212. Complement of A183213, obtained from the tree at A183211.

Sequence A117943 is the characteristic sequence of this one. - M. F. Hasler, Mar 07 2015

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A183213, A183212.

nn=200; t={2}; t0=t; While[t=Select[Union[t,Floor[(3*t-1)/2],3*t], #<=nn &]; t0 != t, t0=t]; t

(See the Mathematica code.)

A117944 Triangle related to powers of 3 partitions of n.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Paul Barry, Apr 05 2006

Inverse is A117945.

Row sums of inverse are A039966.

			Triangle begins
  1;
  0, 1;
  1, 0, 1;
  0, 0, 0, 1;
  0, 0, 0, 0, 1;
  0, 0, 0, 1, 0, 1;
  1, 0, 0, 0, 0, 0, 1;
  0, 1, 0, 0, 0, 0, 0, 1;
  1, 0, 1, 0, 0, 0, 1, 0, 1;
  0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A039966, A117939, A117943, A117945.

T[n_, k_]:= Mod[Sum[JacobiSymbol[Binomial[n, j], 3]*JacobiSymbol[Binomial[n-j, k], 3], {j,0,n}], 2];
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 29 2021 *)

def A117944(n, k): return ( sum(jacobi_symbol(binomial(n, j), 3)*jacobi_symbol(binomial(n-j, k), 3) for j in (0..n)) )%2
flatten([[A117944(n, k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Oct 29 2021

Triangle T(n,k) = Sum_{j=0..n} L(C(n,j)/3)*L(C(n-j,k)/3) mod 2, where L(j/p) is the Legendre symbol of j and p.
T(n, k) = A117939(n,k) mod 2.
T(n, k) = A117939^(-1)(n,k) mod 2.
Sum_{k=0..n} T(n, k) = A117943(n).

A255828 a(n) = n for n < 8; a(8n) = a(n); if every 8th term (a(8), a(16), a(24),...) is deleted, this gives back the original sequence.

Original entry on oeis.org

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

Offset: 1

M. F. Hasler, Mar 07 2015

nonn

A self-generating sequence. This is the m=8 analog of the m=10 variant A126616. Sequence A117943 is the m=3 analog with all terms decreased by 1.

Cf. A117943, A255824 - A255829, A126616.

a(n, m=8)=while(n>=m, if(n%m, n-=n\m, n\=m)); n \\ M. F. Hasler, Mar 07 2015

a(n) = a(n/8) if n==0 (mod 8); a(n) = a(n - floor(n/8)) otherwise.

A255825 A self-generating sequence: a(n) = n for n < 5; a(5n) = a(n); if every 5th term (a(5), a(10), a(15),...) is deleted, this gives back the original sequence.

Original entry on oeis.org

1, 2, 3, 4, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 3, 1, 2, 3, 1, 4, 2, 3, 1, 4, 1, 2, 3, 1, 4, 1, 1, 2, 3, 1, 1, 4, 1, 1, 2, 1, 3, 1, 1, 4, 1, 1, 1, 2, 1, 2, 3, 1, 1, 4, 1, 1, 1, 1, 2, 2, 1, 2, 3, 1, 1, 1, 4, 1, 1, 2, 1, 1, 2, 2, 3, 1, 2, 3, 1, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 3, 2, 2, 3, 1, 1, 2, 3, 1, 1, 4

Offset: 1

M. F. Hasler, Mar 07 2015

nonn

This is the m=5 analog of the m=10 variant A126616. Sequence A117943 is the m=3 analog with all terms decreased by 1.

Cf. A117943, A255824 - A255829, A126616.

a(n, m=5)=while(n>=m, if(n%m, n-=n\m, n\=m)); n \\ M. F. Hasler, Mar 07 2015

a(n) = a(n/5) if n==0 (mod 5); a(n) = a(n - floor(n/5)) otherwise.

A255826 a(n) = n for n < 6; a(6n) = a(n); if every 6th term (a(6), a(12), a(18),...) is deleted, this gives back the original sequence.

Original entry on oeis.org

1, 2, 3, 4, 5, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 2, 1, 3, 2, 1, 4, 3, 2, 1, 4, 3, 5, 2, 1, 4, 3, 5, 1, 2, 1, 4, 3, 5, 1, 1, 2, 1, 4, 3, 1, 5, 1, 1, 2, 1, 1, 4, 3, 1, 5, 1, 1, 1, 2, 1, 1, 4, 1, 3, 1, 5, 1, 1, 2, 1, 2, 1, 1, 4, 1, 1, 3, 1, 5, 1, 2, 1, 2, 1, 2, 1, 1, 1, 4, 1, 1, 3, 2, 1, 5, 1, 2

Offset: 1

M. F. Hasler, Mar 07 2015

nonn

A self-generating sequence. This is the m=6 analog of the m=10 variant A126616. Sequence A117943 is the m=3 analog with all terms decreased by 1.

Cf. A117943, A255824 - A255829, A126616.

a(n, m=6)=while(n>=m, if(n%m, n-=n\m, n\=m)); n \\ M. F. Hasler, Mar 07 2015

a(n) = a(n/6) if n==0 (mod 6); a(n) = a(n - floor(n/6)) otherwise.

A255827 a(n) = n for n < 7; a(7n) = a(n); if every 7th term (a(7), a(14), a(21),...) is deleted, this gives back the original sequence.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 3, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 5, 3, 4, 1, 2, 5, 3, 6, 4, 1, 2, 5, 3, 6, 1, 4, 1, 2, 5, 3, 6, 1, 1, 4, 1, 2, 5, 3, 1, 6, 1, 1, 4, 1, 2, 1, 5, 3, 1, 6, 1, 1, 1, 4, 1, 2, 1, 5, 3, 1, 1, 6, 1, 1, 1, 4, 1, 1, 2, 1, 5, 3, 1, 2, 1, 6

Offset: 1

M. F. Hasler, Mar 07 2015

nonn

A self-generating sequence. This is the m=7 analog of the m=10 variant A126616. Sequence A117943 is the m=3 analog with all terms decreased by 1.

Cf. A117943, A255824 - A255829, A126616.

a(n, m=7)=while(n>=m, if(n%m, n-=n\m, n\=m)); n \\ M. F. Hasler, Mar 07 2015

a(n) = a(n/7) if n==0 (mod 7); a(n) = a(n - floor(n/7)) otherwise.

cp's OEIS Frontend

A255822 Number of nonzero terms in A117943 up to index 2^n: a(n) = Sum_{k=1..2^n} A117943(k).

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A126616 a(n) = n for n < 10, a(10*n) = a(n), and if the terms a(10), a(20), a(30), ... are deleted, one gets back the original sequence.

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A255824 a(n) = n for n < 4; a(4n) = a(n); if every 4th term (a(4), a(8), a(12), ...) is deleted, this gives back the original sequence.

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A255829 a(n) = n for n < 9; a(9n) = a(n); if every 9th term (a(9), a(18), a(27),...) is deleted, this gives back the original sequence.

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A178931 This sequence S is generated by the following rules: 2 is in S, and if n is in S, then floor[(3n-1)/2] and 3n are in S.

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A117944 Triangle related to powers of 3 partitions of n.

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A255828 a(n) = n for n < 8; a(8n) = a(n); if every 8th term (a(8), a(16), a(24),...) is deleted, this gives back the original sequence.

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A255825 A self-generating sequence: a(n) = n for n < 5; a(5n) = a(n); if every 5th term (a(5), a(10), a(15),...) is deleted, this gives back the original sequence.

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