A020989 -id:A020989 - cp's OEIS frontend

A180032 Eight white queens and one red queen on a 3 X 3 chessboard. G.f.: (1+x)/(1-5x-7x^2).

1, 6, 37, 227, 1394, 8559, 52553, 322678, 1981261, 12165051, 74694082, 458625767, 2815987409, 17290317414, 106163498933, 651849716563, 4002393075346, 24574913392671, 150891318490777, 926480986202582, 5688644160448349

Offset: 0

Johannes W. Meijer, Aug 09 2010

The a(n) represent the number of n-move routes of a fairy chess piece starting in a given corner or side square (m = 1, 3, 7, 9; 2, 4, 6, 8) on a 3 X 3 chessboard. This fairy chess piece behaves like a white chess queen on the eight side and corner squares but on the central square the queen explodes with fury and turns into a red queen.
On a 3 X 3 chessboard there are 2^9 = 512 ways to explode with fury on the central square (we assume here that a red queen might behave like a white queen). The red queen is represented by the A[5] vector in the fifth row of the adjacency matrix A, see the Maple program. For the corner and side squares the 512 red queens lead to 17 red queen sequences, see the cross-references for the complete set.
The sequence above corresponds to 8 red queen vectors, i.e., A[5] vectors, with decimal values 239, 367, 431, 463, 487, 491, 493 and 494. The central square leads for these vectors to A152240.
This sequence belongs to a family of sequences with g.f. (1+x)/(1 - 5*x - k*x^2). The members of this family that are red queen sequences are A180030 (k=8), A180032 (k=7; this sequence), A000400 (k=6), A180033 (k=5), A126501 (k=4), A180035 (k=3), A180037 (k=2) A015449 (k=1) and A003948 (k=0). Other members of this family are A030221 (k=-1), A109114 (k=-3), A020989 (k=-4), A166060 (k=-6).
Inverse binomial transform of A054413.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Wikipedia, Alice in Wonderland (2010 film)
Index entries for linear recurrences with constant coefficients, signature (5, 7).

Cf. A180028 (Central square).

Cf. Red queen sequences corner and side squares [decimal value A[5]]: A090018 [511], A135030 [255], A180030 [495], A005668 [127], A180032 [239], A000400 [63], A180033 [47], A001109 [31], A126501 [15], A154244 [23], A180035 [7], A138395 [19], A180037 [3], A084326 [17], A015449 [1], A003463 [16], A003948 [0].

I:=[1,6]; [n le 2 select I[n] else 5*Self(n-1)+7*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 15 2011

with(LinearAlgebra): nmax:=20; m:=1; A[5]:= [1,1,1,1,0,1,1,1,0]: A:=Matrix([[0,1,1,1,1,0,1,0,1], [1,0,1,1,1,1,0,1,0], [1,1,0,0,1,1,1,0,1], [1,1,0,0,1,1,1,1,0], A[5], [0,1,1,1,1,0,0,1,1], [1,0,1,1,1,0,0,1,1], [0,1,0,1,1,1,1,0,1], [1,0,1,0,1,1,1,1,0]]): for n from 0 to nmax do B(n):=A^n: a(n):= add(B(n)[m,k],k=1..9): od: seq(a(n), n=0..nmax);

LinearRecurrence[{5,7},{1,6},40] (* Vincenzo Librandi, Nov 15 2011 *)
CoefficientList[Series[(1+x)/(1-5x-7x^2),{x,0,30}],x] (* Harvey P. Dale, Apr 04 2024 *)

G.f.: (1+x)/(1 - 5*x - 7*x^2).
a(n) = 5*a(n-1) + 7*a(n-2) with a(0) = 1 and a(1) = 6.
a(n) = ((7+9*A)*A^(-n-1) + (7+9*B)*B^(-n-1))/53 with A = (-5+sqrt(53))/14 and B = (-5-sqrt(53))/14.

A083597 a(n) = (7*4^n - 4)/3.

Original entry on oeis.org

1, 8, 36, 148, 596, 2388, 9556, 38228, 152916, 611668, 2446676, 9786708, 39146836, 156587348, 626349396, 2505397588, 10021590356, 40086361428, 160345445716, 641381782868, 2565527131476, 10262108525908, 41048434103636

Offset: 0

Paul Barry, May 02 2003

Binomial transform of A082541.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-4).

[(7*4^n-4)/3: n in [0..25]]; // Vincenzo Librandi, Jul 24 2011

(7*4^Range[0,25]-4)/3 (* or *) LinearRecurrence[{5,-4},{1,8},26] (* Harvey P. Dale, Jul 23 2011 *)
CoefficientList[Series[(1 + 3 x)/((1 - 4 x) (1 - x)), {x, 0, 22}], x] (* Michael De Vlieger, Mar 03 2017 *)

a(n)=(7*4^n-4)/3 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = (7*4^n-4)/3.
G.f.: (1+3*x)/((1-4*x)*(1-x)).
E.g.f.: (7*exp(4*x)-4*exp(x))/3.
a(n) = 4*a(n-1) + 4, n > 0. - Gary Detlefs, Jun 23 2010
a(0)=1, a(1)=8, a(n) = 5*a(n-1) - 4*a(n-2). - Harvey P. Dale, Jul 23 2011
a(n) = A020988(n) + A020989(n), n >= 0. - Yosu Yurramendi, Mar 03 2017

A119726 Triangular array read by rows: T(n,1) = T(n,n) = 1, T(n,k) = 4T(n-1, k-1) + 2T(n-1, k).

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 16, 26, 1, 1, 36, 116, 106, 1, 1, 76, 376, 676, 426, 1, 1, 156, 1056, 2856, 3556, 1706, 1, 1, 316, 2736, 9936, 18536, 17636, 6826, 1, 1, 636, 6736, 30816, 76816, 109416, 84196, 27306, 1, 1, 1276, 16016, 88576, 276896, 526096, 606056, 391396, 109226, 1

Offset: 1

Zerinvary Lajos, Jun 14 2006

Second column is A048487.

Second diagonal is A020989.

			Triangle begins as:
  1;
  1,    1;
  1,    6,     1;
  1,   16,    26,     1;
  1,   36,   116,   106,      1;
  1,   76,   376,   676,    426,      1;
  1,  156,  1056,  2856,   3556,   1706,      1;
  1,  316,  2736,  9936,  18536,  17636,   6826,      1;
  1,  636,  6736, 30816,  76816, 109416,  84196,  27306,      1;
  1, 1276, 16016, 88576, 276896, 526096, 606056, 391396, 109226, 1;

TERMESZET VILAGA XI.TERMESZET-TUDOMANY DIAKPALYAZAT 133.EVF. 6.SZ. jun. 2002. Vegh Lea (and Vegh Erika): "Pascal-tipusu haromszogek" http://www.kfki.hu/chemonet/TermVil/tv2002/tv0206/tartalom.html

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Cf. A007318, A020989, A048483, A048487, A119725, A119727, A123208.

function T(n,k)
  if k eq 1 or k eq n then return 1;
  else return 4*T(n-1,k-1) + 2*T(n-1,k);
  end if;
  return T;
end function;
[T(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Nov 18 2019

T:= proc(n, k) option remember;
      if k=1 and k=n then 1
    else 4*T(n-1, k-1) + 2*T(n-1, k)
      fi
end: seq(seq(T(n, k), k=1..n), n=1..12); # G. C. Greubel, Nov 18 2019

T[n_, k_]:= T[n, k]= If[k==1 || k==n, 1, 4*T[n-1, k-1] + 2*T[n-1, k]]; Table[T[n,k], {n,10}, {k,n}]//Flatten (* G. C. Greubel, Nov 18 2019 *)

T(n,k) = if(k==1 || k==n, 1, 4*T(n-1,k-1) + 2*T(n-1,k));

@CachedFunction
def T(n, k):
    if (k==1 or k==n): return 1
    else: return 4*T(n-1, k-1) + 2*T(n-1, k)
[[T(n, k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Nov 18 2019

Edited by Don Reble, Jul 24 2006

A238275 a(n) = (4*7^n - 1)/3.

Original entry on oeis.org

1, 9, 65, 457, 3201, 22409, 156865, 1098057, 7686401, 53804809, 376633665, 2636435657, 18455049601, 129185347209, 904297430465, 6330082013257, 44310574092801, 310174018649609, 2171218130547265, 15198526913830857, 106389688396816001, 744727818777712009

Offset: 0

Philippe Deléham, Feb 21 2014

Sum of n-th row of triangle of powers of 7: 1; 1 7 1; 1 7 49 7 1; 1 7 49 343 49 7 1; ...

Number of cubes in the crystal structure cubic carbon CCC(n+1), defined in the Baig et al. and in the Gao et al. references. - Emeric Deutsch, May 28 2018

			a(0) = 1;
a(1) = 1 + 7 + 1 = 9;
a(2) = 1 + 7 + 49 + 7 + 1 = 65;
a(3) = 1 + 7 + 49 + 343 + 49 + 7 + 1 = 457; etc.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
A. Q. Baig, M. Imran, W. Khalid, and M. Naeem, Molecular description of carbon graphite and crystal cubic carbon structures, Canadian J. Chem., 95, 674-686, 2017.
W. Gao, M. K. Siddiqui, M. Naeem and N. A. Rehman, Topological characterization of carbon graphite and crystal cubic carbon structures, Molecules, 22, 1496, 1-12, 2017.
Index entries for linear recurrences with constant coefficients, signature (8,-7).

Cf. Similar sequences: A151575, A000012, A040000, A005408, A033484, A048473, A020989, A057651, A061801, this sequence, A238276, A138894, A090843, A199023.

Cf. A112468, A112739.

[(4*7^n - 1)/3: n in [0..30]]; // Vincenzo Librandi, Feb 23 2014

Table[(4 7^n - 1)/3, {n, 0, 40}] (* Vincenzo Librandi, Feb 23 2014 *)

G.f.: (1+x)/((1-x)*(1-7*x)).
a(n) = 7*a(n-1) + 2, a(0) = 1.
a(n) = 8*a(n-1) - 7*a(n-2), a(0) = 1, a(1) = 9.
a(n) = Sum_{k=0..n} A112468(n,k)*8^k.
E.g.f.: exp(x)*(4*exp(6*x) - 1)/3. - Stefano Spezia, Feb 12 2025

A238276 a(n) = (9*8^n - 2)/7.

Original entry on oeis.org

1, 10, 82, 658, 5266, 42130, 337042, 2696338, 21570706, 172565650, 1380525202, 11044201618, 88353612946, 706828903570, 5654631228562, 45237049828498, 361896398627986, 2895171189023890, 23161369512191122, 185290956097528978, 1482327648780231826

Offset: 0

Philippe Deléham, Feb 21 2014

Sum of n-th row of triangle of powers of 8: 1; 1 8 1; 1 8 64 8 1; 1 8 64 512 64 8 1; ...

			a(0) = 1;
a(1) = 1 + 8 + 1 = 10;
a(2) = 1 + 8 + 64 + 8 + 1 = 82;
a(3) = 1 + 8 + 64 + 512 + 64 + 8 + 1 = 658; etc.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (9, -8).

Cf. Similar sequences: A151575, A000012, A040000, A005408, A033484, A048473, A020989, A057651, A061801, A238275, this sequence, A138894, A090843, A199023.

Cf. A112468, A112739.

[(9*8^n - 2)/7: n in [0..30]]; // Vincenzo Librandi, Feb 23 2014

Table[(9 8^n - 2)/7, {n, 0, 50}] (* Vincenzo Librandi, Feb 23 2014 *)

G.f.: (1+x)/((1-x)*(1-8*x)).
a(n) = 8*a(n-1) + 2, a(0) = 1.
a(n) = 9*a(n-1) - 8*a(n-2), a(0) = 1, a(1) = 10.
a(n) = Sum_{k=0..n} A112468(n,k)*9^k.

Corrected by Vincenzo Librandi, Feb 23 2014

A356327 Replace 2^k in binary expansion of n with A039834(1+k).

Original entry on oeis.org

0, 1, -1, 0, 2, 3, 1, 2, -3, -2, -4, -3, -1, 0, -2, -1, 5, 6, 4, 5, 7, 8, 6, 7, 2, 3, 1, 2, 4, 5, 3, 4, -8, -7, -9, -8, -6, -5, -7, -6, -11, -10, -12, -11, -9, -8, -10, -9, -3, -2, -4, -3, -1, 0, -2, -1, -6, -5, -7, -6, -4, -3, -5, -4, 13, 14, 12, 13, 15, 16

Offset: 0

Rémy Sigrist, Aug 03 2022

This sequence has similarities with A022290, and is related to negaFibonacci representations.

			For n = 13:
- 13 = 2^3 + 2^2 + 2^0,
- so a(13) = A039834(4) + A039834(3) + A039834(1) = -3 + 2 + 1 = 0.

Rémy Sigrist, Table of n, a(n) for n = 0..8191
Wikipedia, NegaFibonacci coding
Index entries for sequences related to Zeckendorf expansion of n

Cf. A000201, A001950, A003714, A004957, A020989, A022290, A026351, A039834, A060144, A072197, A189663, A215024, A215025, A309076, A356325, A356326.

Table[Reverse[#].Fibonacci[-Range[Length[#]]] &@ IntegerDigits[n, 2], {n, 0, 69}] (* Rémy Sigrist, Aug 05 2022 *)

a(n) = { my (v=0, k); while (n, n-=2^k=valuation(n, 2); v+=fibonacci(-1-k)); return (v) }

from sympy import fibonacci
def A356327(n): return sum(fibonacci(-a)*int(b) for a, b in enumerate(bin(n)[:1:-1],start=1)) # Chai Wah Wu, Aug 31 2022

a(n) = Sum_{k>=0} A030308(n,k)*A039834(1+k).
a(A215024(n)) = n.
a(A215025(n)) = -n.
a(A003714(n)) = A309076(n).
Empirically:
- a(n) = 0 iff n = 0 or n belongs to A072197,
- a(n) = 1 iff n belongs to A020989,
- a(2*A215024(n)) = -A000201(n) for n > 0,
- a(3*A215024(n)) = -A060143(n),
- a(floor(A215024(n)/2)) = -A060143(n),
- a(4*A215024(n)) = A001950(n) for n > 0,
- a(floor(A215024(n)/4)) = A189663(n) for n > 0,
- a(2*A215025(n)) = A026351(n),
- a(3*A215025(n)) = A019446(n) for n > 0,
- a(floor(A215025(n)/2)) = A019446(n) for n > 0,
- a(4*A215025(n)) = -A004957(n),
- a(floor(A215025(n)/4)) = -A060144(n+1) for n >= 0.

A176965 a(n) = 2^(n-1) - (2^n*(-1)^n + 2)/3.

Original entry on oeis.org

1, 0, 6, 2, 26, 10, 106, 42, 426, 170, 1706, 682, 6826, 2730, 27306, 10922, 109226, 43690, 436906, 174762, 1747626, 699050, 6990506, 2796202, 27962026, 11184810, 111848106, 44739242, 447392426, 178956970, 1789569706, 715827882, 7158278826

Offset: 1

Roger L. Bagula, Apr 29 2010

The ratio a(n+1)/a(n) approaches 10 for even n and 2/5 for odd n as n->infinity.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,4,-4).

Merger of A020988 (even n) and A020989 (odd n).

List([1..30], n-> (3*2^(n-1) -(-2)^n -2)/3); # G. C. Greubel, Dec 28 2019

[(3*2^(n-1) -(-2)^n -2)/3: n in [1..30]]; // G. C. Greubel, Dec 28 2019

seq( (3*2^(n-1) -(-2)^n -2)/3, n=1..30); # G. C. Greubel, Dec 28 2019

a[n_]:= a[n]= 2^(n-1)*If[n==1, 1, a[n-1]/2 +(-1)^(n-1)*Sqrt[(5 +4*(-1)^(n-1) )]/2]; Table[a[n], {n,30}]
LinearRecurrence[{1,4,-4}, {1,0,6}, 30] (* G. C. Greubel, Dec 28 2019 *)

vector(30, n, (3*2^(n-1) -(-2)^n -2)/3 ) \\ G. C. Greubel, Dec 28 2019

[(3*2^(n-1) -(-2)^n -2)/3 for n in (1..30)] # G. C. Greubel, Dec 28 2019

From R. J. Mathar, Apr 30 2010: (Start)
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3).
G.f.: x*(1 - x + 2*x^2)/( (1-x)*(1+2*x)*(1-2*x) ). (End)
a(n) = A087231(n), n > 2. - R. J. Mathar, May 03 2010
a(2n-1) = A061547(2n), a(2n) = A061547(2n-1), n > 0. - Yosu Yurramendi, Dec 23 2016
a(n+1) = 2*A096773(n), n > 0. - Yosu Yurramendi, Dec 30 2016
a(2n-1) = A020989(n-1), a(2n) = A020988(n-1), n > 0. - Yosu Yurramendi, Jan 03 2017
a(2n-1) = (A083597(n-1) + A000302(n-1))/2, a(2n) = (A083597(n-1) - A000302(n-1))/2, n > 0. - Yosu Yurramendi, Mar 04 2017
a(n+2) = 4*a(n) + 2, a(1) = 1, a(2) = 0, n > 0. - Yosu Yurramendi, Mar 07 2017
a(n) = (-16 + (9 - (-1)^n) * 2^(n - (-1)^n))/24. - Loren M. Pearson, Dec 28 2019
E.g.f.: (3*exp(2*x) - 4*exp(x) + 3 - 2*exp(-2*x))/6. - G. C. Greubel, Dec 28 2019
a(n) = (2^n*5^(n mod 2) - 4)/6. - Heinz Ebert, Jun 29 2021

A180031 Number of n-move paths on a 3 X 3 chessboard of a queen starting or ending in the central square.

Original entry on oeis.org

1, 8, 48, 304, 1904, 11952, 74992, 470576, 2952816, 18528688, 116265968, 729559344, 4577924464, 28726097072, 180253881072, 1131078181936, 7097421958256, 44535735246768, 279458051899888, 1753576141473584

Offset: 0

Johannes W. Meijer, Aug 09 2010

The a(n) represent the number of n-move paths of a chess queen starting or ending in the central square (m = 5) on a 3 X 3 chessboard. The other squares lead to A180030.
To determine the a(n) we can either sum the components of the column vector A^n[k,m], with A the adjacency matrix of the queen's graph, or we can sum the components of the row vector A^n[m,k], see the Maple program.
Closely related with this sequence are the red queen sequences, see A180028 and A180032.
This sequence belongs to a family of sequences with g.f. (1+k*x)/(1 - 5*x - (k+5)*x^2). The members of this family that are red queen sequences are A180031 (k=3; this sequence), A152240 (k=2), A000400 (k=1), A057088 (k=0), A122690 (k=-1), A180036 (k=-2), A180038 (k=-3), A015449 (k=-4) and A000007 (k=-5). Other members of this family are A030221 (k= -6), 3*A109114 (k=-8), 4*A020989 (k=-9), 6*A166060 (k=-11).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (5, 8).

I:=[1,8]; [n le 2 select I[n] else 5*Self(n-1)+8*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 15 2011

with(LinearAlgebra): nmax:=19; m:=5; A[5]:= [1,1,1,1,0,1,1,1,1]: A:=Matrix([[0,1,1,1,1,0,1,0,1], [1,0,1,1,1,1,0,1,0], [1,1,0,0,1,1,1,0,1], [1,1,0,0,1,1,1,1,0], A[5], [0,1,1,1,1,0,0,1,1], [1,0,1,1,1,0,0,1,1], [0,1,0,1,1,1,1,0,1], [1,0,1,0,1,1,1,1,0]]): for n from 0 to nmax do B(n):=A^n: a(n):= add(B(n)[m,k],k=1..9): od: seq(a(n), n=0..nmax);

LinearRecurrence[{5,8},{1,8},50] (* Vincenzo Librandi, Nov 15 2011 *)

G.f.: (1+3*x)/(1 - 5*x - 8*x^2).
a(n) = 5*a(n-1) + 8*a(n-2) with a(0) = 1 and a(1) = 8.
a(n) = ((A+11)*A^(-n-1) + (B+11)*B^(-n-1))/57 with A = (-5+sqrt(57))/16 and B = (-5-sqrt(57))/16.

A238303 Triangle T(n,k), 0<=k<=n, read by rows given by T(n,0) = 1, T(n,k) = 2 if k>0.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 0

Philippe Deléham, Feb 24 2014

Row sums are A005408(n).
Diagonals sums are A109613(n).
Sum_{k=0..n} T(n,k)*x^k = A033999(n), A000012(n), A005408(n), A036563(n+2), A058481(n+1), A083584(n), A137410(n), A233325(n), A233326(n), A233328(n), A211866(n+1), A165402(n+1) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 respectively.
Sum_{k=0..n} T(n,k)*x^(n-k) = A151575(n), A000012(n), A040000(n), A005408(n), A033484(n), A048473(n), A020989(n), A057651(n), A061801(n), A238275(n), A238276(n), A138894(n), A090843(n), A199023(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 respectively.
Sum_{k=0..n} T(n,k)^x = A000027(n+1), A005408(n), A016813(n), A017077(n) for x = 0, 1, 2, 3 respectively.
Sum_{k=0..n} k*T(n,k) = A002378(n).
Sum_{k=0..n} A000045(k)*T(n,k) = A019274(n+2).
Sum_{k=0..n} A000142(k)*T(n,k) = A066237(n+1).

			Triangle begins:
1;
1, 2;
1, 2, 2;
1, 2, 2, 2;
1, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2;
1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2;
...

Antti Karttunen, Table of n, a(n) for n = 0..22154; the first 210 rows of triangle

Cf. Diagonals: A040000.
Cf. Columns: A000012, A007395.
First differences of A001614.

A238303(n) = (2-ispolygonal(n,3)); \\ Antti Karttunen, Jan 19 2025

T(n,0) = A000012(n) = 1, T(n+k,k) = A007395(n) = 2 for k>0.

Data section extended to a(104) by Antti Karttunen, Jan 19 2025

A308551 Start with an empty stack S; for n = 1, 2, 3, ..., interpret the binary representation of n from left to right as follows: in case of bit 1, push the number 1 on top of S, in case of bit 0, replace the two numbers on top of S with their sum; a(n) gives the number on top of S after processing n.

Original entry on oeis.org

1, 2, 1, 3, 1, 2, 1, 4, 1, 3, 1, 3, 1, 2, 1, 5, 1, 12, 1, 4, 1, 2, 1, 4, 1, 3, 1, 3, 1, 2, 1, 6, 1, 15, 1, 23, 1, 2, 1, 5, 1, 4, 1, 4, 1, 2, 1, 5, 1, 4, 1, 4, 1, 2, 1, 4, 1, 3, 1, 3, 1, 2, 1, 7, 1, 19, 1, 30, 1, 2, 1, 47, 1, 57, 1, 5, 1, 2, 1, 6, 1, 20, 1, 5

Offset: 1

Rémy Sigrist, Jun 07 2019

After processing n, S has A268289(n) elements, the sum of which is A000788(n).
Every positive integer appears infinitely many times in the sequence.
The sequence has the same shape when represented at different scales.

			The first terms, alongside the binary representation of n and the evolution of stack S, are:
  n  a(n)  bin(n)  S
  -  ----  ------  -------------------------------------------------
  1     1       1  () -> (1)
  2     2      10  (1) -> (1,1) -> (2)
  3     1      11  (2) -> (2,1) -> (2,1,1)
  4     3     100  (2,1,1) -> (2,1,1,1) -> (2,1,2) -> (2,3)
  5     1     101  (2,3) -> (2,3,1) -> (2,4) -> (2,4,1)
  6     2     110  (2,4,1) -> (2,4,1,1) -> (2,4,1,1,1) -> (2,4,1,2)

Rémy Sigrist, Table of n, a(n) for n = 1..8192
Sean A. Irvine, Java program (github)
Rémy Sigrist, PARI program
Wikipedia, Stack (abstract data type)

Cf. A000788, A020989, A268289.

Java
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See Links section.
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PARI
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See Links section.
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a(n) = 1 iff n is odd.
a(A020989(k)) = k + 1 for any k >= 0.
If n is in A014486, then a(n) = a(n-1) + A000120(n) = 1 + A000120(n). - Charlie Neder, Jun 07 2019

cp's OEIS Frontend

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