A147880 -id:A147880 - cp's OEIS frontend

A005229 a(1) = a(2) = 1; for n > 2, a(n) = a(a(n-2)) + a(n - a(n-2)).

1, 1, 2, 3, 3, 4, 5, 6, 6, 7, 7, 8, 9, 10, 10, 11, 12, 12, 13, 14, 15, 16, 16, 17, 17, 18, 19, 19, 20, 20, 21, 22, 23, 24, 24, 25, 26, 26, 27, 28, 29, 29, 30, 30, 30, 31, 32, 33, 34, 35, 36, 36, 37, 37, 38, 39, 39, 40, 41, 42, 43, 43, 44, 45, 45, 45, 46

Offset: 1

N. J. A. Sloane, Simon Plouffe

By induction a(n) <= n, but an exact rate of growth is not known.

J. Arkin, D. C. Arney, L. S. Dewald, and W. E. Ebel, Jr., Families of recursive sequences, J. Rec. Math., 22 (No. 22, 1990), 85-94.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..1000
Altug Alkan, On a Generalization of Hofstadter's Q-Sequence: A Family of Chaotic Generational Structures, Complexity (2018), Article ID 8517125.
Nick Hobson, Python program for this sequence
C. L. Mallows, Conway's challenge sequence, Amer. Math. Monthly, 98 (1991), 5-20.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
Eric Weisstein's World of Mathematics, Mallows' Sequence.

Cf. A004001, A051105, A087758, A116591, A147880, A169638.

import Data.Function (on)
a005229 n = a005229_list !! (n-1)
a005229_list = 1 : 1 : zipWith ((+) `on` a005229)
                       a005229_list (zipWith (-) [3..] a005229_list)
-- Reinhard Zumkeller, Jan 17 2014

A005229:= proc(n) option remember;
     if n<=2 then 1 else A005229(A005229(n-2)) +A005229(n-A005229(n-2));
     fi; end;
seq(A005229(n), n=1..70)

a[1] = a[2] = 1; a[n_] := a[n] = a[a[n-2]] + a[n - a[n-2]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Sep 06 2013 *)

a(n)=an[n]; an=vector(100,n,1); for(n=3,100,an[n]=a(a(n-2))+a(n-a(n-2)))

@CachedFunction
def a(n): # A005229
    if (n<3): return 1
    else: return a(a(n-2)) + a(n-a(n-2))
[a(n) for n in (1..100)] # G. C. Greubel, Mar 27 2022

Typo in definition corrected by Nick Hobson, Feb 21 2007

A147871 Expansion of Product_{k > 0} (1 + A147665(k)*x^k).

Original entry on oeis.org

1, 1, 1, 3, 4, 7, 10, 15, 24, 37, 49, 73, 105, 142, 208, 294, 391, 538, 752, 988, 1359, 1812, 2410, 3232, 4270, 5598, 7454, 9721, 12639, 16625, 21445, 27649, 35793, 46235, 59141, 76215, 96975, 123262, 157671, 199625, 252591, 319792, 403262, 507682

Offset: 0

Roger L. Bagula, Nov 16 2008

nonn

			From _Petros Hadjicostas_, Apr 11 2020: (Start)
Let f(m) = A147665(m). Using the strict partitions of each n (see A000009), we get
a(1) = f(1) = 1,
a(2) = f(2) = 1,
a(3) = f(3) + f(1)*f(2) = 2 + 1*1 = 3,
a(4) = f(4) + f(1)*f(3) = 2 + 1*2 = 4,
a(5) = f(5) + f(1)*f(4) + f(2)*f(3) = 3 + 1*2 + 1*2 = 7,
a(6) = f(6) + f(1)*f(5) + f(2)*f(4) + f(1)*f(2)*f(3) = 3 + 1*3 + 1*2 + 1*1*2 = 10,
a(7) = f(7) + f(1)*f(6) + f(2)*f(5) + f(3)*f(4) + f(1)*f(2)*f(4) = 3 + 1*3 + 1*3 + 2*2 + 1*1*2 = 15. (End)

Cf. A000009, A004001, A147559, A147654, A147655, A147665, A147880.

(*A147665*) f[0] = 0; f[1] = 1; f[2] = 1; f[n_] := f[n] = f[f[n - 1]] + If[Mod[ n, 3] == 0, f[f[n/3]], If[Mod[n, 3] == 1, f[f[(n - 1)/3]], f[n - f[(n - 2)/3]]]];
P[x_, n_] := P[x, n] = Product[1 + f[m]*x^m, {m, 0, n}];
Take[CoefficientList[P[x, 45], x], 45] (* Program simplified by Petros Hadjicostas, Apr 13 2020 *)

a(n) = [x^n] Product_{k > 0} (1 + A147665(k)*x^k).

a(n) = Sum_{(b_1,...,b_n)} f(1)^b_1 * f(2)^b_2 * ... * f(n)^b_n, where f(m) = A147665(m), and the sum is taken over all lists (b_1,...,b_n) with b_j in {0,1} and Sum_{j=1..n} j*b_j = n. - Petros Hadjicostas, Apr 11 2020

Various sections edited by Petros Hadjicostas, Apr 11 2020

A147869 Expansion of Product_{k>0} (1 + A004001(k)*x^k).

Original entry on oeis.org

1, 1, 1, 3, 4, 7, 11, 17, 25, 41, 59, 86, 125, 180, 263, 382, 536, 738, 1073, 1466, 2028, 2841, 3889, 5275, 7211, 9800, 13249, 17860, 23948, 31921, 42864, 56802, 75115, 99788, 131239, 172870, 226789, 296404, 386745, 504939, 655227, 849628, 1101270

Offset: 0

Roger L. Bagula, Nov 16 2008

nonn

			From _Petros Hadjicostas_, Apr 11 2020: (Start)
Let f(m) = A004001(m). Using the strict partitions of each n (see A000009), we get
a(1) = f(1) = 1,
a(2) = f(2) = 1,
a(3) = f(3) + f(1)*f(2) = 2 + 1*1 = 3,
a(4) = f(4) + f(1)*f(3) = 2 + 1*2 = 4,
a(5) = f(5) + f(1)*f(4) + f(2)*f(3) = 3 + 1*2 + 1*2 = 7,
a(6) = f(6) + f(1)*f(5) + f(2)*f(4) + f(1)*f(2)*f(3) = 4 + 1*3 + 1*2 + 1*1*2 = 11,
a(7) = f(7) + f(1)*f(6) + f(2)*f(5) + f(3)*f(4) + f(1)*f(2)*f(4) = 4 + 1*4 + 1*3 + 2*2 + 1*1*2 = 17. (End)

Cf. A000009, A004001, A147654, A147655, A147559, A147880.

f[0] = 0; f[1] = 1; f[2] = 1; f[n_] := f[n] = f[f[n - 1]] + f[n - f[n - 1]];
P[x_, n_] := P[x, n] = Product[1 + f[m]*x^m, {m, 0, n}];
Take[CoefficientList[P[x, 45], x], 45]

a(n) = [x^n] Product_{k > 0} (1 + A004001(k)*x^k).

a(n) = Sum_{(b_1,...,b_n)} f(1)^b_1 * f(2)^b_2 * ... * f(n)^b_n, where f(m) = A004001(m), and the sum is taken over all lists (b_1,...,b_n) with b_j in {0,1} and Sum_{j=1..n} j*b_j = n. - Petros Hadjicostas, Apr 11 2020

Various sections edited by Petros Hadjicostas, Apr 11 2020

A147953 Expansion of Product_{k > 0} (1 + f(k)*x^k), where f(n) = A147952(n).

Original entry on oeis.org

1, 1, 1, 3, 4, 7, 9, 14, 22, 32, 43, 61, 89, 118, 167, 235, 312, 417, 572, 748, 1006, 1326, 1744, 2283, 2982, 3878, 5048, 6518, 8355, 10786, 13727, 17436, 22173, 28250, 35561, 45008, 56651, 70818, 88992, 111280, 138431, 172284, 214019, 265166, 328127

Offset: 0

Roger L. Bagula, Nov 17 2008

nonn

Cf. A147654, A147655, A147665, A147869, A147871, A147880, A147952.

f[0] = 0; f[1] = 1; f[2] = 1;
f[n_] := f[n] = f[f[n - 2]] + If[Mod[n, 3] == 0,f[f[n/3]], If[Mod[n, 3] == 1, f[f[(n - 1)/3]], f[n - f[(n - 2)/3]]]];
P[x_, n_] := P[x, n] = Product[1 + f[m] x^m, {m, 0, n}];
Take[CoefficientList[P[x, 45], x],45]
(* Program edited and corrected by Petros Hadjicostas, Apr 12 2020 *)

a(n) = [x^n] Product_{k > 0} (1 + f(k)*x^k), where f(1) = f(2) = 1, and for m >= 3, f(m) = f(f(m-2)) + r(m), where r(m) = f(f(floor(m/3)) when m == 0 or 1 (mod 3) and = f(m - f(floor(m/3))) when m == 2 (mod 3).

Various sections edited by Petros Hadjicostas, Apr 12 2020

A152006 Expansion of Product_{k > 0} (1 + f(k)*x^k), where f(1) = 1 and f(m) = prime(m-1) for m >= 2.

Original entry on oeis.org

1, 1, 2, 5, 8, 18, 34, 63, 102, 203, 336, 589, 999, 1675, 2799, 4768, 7561, 12224, 20513, 31724, 51621, 81976, 128560, 199192, 312536, 482806, 744847, 1147952, 1755931, 2649474, 4051413, 6069450, 9105323, 13747364, 20335077, 30508629, 45198631

Offset: 0

Roger L. Bagula, Nov 19 2008

nonn

Cf. A147557, A147654, A147655, A147665, A147869, A147871, A147880, A147953.

f[n_] = If[n < 2, n, Prime[n - 1]];
P[x_, n_] := P[x, n] = Product[1 + f[m]*x^m, {m, 0, n}];
Take[CoefficientList[P[x, 37], x],37]
(* Program edited and corrected by Petros Hadjicostas, Apr 12 2020 *)

a(n) = [x^n] Product_{k > 0} (1 + f(k)*x^k), where f(1) = 1 and f(m) = prime(m-1) for m >= 2.

Various sections edited by Petros Hadjicostas, Apr 12 2020

cp's OEIS Frontend

A005229 a(1) = a(2) = 1; for n > 2, a(n) = a(a(n-2)) + a(n - a(n-2)).

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A147871 Expansion of Product_{k > 0} (1 + A147665(k)*x^k).

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A147869 Expansion of Product_{k>0} (1 + A004001(k)*x^k).

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A147953 Expansion of Product_{k > 0} (1 + f(k)*x^k), where f(n) = A147952(n).

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A152006 Expansion of Product_{k > 0} (1 + f(k)*x^k), where f(1) = 1 and f(m) = prime(m-1) for m >= 2.

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