A023116 -id:A023116 - cp's OEIS frontend

A022778 Place where n-th 1 occurs in A023116.

1, 2, 4, 6, 9, 12, 16, 21, 26, 32, 38, 45, 52, 60, 69, 78, 88, 98, 109, 120, 132, 145, 158, 172, 186, 201, 217, 233, 250, 267, 285, 303, 322, 342, 362, 383, 404, 426, 448, 471, 495, 519, 544, 569, 595, 621, 648, 676, 704, 733, 762, 792, 823, 854

Offset: 1

Clark Kimberling

nonn

Also place where first n appears in A007337. - Franklin T. Adams-Watters, Nov 10 2006

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A022776, A244977.

Table[n + 1 + Sum[Floor[(n - k)/Sqrt[3]], {k, 0, n}], {n, 0, 200}] (* A022778 *)
(* Clark Kimberling, Mar 14 2015 *)

a(n) = n + 1 + Sum{floor[(n - k)/sqrt(3)], k = 0..n}.

A084822 Signature sequence of x, where x=0.577284608955710564894... (A084823) is the unique number between 0 and 1 having the property that the signature sequence of x is equal to the continued fraction expansion of x.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 3, 2, 1, 3, 2, 1, 4, 3, 2, 1, 4, 3, 2, 5, 1, 4, 3, 2, 5, 1, 4, 3, 6, 2, 5, 1, 4, 3, 6, 2, 5, 1, 4, 7, 3, 6, 2, 5, 1, 4, 7, 3, 6, 2, 5, 1, 8, 4, 7, 3, 6, 2, 5, 1, 8, 4, 7, 3, 6, 2, 9, 5, 1, 8, 4, 7, 3, 6, 2, 9, 5, 1, 8, 4, 7, 3, 10, 6, 2, 9, 5, 1, 8, 4, 7, 3, 10, 6, 2, 9, 5, 1, 8, 4, 11, 7, 3

Offset: 1

Paul D. Hanna, Jun 04 2003

The initial terms are close to those of A023116 since x is close to 1/sqrt(3)=A020760. (See A023116.) - R. J. Mathar, Sep 17 2008
Where do they first differ? - N. J. A. Sloane, Jan 20 2023
The first difference is a(2793) = 1 and A023116(2793) = 57. - Paul D. Hanna and Michael S. Branicky, Jan 21 2023

			Given x = 0.5772846089557105648944851585150204938530... (A084823), the continued fraction of x equals the signature sequence of x.
To obtain the signature sequence of x, arrange the numbers i+j*x (i,j >= 1) in increasing order like so:
[1+1*x, 1+2*x, 2+1*x, 1+3*x, 2+2*x, 1+4*x, 3+1*x, 2+3*x, 1+5*x, 3+2*x, 2+4*x, 1+6*x, 4+1*x, 3+3*x, 2+5*x, 1+7*x, 4+2*x, 3+4*x, 2+6*x, 5+1*x, 1+8*x, 4+3*x, 3+5*x, 2+7*x, 5+2*x, 1+9*x, 4+4*x, 3+6*x, 6+1*x, 2+8*x, 5+3*x, 1+10*x, 4+5*x, 3+7*x, 6+2*x, 2+9*x, 5+4*x, 1+11*x, 4+6*x, 7+1*x, 3+8*x, 6+3*x, 2+10*x, 5+5*x, 1+12*x, 4+7*x, 7+2*x, 3+9*x, 6+4*x, 2+11*x, ...];
then the sequence of i's is the signature of x, and forms this sequence.

Paul D. Hanna, Table of n, a(n) for n = 1..2000
Index entries for sequences related to signature sequences

Cf. A084823 (decimal expansion).

A022777 Place where n-th 1 occurs in A007337.

Original entry on oeis.org

1, 3, 7, 13, 20, 29, 40, 53, 67, 83, 101, 121, 142, 165, 190, 216, 244, 274, 306, 339, 374, 411, 450, 490, 532, 576, 622, 669, 718, 769, 821, 875, 931, 989, 1048, 1109, 1172, 1237, 1303, 1371, 1441, 1513, 1586, 1661, 1738, 1816, 1896, 1978, 2062

Offset: 1

Clark Kimberling

nonn

Also place where first n appears in A023116. First differences minus 1 are A022838. - Franklin T. Adams-Watters, Nov 10 2006

cp's OEIS Frontend

A022778 Place where n-th 1 occurs in A023116.

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A084822 Signature sequence of x, where x=0.577284608955710564894... (A084823) is the unique number between 0 and 1 having the property that the signature sequence of x is equal to the continued fraction expansion of x.

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A022777 Place where n-th 1 occurs in A007337.

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