A020944 -id:A020944 - cp's OEIS frontend

A020948 Least k such that b(k) = n, where b( ) is sequence A020944.

1, 8, 16, 32, 34, 128, 66, 68, 130, 134, 132, 140, 138, 260, 264, 284, 268, 266, 274, 572, 276, 524, 522, 536, 534, 530, 538, 1034, 532, 548, 556, 1046, 1042, 554, 1050, 1082, 1044, 2072, 1092, 1064, 1060, 2114, 1068, 1076, 1132, 1100, 1066, 2102, 1098, 1106, 2088, 2084, 2098, 4346, 1108

Offset: 1

Clark Kimberling

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A020943-A020945, A002487, A020947, A020948, A020949, A020950.

a[n_] := a[n] = Which[n < 2, Boole[n == 1] - Boole[n == 0], OddQ[n], Abs[a[n - 1] - a[n - 2]], True, a[n/2] + a[n/2 - 1]]; s = Array[a[#] &, 4200]; Array[FirstPosition[s, #][[1]] &, LengthWhile[Differences@ Union@ s, # == 1 &]] (* Michael De Vlieger, Feb 18 2022, after Michael Somos at A020944 *)

More terms from Seiichi Manyama, Feb 18 2022

A153893 a(n) = 3*2^n - 1.

Original entry on oeis.org

2, 5, 11, 23, 47, 95, 191, 383, 767, 1535, 3071, 6143, 12287, 24575, 49151, 98303, 196607, 393215, 786431, 1572863, 3145727, 6291455, 12582911, 25165823, 50331647, 100663295, 201326591, 402653183, 805306367, 1610612735, 3221225471

Offset: 0

Vladimir Joseph Stephan Orlovsky, Jan 03 2009

A020944(a(n)) = 0. - Reinhard Zumkeller, Mar 13 2011
a(n) + a(n-1)^2 is a perfect square. - Vincenzo Librandi, Oct 28 2011
Number of distinct continued fractions of n terms chosen from {1,2}. - Clark Kimberling, Jul 20 2015
Also, the decimal representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 643", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. See A283508. - Robert Price, Mar 09 2017
This sequence has been used by the ninth-century mathematician Thabit ibn Qurra to devise the first method to construct amicable pairs (see Tattersall). - Stefano Spezia, Jul 18 2025

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 138.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Gennady Eremin, Partitioning the set of natural numbers into Mersenne trees and into arithmetic progressions; Natural Matrix and Linnik's constant, arXiv:2405.16143 [math.CO], 2024. See pp. 2-5, 14.
Gennady Eremin, Infinite matrix of odd natural numbers. A bit about Sophie Germain prime numbers, arXiv:2501.17090 [math.GM], 2025. See pp. 3, 11.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A283508.

[3*2^n-1: n in [0..30]]; // Vincenzo Librandi, Oct 28 2011

Table[3*2^n - 1 , {n,0,25}] (* G. C. Greubel, Sep 01 2016 *)
LinearRecurrence[{3,-2},{2,5},40] (* Harvey P. Dale, Mar 01 2024 *)

a(n)=3*2^n-1 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = a(n-1)*2 + 1, a(0)=2.
a(n) = A083329(n+1).
a(n) = A055010(n+1).
G.f.: (2 - x)/((1-x)(1-2x)). - R. J. Mathar, Feb 13 2009
a(n) = A083416(2n) = A033484(n) + 1. - Philippe Deléham, Apr 14 2013
From G. C. Greubel, Sep 01 2016: (Start)
a(n) = 3*a(n-1) - 2*a(n-2).
E.g.f.: 3*exp(2*x) - exp(x). (End)

Edited by N. J. A. Sloane, Feb 14 2009

A213369 The twisted Stern sequence: a(0) = 0, a(1) = 1 and a(2n) = -a(n), a(2n + 1) = -a(n)-a(n + 1) for n>=1.

Original entry on oeis.org

0, 1, -1, 0, 1, 1, 0, -1, -1, -2, -1, -1, 0, 1, 1, 2, 1, 3, 2, 3, 1, 2, 1, 1, 0, -1, -1, -2, -1, -3, -2, -3, -1, -4, -3, -5, -2, -5, -3, -4, -1, -3, -2, -3, -1, -2, -1, -1, 0, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7

Offset: 0

N. J. A. Sloane, Jun 13 2012

Bruno Berselli, Table of n, a(n) for n = 0..10000
Jean-Paul Allouche, On the Stern sequence and its twisted version, arXiv preprint arXiv:1202.4171 [math.NT], 2012.
Roland Bacher, Twisting the Stern sequence, arXiv:1005.5627v1 [math.CO], 2010.
Peter Bundschuh & Keijo Väänänen, Algebraic independence of the generating functions of Stern's sequence and of its twist, Journal de théorie des nombres de Bordeaux, 25 no. 1 (2013), p. 43-57, doi: 10.5802/jtnb.824.
Michael Coons, On Some Conjectures concerning Stern's Sequence and its Twist, arXiv:1008.0193v3 [math.NT], 2010.

Cf. A002487. Absolute values give A020944.

a[0]=0; a[1]=1; a[n_] := a[n] = If[EvenQ[n], -a[n/2], -a[(n-1)/2]-a[(n+1)/2 ]]; Table[a[n], {n, 0, 77}] (* Jean-François Alcover, Oct 02 2018 *)

a[0]:0$ a[1]:1$ a[n]:=-a[floor(n/2)]-(1-(-1)^n)*a[floor((n-1)/2)+1]/2$ makelist(a[n],n,0,77); /* Bruno Berselli, Jun 15 2012 */

a(n) = A287729(n) - A287730(n) for n > 0. - Michel Marcus & I. V. Serov, May 28 2019

A318163 a(0) = a(3) = 0, a(1) = a(2) = 1; for n >= 2, a(2n) = -a(n-1) and a(2n+1) = -a(n-1)-a(n).

Original entry on oeis.org

0, 1, 1, 0, -1, -2, -1, -1, 0, 1, 1, 3, 2, 3, 1, 2, 1, 1, 0, -1, -1, -2, -1, -4, -3, -5, -2, -5, -3, -4, -1, -3, -2, -3, -1, -2, -1, -1, 0, 1, 1, 2, 1, 3, 2, 3, 1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 1, 4, 3, 5, 2, 5, 3, 4, 1, 3, 2, 3, 1, 2, 1, 1, 0, -1, -1, -2, -1, -3, -2, -3, -1, -4, -3, -5, -2, -5, -3, -4, -1, -6, -5, -9, -4, -11, -7

Offset: 0

Altug Alkan, Aug 19 2018

Inspired by A002487.
Alternatively, a(0) = 0, a(1) = 1; for n >= 1, a(2*n) = a(2*n-1) - a(2*n-2), a(2*n+1) = a(2*n) - a(n). Note that if b(0) = 0, b(1) = 1; for n >= 1, b(2*n) = b(2*n-1) - b(n), b(2*n+1) = b(2*n) - b(2*n-1), then b(n) + A213369(n+1) = 0 for all n >= 1.
The main block structure of this sequence is described by A020714.

Altug Alkan, Table of n, a(n) for n = 0..20480
Altug Alkan, A scatterplot of a(n) for n <= 5*2^12
Altug Alkan, A scatterplot of first differences of a(n) for n <= 5*2^13

Cf. A002487, A020944, A213369, A303404.

a[0]=a[3]=0; a[1]=a[2]=1; a[n_] := a[n] = If[EvenQ[n], -a[n/2-1], -a[(n-1)/2 - 1] - a[(n-1)/2]]; Array[a, 101, 0] (* Giovanni Resta, Aug 27 2018 *)

a = vector(100); print1(0", "); for(k=1, #a, print1 (a[k]=if(k<=2,1, my (n=k\2); if (k%2==0, -a[n-1], a[2*n]-a[n]))", "));

a(5*2^k-2) = 0 for all k >= 0.

cp's OEIS Frontend

A020948 Least k such that b(k) = n, where b( ) is sequence A020944.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Extensions

A153893 a(n) = 3*2^n - 1.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A213369 The twisted Stern sequence: a(0) = 0, a(1) = 1 and a(2n) = -a(n), a(2n + 1) = -a(n)-a(n + 1) for n>=1.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Maxima

Formula

A318163 a(0) = a(3) = 0, a(1) = a(2) = 1; for n >= 2, a(2*n) = -a(n-1) and a(2*n+1) = -a(n-1)-a(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A318163 a(0) = a(3) = 0, a(1) = a(2) = 1; for n >= 2, a(2n) = -a(n-1) and a(2n+1) = -a(n-1)-a(n).