cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A358504 Number of genetic relatives of a person M in a genealogical tree extending back n generations and where everyone has 3 children down to the generation of M.

Original entry on oeis.org

1, 5, 25, 137, 793, 4697, 28057, 168089, 1008025, 6047129, 36280729, 217680281, 1306073497, 7836424601, 47018514841, 282111023513, 1692666010009, 10155995797913, 60935974263193, 365615844530585, 2193695065086361, 13162170386323865, 78973022309554585
Offset: 0

Views

Author

Hans Braxmeier, Nov 19 2022

Keywords

Comments

M has 2 parents, 4 grandparents, and so on up to 2^n ancestors at the top of the tree.
The genetic relatives of M are all descendants of those ancestors.
M is a genetic relative of himself or herself.

Examples

			For n=2, the tree comprises a(2) = 25 people,
      G-------G       G-------G       G = 4 grandparents
     /    |    \     /    |    \      P = 2 parents
    U     U     P---P     U     U     S = 2 siblings
   /|\   /|\     /|\     /|\   /|\    U = 4 uncles (or aunts)
  C C C C C C   S M S   C C C C C C   C = 12 cousins
The spouses of U are not shown and are not genetic relatives of M.

Links

Crossrefs

Cf. A154407.
Other numbers of children: A076024 (2), A358598 (4), A358599 (5), A358600 (6), A358601 (7).

Programs

  • Mathematica
    A358504[n_] := 2^n + 3*(6^n-1)/5; Array[A358504, 25, 0] (* or *)
LinearRecurrence[{9, -20, 12}, {1, 5, 25}, 25] (* Paolo Xausa, Feb 09 2024 *)
  • PARI
    a(n) = (3^(n+1)+5)<Kevin Ryde, Nov 23 2022
  • Python
    for n in range(0,23): print(2**n+3*(6**n-1)//5)

Formula

a(n) = 2^n + 3*(6^n - 1)/5.
a(n) = 2*(A154407(n) + 1)/5 - 1. - Hugo Pfoertner, Nov 22 2022

A358598 Number of genetic relatives of a person M in a genealogical tree extending back n generations and where everyone has 4 children down to the generation of M.

Original entry on oeis.org

1, 6, 40, 300, 2356, 18756, 149860, 1198500, 9587236, 76696356, 613567780, 4908536100, 39268276516, 314146187556, 2513169451300, 20105355512100, 160842843900196, 1286742750808356, 10293942005680420, 82351536043870500, 658812288347818276, 5270498306776254756
Offset: 0

Views

Author

Hans Braxmeier, Nov 19 2022

Keywords

Comments

M has 2 parents, 4 grandparents, and so on up to 2^n ancestors at the top of the tree.
The genetic relatives of M are all descendants of the ancestors.
M is a genetic relative of himself or herself.

Links

Crossrefs

Cf. A000079, A108019.
Other numbers of children: A076024 (2), A358504 (3), A358599 (5), A358600 (6), A358601 (7).

Programs

  • Mathematica
    A358598[n_] := 2^n + 4*(8^n-1)/7; Array[A358598, 25, 0] (* or *)
LinearRecurrence[{11, -26, 16}, {1, 6, 40}, 25] (* Paolo Xausa, Feb 09 2024 *)
  • Python
    for n in range(0,10): print(2**n+4*(8**n-1)//7)

Formula

a(n) = 2^n + 4*(8^n - 1)/7.
a(n) = A000079(n) + A108019(n). - Michel Marcus, Nov 25 2022
From Stefano Spezia, Nov 25 2022: (Start)
O.g.f.: (1 - 5*x)/((1 - x)*(1 - 2*x)*(1 - 8*x)).
E.g.f.: exp(x)*(4*(exp(7*x) - 1) + 7*exp(x))/7.
a(n) = 11*a(n-1) - 26*a(n-2) + 16*a(n-3) for n > 2. (End)

A358601 Number of genetic relatives of a person M in a genealogical tree extending back n generations and where everyone has 7 children down to the generation of M.

Original entry on oeis.org

1, 9, 109, 1485, 20701, 289629, 4054429, 56761245, 794655901, 11125179549, 155752507549, 2180535093405, 30527491283101, 427384877914269, 5983388290701469, 83767436069623965, 1172744104974342301, 16418417469640005789, 229857844574958508189
Offset: 0

Views

Author

Hans Braxmeier, Nov 23 2022

Keywords

Comments

M has 2 parents, 4 grandparents, and so on up to 2^n top ancestors at the top of the tree.
The genetic relatives of M are all descendants of those ancestors.
M is a genetic relative of himself or herself.

Links

Crossrefs

Other numbers of children: A076024 (2), A358504 (3), A358598 (4), A358599 (5), A358600 (6).

Programs

  • Mathematica
    A358601[n_] := 2^n + 7*(14^n-1)/13; Array[A358601, 25, 0] (* or *)
LinearRecurrence[{17, -44, 28}, {1, 9, 109}, 25] (* Paolo Xausa, Feb 09 2024 *)
  • Python
    print([2**n+7*(14**n-1)//13 for n in range(10)])

Formula

a(n) = 2^n + 7*(14^n - 1)/13.
G.f.: (8*x-1)/((x-1)*(2*x-1)*(14*x-1)). - Alois P. Heinz, Dec 04 2022

A183060 Number of "ON" cells at n-th stage in a simple 2-dimensional cellular automaton (see Comments for precise definition).

Original entry on oeis.org

0, 1, 4, 7, 14, 17, 24, 31, 50, 53, 60, 67, 86, 93, 112, 131, 186, 189, 196, 203, 222, 229, 248, 267, 322, 329, 348, 367, 422, 441, 496, 551, 714, 717, 724, 731, 750, 757, 776, 795, 850, 857, 876, 895, 950, 969, 1024, 1079, 1242, 1249, 1268, 1287
Offset: 0

Views

Author

Omar E. Pol, Feb 20 2011

Keywords

Comments

On the semi-infinite square grid, start with all cells OFF.
Turn a single cell to the ON state in row 1.
At each subsequent step, each cell with exactly one neighbor ON is turned ON, and everything that is already ON remains ON.
The sequence gives the number of "ON" cells after n stages. A183061 (the first differences) gives the number of cells turned "ON" at the n-th stage.
Note that this is just half plus the rest of the center line of the cellular automaton described in A147562.
After 2^k stages the structure resembles an isosceles right triangle. For a three-dimensional version using cubes see A186410. For more information see A147562.

Examples

			Illustration of the structure after eight stages in which we label the generations of cells turned ON by consecutive numbers:
         8
        878
       8 6 8
      8765678
     8 8 4 8 8
    878 434 878
   8 6 4 2 4 6 8
  876543212345678
...................
There are 50 "ON" cells so a(8) = 50.

Links

Crossrefs

Cf. A139250, A147562, A147582, A147610, A151920, A183061, A186410.

Programs

  • Mathematica
    A183060[0] = 0; A183060[n_] := Total[With[{m = n - 1}, CellularAutomaton[{4042387958, 2, {{0, 1}, {-1, 0}, {0, 0}, {1, 0}, {0, -1}}}, {{{1}}, 0}, {{{m}}, -m}]], 2] (* JungHwan Min, Jan 24 2016 *)
A183060[0] = 0; A183060[n_] := Total[With[{m = n - 1}, CellularAutomaton[{686, {2, {{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}}, {1, 1}}, {{{1}}, 0}, {{{m}}, -m}]], 2] (* JungHwan Min, Jan 24 2016 *)

Formula

a(n) = n + (A147562(n) - 1)/2, n >= 1.
a(n) = n + 2*A151920(n-2), n >= 2.
a(2^n) = A076024(n+1). - Nathaniel Johnston, Mar 14 2011

Extensions

Comments edited by Omar E. Pol, Mar 19 2011 at the suggestion of John W. Layman and Franklin T. Adams-Watters

A358599 Number of genetic relatives of a person M in a genealogical tree extending back n generations and where everyone has 5 children down to the generation of M.

Original entry on oeis.org

1, 7, 59, 563, 5571, 55587, 555619, 5555683, 55555811, 555556067, 5555556579, 55555557603, 555555559651, 5555555563747, 55555555571939, 555555555588323, 5555555555621091, 55555555555686627, 555555555555817699, 5555555555556079843, 55555555555556604131
Offset: 0

Views

Author

Hans Braxmeier, Nov 23 2022

Keywords

Comments

M has 2 parents, 4 grandparents, and so on up to 2^n top ancestors at the top of the tree.
The genetic relatives of M are all descendants of those ancestors.
M is a genetic relative of himself or herself.

Links

Crossrefs

Other numbers of children: A076024 (2), A358504 (3), A358598 (4), A358600 (6), A358601 (7).
Cf. A000079, A002279.

Programs

  • Mathematica
    LinearRecurrence[{13, -32, 20}, {1, 7, 59}, 21] (* Hugo Pfoertner, Dec 05 2022 *)
  • Python
    print([2**n+5*(10**n-1)//9 for n in range(10)])

Formula

a(n) = 2^n + 5*(10^n - 1)/9.
a(n) = A000079(n) + A002279(n).
G.f.: (6*x-1)/((x-1)*(2*x-1)*(10*x-1)). - Alois P. Heinz, Dec 05 2022
a(n) = 13*a(n-1) - 32*a(n-2) + 20*a(n-3). - Wesley Ivan Hurt, Jun 19 2025

A358600 Number of genetic relatives of a person M in a genealogical tree extending back n generations and where everyone has 6 children down to the generation of M.

Original entry on oeis.org

1, 8, 82, 950, 11326, 135758, 1628782, 19544750, 234535726, 2814426158, 33773108782, 405277295150, 4863327521326, 58359930214958, 700319162497582, 8403829949807150, 100845959397358126, 1210151512767642158, 14521818153210395182, 174261817838522120750
Offset: 0

Views

Author

Hans Braxmeier, Nov 23 2022

Keywords

Comments

M has 2 parents, 4 grandparents, and so on up to 2^n top ancestors at the top of the tree.
The genetic relatives of M are all descendants of those ancestors.
M is a genetic relative of himself or herself.

Links

Crossrefs

Other numbers of children: A076024 (2), A358504 (3), A358598 (4), A358599 (5), A358601 (7).

Programs

  • Mathematica
    LinearRecurrence[{15, -38, 24}, {1, 8, 82}, 20] (* Hugo Pfoertner, Dec 05 2022 *)
  • Python
    print([2**n+6*(12**n-1)//11 for n in range(10)])

Formula

a(n) = 2^n + 6*(12^n - 1)/11.
G.f.: (1 - 7*x)/((1 - x)*(1 - 2*x)*(1 - 12*x)). - Stefano Spezia, Dec 05 2022

A346295 a(n) = Sum_{k=0..n} (2^k + 1) * (2^k + 2) / 2.

Original entry on oeis.org

3, 9, 24, 69, 222, 783, 2928, 11313, 44466, 176307, 702132, 2802357, 11197110, 44763831, 179006136, 715926201, 2863508154, 11453639355, 45813770940, 183253510845, 733010897598, 2932037298879, 11728136612544, 46912521284289, 187650034805442, 750600038558403
Offset: 0

Views

Author

Paul Weisenhorn, Jul 13 2021

Keywords

Comments

All terms are multiples of 3.

Links

Crossrefs

Cf. A028401 (first differences).
Cf. A006095, A076024.

Programs

  • Maple
    a:= proc(n) option remember:
if n=0 then 3 else (2^n+1)*(2^n+2)/2+procname(n-1) fi:
end proc:
seq(a(n), n=0..30);
  • Mathematica
    Accumulate @ Table[(2^k + 1)*(2^k + 2)/2, {k, 0, 25}] (* Amiram Eldar, Jul 27 2021 *)
LinearRecurrence[{8,-21,22,-8},{3,9,24,69},30] (* Harvey P. Dale, Nov 21 2021 *)
  • PARI
    a(n)=sum(k=0, n, (2^k+1)*(2^k+2)/2); \\ Michel Marcus, Jul 16 2021

Formula

a(n) = (2^(n+1) + 4) * (2^(n+1) + 5) / 6 - 4 + n.
More generally: let f(n, b) be the triangular sum Sum_{k=0..n} (2^k+b) * (2^k+b+1) / 2.
f(n, b) = (2^(n+1) + 3*b + 1) * (2^(n+1) + 3*b + 2) / 6 - (b + 1)^2 + b*(b + 1)*n / 2.
G.f.: ((b^2+3*b+2)/2 - (3*b^2+8*b+4)*x + (4*b^2+8*b+3)*x^2) / ((4*x-1) * (2*x-1) * (x-1)^2).
E.g.f.: exp(x) * ((6*b+3)*exp(x) + 2*exp(3*x) + 3(b^2+b)*x/2 + (3*b^2-3*b-4) / 2) / 3.
Then b = -1 gives A006095, b = 0 gives A076024, b = 1 gives A346295, b = 2 gives A346375.
G.f.: 3*(5*x^2 - 5*x + 1) / ((4*x - 1) * (2*x - 1) * (x - 1)^2).
a(n) = 8*a(n-1) - 21*a(n-2) + 22*a(n-3) - 8*a(n-4) for n > 3.
This recurrence is valid for all sequences f(n,b).
E.g.f.: exp(x) * (9*exp(x) + 2*exp(3*x) + 3*x - 2) / 3. - Stefano Spezia, Aug 13 2021

A346375 a(n) = Sum_{k=0..n} (2^k + 2) * (2^k + 3) / 2.

Original entry on oeis.org

6, 16, 37, 92, 263, 858, 3069, 11584, 44995, 177350, 704201, 2806476, 11205327, 44780242, 179038933, 715991768, 2863639259, 11453901534, 45814295265, 183254559460, 733012994791, 2932041493226, 11728145001197, 46912538061552, 187650068359923, 750600105667318, 3002400087124729
Offset: 0

Views

Author

Paul Weisenhorn, Jul 14 2021

Keywords

Links

Crossrefs

Cf. A006095, A076024.

Programs

  • Maple
    a:= proc(n) option remember:
     if n=0 then 6 else procname(n-1)+(2^n+3)*(2^n+2)/2 fi:
    end proc:
seq(a(n), n=0..26);
  • Mathematica
    a[n_]:=Sum[(2^k+2)*(2^k+3)/2,{k,0,n}];Array[a,30,0] (* Giorgos Kalogeropoulos, Jul 27 2021 *)
  • PARI
    a(n) = sum(k=0, n, (2^k+2)*(2^k+3)/2); \\ Michel Marcus, Jul 28 2021

Formula

a(n) = Sum_{k=0..n} (2^k + 2) * (2^k + 3) / 2.
a(n) = (2^(n+1) + 7) * (2^(n+1) + 8)/6 - 9 + 3*n.
More generally: let f(n, b) = Sum_{k=0..n} (2^k + b) * (2^k + b + 1)/2 then f(n, b) = (2^(n+1) + 3*b + 1) * (2^(n+1) + 3*b + 2) / 6 - (b + 1)^2 + b*(b + 1)*n/2.
G.f.: ((b^2+3*b+2)/2 - (3*b^2+8*b+4)*x + (4*b^2+8*b+3)*x^2) / ((4*x-1) * (2*x-1) * (x-1)^2).
E.g.f.: exp(x)*((6*b+3)*exp(x) + 2*exp(3*x) + 3*(b^2+b)*x/2 +(3*b^2-3*b-4) / 2) / 3.
Then b = -1 gives A006095, b = 0 gives A076024, b = 1 gives A346295, b = 2 gives A346375.
a(n) = 8*a(n-1) - 21*a(n-2) + 22*a(n-3) - 8*a(n-4) with n > 3.
This recurrence is valid for all sequences f(n, b).
G.f.: (35*x^2 - 32*x + 6) / ((4*x - 1) * (2*x - 1) * (x - 1)^2).
E.g.f.: exp(x) * (1 + 15*exp(x) + 2*exp(3*x) + 9*x)/3. - Stefano Spezia, Aug 15 2021
Showing 1-8 of 8 results.

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