A147665 -id:A147665 - cp's OEIS frontend

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

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

A166497 A Per Bak sand pile collapse sequence using A147665 in the A153112 form.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 4, 3, 2, 4, 3, 2, 4, 3, 2, 5, 1, 2, 2, 2, 2, 3, 2, 2, 4, 3, 2, 4, 3, 2, 4, 3, 2, 5, 4, 3, 5, 4, 2, 5, 5, 5, 5, 4, 3, 6, 3, 2, 4, 4, 4, 4, 3, 2, 5, 4, 3, 5, 5, 5, 6, 3, 2, 7, 3, 2, 4, 5, 6, 5, 4, 3, 4, 3, 2, 4, 3, 2, 4, 1, 2, 2, 2, 2, 3, 2, 2, 4, 3, 2, 4, 3, 2, 4

Offset: 0

Roger L. Bagula, Oct 15 2009

Per Bak, "How nature works, the science of self-organized criticality", Springer-Verlag, New York, 1996, pages 49-64

Cf. A153112, A147665.

f[0] = 1; f[1] = 1; f[2] = 1;
f[n_] := f[n] = If[Mod[Floor[Sum[f[i], {i, 0, n - 1}]/2], 2^(4 + Mod[n, 3])] == 1, 1 + Mod[n, 3], 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]]]]];
Table[f[n], {n, 0, 200}]

A147880 Expansion of Product_{k > 0} (1 + A005229(k)*x^k).

Original entry on oeis.org

1, 1, 1, 3, 5, 8, 12, 21, 30, 50, 75, 110, 169, 249, 361, 539, 757, 1076, 1583, 2207, 3121, 4415, 6184, 8468, 11775, 16274, 22314, 30601, 41745, 56412, 77008, 103507, 138383, 186928, 249855, 333375, 443898, 588402, 778276, 1031126, 1356945, 1780645

Offset: 0

Roger L. Bagula, Nov 16 2008

nonn

			From _Petros Hadjicostas_, Apr 10 2020: (Start)
Let f(m) = A005229(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) = 3 + 1*2 = 5,
a(5) = f(5) + f(1)*f(4) + f(2)*f(3) = 3 + 1*3 + 1*2 = 8,
a(6) = f(6) + f(1)*f(5) + f(2)*f(4) + f(1)*f(2)*f(3) = 4 + 1*3 + 1*3 + 1*1*2 = 12,
a(7) = f(7) + f(1)*f(6) + f(2)*f(5) + f(3)*f(4) + f(1)*f(2)*f(4) = 5 + 1*4 + 1*3 + 2*3 + 1*1*3 = 21. (End)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A000009, A004001, A005229, A147559, A147654, A147655, A147665.

(*A005229*) f[n_Integer?Positive] := f[n] = f[ f[n - 2]] + f[n - f[n - 2]]; f[0] = 0; f[1] = f[2] = 1;
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 *)

\\ here B(n) is A005229 as vector.
B(n)={my(a=vector(n, i, 1)); for(n=3, n, a[n] = a[a[n-2]] + a[n-a[n-2]]); a}
seq(n)={my(v=B(n)); Vec(prod(k=1, n, 1 + v[k]*x^k + O(x*x^n)))} \\ Andrew Howroyd, Apr 10 2020

G.f.: Product_{k > 0} (1 + A005229(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) = A005229(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.

Various sections edited by Joerg Arndt and Petros Hadjicostas, Apr 10 2020

A147952 a(0) = 0, a(1) = a(2) = 1, and for n >= 3, a(n) = a(a(n-2)) + r(n), where r(n) = a(a(floor(n/3))) when n == 0 or 1 (mod 3) and = a(n - a(floor(n/3))) when n == 2 (mod 3).

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 2, 3, 4, 3, 3, 5, 3, 4, 5, 4, 5, 7, 4, 4, 6, 4, 4, 8, 4, 6, 6, 4, 4, 8, 4, 6, 10, 5, 6, 7, 4, 5, 9, 5, 5, 8, 6, 7, 7, 5, 5, 10, 6, 6, 7, 5, 6, 8, 4, 6, 8, 4, 6, 8, 4, 6, 10, 4, 5, 8, 5, 6, 8, 6, 8, 6, 6, 4, 10, 4, 5, 8, 5, 6, 13, 4, 6, 8, 4, 6, 8, 6, 8, 6, 6, 4, 10, 4, 5, 8, 6, 7, 10, 6, 6

Offset: 0

Roger L. Bagula, Nov 17 2008

nonn

Cf. A147665, A147953, A147981.

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]]]]; Table[f[n], {n, 0, 100}]

a(n) = a(a(n - 2)) + If[Mod[n, 3] == 0, a(a(n/3)), If[Mod[n, 3] == 1, a(a((n - 1)/3)), a(n - a((n - 2)/3))] for n >= 3 with a(0) = 0 and a(1) = a(2) = 1. [edited by Petros Hadjicostas, Apr 13 2020]

Name edited by Petros Hadjicostas, Apr 13 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

A147954 a(0) = 0, a(1) = a(2) = 1, a(n) = a(a(n-1)) + a(n-a(n-1)) for 3 <= n <= 5, and a(n) = a(a(n-1)) + r(n) for n >= 6, where r(n) = a(a(floor(n/6))) for n == 0, 1, 2, 3, 4 (mod 6), and r(n) = a(n - a(floor(n/6))) for n == 5 (mod 6).

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 5, 4, 3, 3, 3, 3, 5, 4, 3, 3, 3, 3, 5, 4, 3, 3, 3, 3, 5, 5, 5, 5, 5, 5, 8, 5, 5, 5, 5, 5, 8, 5, 5, 5, 5, 5, 8, 5, 5, 5, 5, 5, 8, 5, 5, 5, 5, 5, 8, 5, 5, 5, 5, 5, 8, 6, 6, 6, 6, 6, 9, 5, 5, 5, 5, 5, 8, 5, 5, 5, 5, 5, 8, 5, 5, 5

Offset: 0

Roger L. Bagula, Nov 17 2008

nonn

Cf. A004001, A147665, A147955.

a := proc(n) local v; option remember;
if n = 0 then v := 0; end if;
if n = 1 or n = 2 then v := 1; end if;
if 3 <= n and n <= 5 then v := a(a(n - 1)) + a(n - a(n - 1)); end if;
if 6 <= n and 5 <> n mod 6 then v := a(a(n - 1)) + a(a(floor(n/6))); end if;
if 6 <= n and 5 = n mod 6 then v := a(a(n - 1)) + a(n - a(floor(n/6))); end if; v; end proc; # Petros Hadjicostas, Apr 21 2020

f[0] = 0; f[1] = 1; f[2] = 1;
f[n_] := f[n] =
  f[f[n - 1]] +
   If[n < 6, f[n - f[n - 1]],
    If[Mod[n, 6] == 0, f[f[n/6]],
     If[Mod[n, 6] == 1, f[f[(n - 1)/6]],
      If[Mod[n, 6] == 2, f[f[(n - 2)/6]],
       If[Mod[n, 6] == 3, f[f[(n - 3)/6]],
        If[Mod[n, 6] == 4, f[f[(n - 4)/6]], f[n - f[(n - 5)/6]]]]]]]];
Table[f[n], {n, 0, 300}]

Name, data, and Mathematica program edited and corrected by Petros Hadjicostas, Apr 21 2020

A147955 Expansion of Product_{k >= 0} (1 + A147954(k)*x^k).

Original entry on oeis.org

1, 1, 1, 3, 4, 7, 10, 15, 22, 34, 46, 65, 93, 123, 175, 245, 324, 425, 592, 764, 1015, 1352, 1750, 2266, 2931, 3793, 4897, 6259, 7930, 10080, 12788, 16047, 20176, 25482, 31641, 39630, 49306, 60932, 75552, 93432, 114597, 141013, 173259, 211595, 258933, 316375, 384359, 466927, 566443

Offset: 0

Roger L. Bagula, Nov 17 2008

nonn

			From _Petros Hadjicostas_, Apr 21 2020: (Start)
Let f(m) = A147954(m). Using the strict partitions of 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, A147655, A147665, A147871, A147954.

f := proc(n) local v; option remember;
if n = 0 then v := 0; end if;
if n = 1 or n = 2 then v := 1; end if;
if 3 <= n and n <= 5 then v := f(f(n - 1)) + f(n - f(n - 1)); end if;
if 6 <= n and 5 <> n mod 6 then v := f(f(n - 1)) + f(f(floor(n/6))); end if;
if 6 <= n and 5 = n mod 6 then v := f(f(n - 1)) + f(n - f(floor(n/6))); end if; v; end proc; # this gives sequence A147954
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +`if`(i>n, 0, b(n-i, i-1)*f(i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50); # Petros Hadjicostas, Apr 21 2020 (using Alois P. Heinz's program from A147655)

f[0] = 0; f[1] = 1; f[2] = 1;
f[n_] := f[n] =
   f[f[n - 1]] +
    If[n < 6, f[n - f[n - 1]],
     If[Mod[n, 6] == 0, f[f[n/6]],
      If[Mod[n, 6] == 1, f[f[(n - 1)/6]],
       If[Mod[n, 6] == 2, f[f[(n - 2)/6]],
        If[Mod[n, 6] == 3, f[f[(n - 3)/6]],
         If[Mod[n, 6] == 4, f[f[(n - 4)/6]], f[n - f[(n - 5)/6]]]]]]]];
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 + A147954(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) = A147954(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 21 2020

Name, data, and Mathematica program edited and corrected by Petros Hadjicostas, Apr 21 2020

A147879 Expansion of Product_{k>=1} (1 + x^k*A005185(k)).

Original entry on oeis.org

1, 1, 1, 3, 5, 8, 12, 21, 29, 49, 73, 105, 162, 236, 338, 502, 706, 984, 1441, 1998, 2800, 3934, 5472, 7407, 10210, 14053, 19066, 25986, 35134, 47010, 63739, 85008, 112610, 150861, 200133, 264838, 349587, 459970, 602763, 792220, 1034136, 1345530

Offset: 0

Roger L. Bagula, Nov 16 2008

nonn

Cf. A004001, A005185, A147559, A147654, A147655, A147665, A147869, A147871.

f[n_Integer?Positive] := f[n] = f[n - f[n - 1]] + f[n - f[n - 2]]; f[0] = 0; f[1] = f[2] = 1; (* A005185 *)
nmax = 41; CoefficientList[Series[Product[(1 + f[k] * x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Georg Fischer, Dec 10 2020 *)

\\ here B(n) is A005185 as vector.
B(n)={my(A=vector(n, k, 1)); for(k=3, n, A[k]= A[k-A[k-1]]+ A[k-A[k-2]]); A}
seq(n)=my(v=B(n)); {Vec(prod(k=1, #v, 1 + x^k*v[k] + O(x*x^n)))} \\ Andrew Howroyd, Dec 10 2020

Definition corrected by Georg Fischer, Dec 10 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

A174231 A chaotic designed sequence with modulo six depth if ladder.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 3, 4, 5, 6, 6, 6, 4, 6, 9, 12, 8, 8, 6, 9, 15, 16, 12, 8, 6, 11, 21, 24, 10, 9, 8, 17, 14, 17, 23, 14, 11, 14, 33, 28, 14, 15, 14, 19, 17, 29, 18, 12, 7, 18, 25, 28, 34, 34, 26, 37, 22, 27, 35, 28, 13, 23, 26, 35, 47, 20, 18, 18, 31, 45, 57, 48, 9, 17, 28, 45, 36, 25, 14

Offset: 0

Roger L. Bagula, Mar 13 2010

Programmed during one of my OEIS banned period and never entered.

Cf. A147665

f[0] = 0; f[1] = 1; f[2] = 1;
f[n_] := f[n] = f[f[n - 1]] +
If[n < 6, f[n - f[(n - 1)]], If[ Mod[n, 6] == 0, f[f[n/6]],
If[Mod[1 + n, 6] == 1, f[f[(n - 1)/6]], If[Mod[2 + n, 6] == 2,
f[f[(n - 2)/6]], If[Mod[3 + n, 6] == 3, f[f[(n - 3)/6]],
If[Mod[4 + n, 6] == 4, f[f[(n - 4)/6]],
If[ Mod[5 + n, 6] == 5, f[f[(n - 5)/6]], f[n - f[(n - 2)]]]]]]]]];
Table[f[n], {n, 0, 100}]

cp's OEIS Frontend

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

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A166497 A Per Bak sand pile collapse sequence using A147665 in the A153112 form.

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A147880 Expansion of Product_{k > 0} (1 + A005229(k)*x^k).

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A147952 a(0) = 0, a(1) = a(2) = 1, and for n >= 3, a(n) = a(a(n-2)) + r(n), where r(n) = a(a(floor(n/3))) when n == 0 or 1 (mod 3) and = a(n - a(floor(n/3))) when n == 2 (mod 3).

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

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A147954 a(0) = 0, a(1) = a(2) = 1, a(n) = a(a(n-1)) + a(n-a(n-1)) for 3 <= n <= 5, and a(n) = a(a(n-1)) + r(n) for n >= 6, where r(n) = a(a(floor(n/6))) for n == 0, 1, 2, 3, 4 (mod 6), and r(n) = a(n - a(floor(n/6))) for n == 5 (mod 6).

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A147955 Expansion of Product_{k >= 0} (1 + A147954(k)*x^k).

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A147879 Expansion of Product_{k>=1} (1 + x^k*A005185(k)).

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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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