A020947 -id:A020947 - cp's OEIS frontend

A020950 a(n) = k-1, where k is smallest number such that A002487(k) = n.

0, 2, 4, 8, 10, 32, 18, 20, 34, 38, 36, 44, 42, 68, 72, 92, 76, 74, 82, 188, 84, 140, 138, 152, 150, 146, 154, 266, 148, 164, 172, 278, 274, 170, 282, 314, 276, 536, 324, 296, 292, 578, 300, 308, 364, 332, 298, 566, 330, 338, 552, 548, 562, 1274, 340, 584, 564, 614, 628

Offset: 1

Clark Kimberling

nonn

			A002487(33) = 6 and this is the first time 6 appears, so a(6) = 33-1 = 32.

Equals A020946 - 1.

Cf. A020943-A020946, A002487, A020947-A020949.

from itertools import count
from functools import reduce
def A020950(n): return next(filter(lambda k:sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(k)[-1:2:-1],(1,0)))==n,count(1)))-1 # Chai Wah Wu, May 05 2023

Corrected and extended by David W. Wilson

A020943 a(2n+1) = |a(2n) - a(2n-1)|, a(2n) = a(n) + a(n-1).

Original entry on oeis.org

0, 1, 1, 1, 0, 2, 2, 2, 0, 1, 1, 2, 1, 4, 3, 4, 1, 2, 1, 1, 0, 2, 2, 3, 1, 3, 2, 5, 3, 7, 4, 7, 3, 5, 2, 3, 1, 3, 2, 2, 0, 1, 1, 2, 1, 4, 3, 5, 2, 4, 2, 4, 2, 5, 3, 7, 4, 8, 4, 10, 6, 11, 5, 11, 6, 10, 4, 8, 4, 7, 3, 5, 2, 4, 2, 4, 2, 5, 3, 4, 1, 2, 1, 1, 0, 2, 2, 3, 1, 3, 2, 5, 3, 7, 4, 8, 4, 7, 3, 6, 3, 6, 3

Offset: 1

Clark Kimberling

Presumably, the positions n such that a(n)=0 are the terms of A097074. - Ivan Neretin, Jul 06 2015

Ivan Neretin, Table of n, a(n) for n = 1..5461

Cf. A020947, A020948, A020949, A020950.

a = {0, 1, 1}; Do[AppendTo[a, a[[n]] + a[[n - 1]]]; AppendTo[a, Abs[a[[-1]] - a[[-2]]]], {n, 2, 51}]; a (* Ivan Neretin, Jul 06 2015 *)

More terms from Henry Bottomley, May 16 2001

A020944 a(2n+1) = |a(2n) - a(2n-1)|, a(2n) = a(n) + a(n-1), a(0) = -1.

Original entry on oeis.org

-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, 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

Offset: 0

Clark Kimberling

a(n) = abs(t(n+1)) if n>0 where t(n) is the twisted Stern sequence defined by R. Bacher and M. Coons. - Michael Somos, Jan 08 2011

a(A153893(n)) = 0. - Reinhard Zumkeller, Mar 13 2011

			G.f. = -1 + x + x^3 + x^4 + x^6 + x^7 + 2*x^8 + x^9 + x^10 + x^12 + x^13 + 2*x^14 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..12287
Roland Bacher, Twisting the Stern sequence, arXiv:1005.5627v1 [math.CO], 2010.
Michael Coons, On Some Conjectures concerning Stern's Sequence and its Twist, arXiv:1008.0193v3 [math.NT], 2010.

Cf. A020947, A020948, A020949, A020950, A153893.

a020944 n = a020944_list !! n
a020944_list = -1 : f [1,0] where f (x:y:xs) = x : f (y:xs ++ [x,x+y])
-- Same list generator function as for a020951_list, cf. A020951.
-- Reinhard Zumkeller, Mar 13 2013

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]]; (* Michael Somos, Jul 25 2018 *)

{a(n) = if( n<2,(n==1) - (n==0),  n%2, abs( a(n-1) - a(n-2) ), a(n/2) + a(n/2 - 1) )}; /* Michael Somos, Jan 08 2011 */

{a(n) = my(A, m); if( n<0, 0, m = 1; A = -1 + O(x); while( m <= n, m*=2; A = 2*x + (1 + x + x^2) * subst( A, x, x^2 ) ); polcoeff( A, n ) )}; /* Michael Somos, Jan 08 2011 */

G.f. A(x) satisfies: A(x) = 2*x + (1 + x + x^2) * A(x^2). - Michael Somos, Jan 08 2011

More terms from Henry Bottomley, May 16 2001

Added a(0) from Michael Somos, Jan 08 2011

A020946 a(n) is the smallest number k such that A002487(k) = n.

Original entry on oeis.org

0, 1, 3, 5, 9, 11, 33, 19, 21, 35, 39, 37, 45, 43, 69, 73, 93, 77, 75, 83, 189, 85, 141, 139, 153, 151, 147, 155, 267, 149, 165, 173, 279, 275, 171, 283, 315, 277, 537, 325, 297, 293, 579, 301, 309, 365, 333, 299, 567, 331, 339, 553, 549, 563, 1275, 341, 585, 565, 615, 629

Offset: 0

N. J. A. Sloane and David W. Wilson, Jun 27 2002

nonn

			A002487(33) = 6 and this is the first time 6 appears, so a(6) = 33.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Equals A020950 + 1.

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

aa = {}; a[0] = 0; a[1] = 1; a[n_] := a[n] = If[EvenQ[n], a[n/2], a[(n - 1)/2] + a[(n + 1)/2]]; Do[k = 0; While[a[k] != p, k++]; AppendTo[aa, k], {p, 0, 100}]; aa (* Artur Jasinski, Dec 06 2010 *)

fusc(n)={my(a=1, b=0);while(n,if(bitand(n, 1), b+=a, a+=b);n>>=1); b};
list(N)={
    my(v=vector(N),k);
    forstep(n=1,9e99,2,
        k=fusc(n);
        if(k<=N && !v[k],
            v[k]=n;
            if(vecmin(v),return(v))
        )
    )
}; \\ Charles R Greathouse IV, Dec 20 2011

from itertools import count
from functools import reduce
def A020946(n): return next(filter(lambda k:sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(k)[-1:2:-1],(1,0)))==n,count(1))) if n else 0 # Chai Wah Wu, May 05 2023

A020945 a(2n+1) = |a(2n) - a(2n-1)|, a(2n) = a(n) + a(n-1).

Original entry on oeis.org

1, 1, 0, 2, 2, 1, 1, 2, 1, 4, 3, 3, 0, 2, 2, 3, 1, 3, 2, 5, 3, 7, 4, 6, 2, 3, 1, 2, 1, 4, 3, 5, 2, 4, 2, 4, 2, 5, 3, 7, 4, 8, 4, 10, 6, 11, 5, 10, 5, 8, 3, 5, 2, 4, 2, 3, 1, 3, 2, 5, 3, 7, 4, 8, 4, 7, 3, 6, 3, 6, 3, 6, 3, 6, 3, 7, 4, 8, 4, 10, 6, 11, 5, 12, 7, 12, 5, 14, 9, 16, 7, 17, 10, 16

Offset: 1

Clark Kimberling

nonn

Cf. A020947-A020950.

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

Original entry on oeis.org

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

A020949 Least k such that A(k) = n, where A( ) is sequence A020945.

Original entry on oeis.org

1, 4, 11, 10, 20, 24, 22, 42, 89, 44, 46, 84, 100, 88, 96, 90, 92, 204, 170, 202, 192, 190, 178, 184, 180, 188, 186, 338, 382, 340, 366, 354, 358, 390, 360, 356, 376, 364, 674, 362, 378, 680, 372, 676, 684, 710, 812, 682, 752, 708, 810, 1346, 732

Offset: 1

Clark Kimberling

nonn

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

cp's OEIS Frontend

A020950 a(n) = k-1, where k is smallest number such that A002487(k) = n.

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A020943 a(2n+1) = |a(2n) - a(2n-1)|, a(2n) = a(n) + a(n-1).

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A020944 a(2n+1) = |a(2n) - a(2n-1)|, a(2n) = a(n) + a(n-1), a(0) = -1.

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A020946 a(n) is the smallest number k such that A002487(k) = n.

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A020945 a(2n+1) = |a(2n) - a(2n-1)|, a(2n) = a(n) + a(n-1).

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

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Mathematica

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A020949 Least k such that A(k) = n, where A( ) is sequence A020945.

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