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 1-9 of 9 results.

A038044 Shifts left under transform T where Ta is a DCONV a.

Original entry on oeis.org

1, 1, 2, 4, 9, 18, 40, 80, 168, 340, 698, 1396, 2844, 5688, 11456, 22948, 46072, 92144, 184696, 369392, 739536, 1479232, 2959860, 5919720, 11842696, 23685473, 47376634, 94753940, 189519576, 379039152, 758102900, 1516205800
Offset: 1

Views

Author

Christian G. Bower

Keywords

Links

Crossrefs

Positions of odd terms are given by A003095. Other self-convolved sequences: A000108, A007460 - A007464, A025192, A061922, A062177.
Column k=1 of A144324 and A144823. - Alois P. Heinz, Nov 04 2012
Cf. A038040.
Cf. A000010.

Programs

  • Haskell
    import Data.Function (on)
a038044 n = a038044_list !! (n-1)
a038044_list = 1 : f 1 [1] where
   f x ys = y : f (x + 1) (y:ys) where
     y = sum $ zipWith ((*) `on` a038044) divs $ reverse divs
         where divs = a027750_row x
-- Reinhard Zumkeller, Jan 21 2014
  • Maple
    with(numtheory); EIGENbyDIRCONV := proc(upto_n) local n,a,j,i,s,m; a := [1]; for i from 1 to upto_n do s := 0; m := convert(divisors(i),set); n := nops(m); for j from 1 to n do s := s+(a[m[j]]*a[m[(n-j)+1]]); od; a := [op(a),s]; od; RETURN(a); end;
  • Mathematica
    dc[b_, c_] := Module[{p}, p[n_] := p[n] = Sum[b[d]*c[n/d], {d, If[n<0, {}, Divisors[n]]}]; p]; A[n_, k_] := Module[{f, b, t}, b[1] = dc[f, f]; For[t = 2, t <= k, t++, b[t] = dc[b[t-1], b[t-1]]]; f = Function[m, If[m == 1, 1, b[k][m-1]]]; f[n]]; a[n_] := A[n, 1]; Array[a, 40] (* Jean-François Alcover, Mar 20 2017, after A144324 *)

Formula

From Benoit Cloitre, Aug 29 2004: (Start)
a(n+1) = Sum_{d|n} a(d)*a(n/d), a(1) = 1.
a(prime(k)+1) = 2*a(prime(k));
a(n) is asymptotic to c*2^n where c=0.353030198... (End)
G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * (1 + Sum_{i>=1} Sum_{j>=1} a(i)*a(j)*x^(i*j)). - Ilya Gutkovskiy, May 01 2019 [modified by Ilya Gutkovskiy, May 09 2019]
a(n+1) = Sum_{k=1..n} a(gcd(n,k))*a(n/gcd(n,k))/phi(n/gcd(n,k)) where phi = A000010. - Richard L. Ollerton, May 19 2021

A144316 Shifts left when Dirichlet convolution (DC:(b,b)->a) applied twice.

Original entry on oeis.org

1, 1, 4, 16, 70, 280, 1168, 4672, 18884, 75632, 303368, 1213472, 4858064, 19432256, 77743040, 310975520, 1243959873, 4975839492, 19903598208, 79614392832, 318458493192, 1273834028832, 5095339755744, 20381359022976, 81525450936496, 326101803775384
Offset: 1

Views

Author

Alois P. Heinz, Sep 17 2008

Keywords

Links

Crossrefs

2nd column of A144324, 3rd column of A144823.

Programs

  • Maple
    k:=2: with(numtheory): dc:= proc(b,c) proc(n) option remember; add(b(d) *c(n/d), d=`if`(n<0,{}, divisors(n))) end end: a:='a': b[1]:= dc(a,a): for t from 2 to k do b[t]:= dc(b[t-1], b[t-1]) od: a:= n-> `if`(n=1, 1, b[k](n-1)): seq(a(n), n=1..30);
  • Mathematica
    dc[b_, c_] := Module[{proc}, proc[n_] := proc[n] = Sum[b[d]*c[n/d], {d, If[n < 0, {}, Divisors[n]]}]; proc];
A[n_, k_] := Module[{a, b, t}, b[1] = dc[a, a]; For[t = 2, t <= k, t++, b[t] = dc[b[t-1], b[t-1]]]; a = Function[m, If[m == 1, 1, b[k][m-1]]]; a[n]];
a[n_] := A[n, 2];
Array[a, 30] (* Jean-François Alcover, Jun 11 2018, after Alois P. Heinz *)

Formula

a(n) ~ c * 4^n, where c = 0.0724091505138381672774074945426621544789572745186499358668403190389... . - Vaclav Kotesovec, Sep 03 2014

A144317 Shifts left when Dirichlet convolution (DC:(b,b)->a) applied 3 times.

Original entry on oeis.org

1, 1, 8, 64, 540, 4320, 35008, 280064, 2244152, 17955008, 143670304, 1149362432, 9195171392, 73561371136, 588492929536, 4707943678208, 37663565234758, 301308521878064, 2410468302643136, 19283746421145088, 154269972376667232
Offset: 1

Views

Author

Alois P. Heinz, Sep 17 2008

Keywords

Links

Crossrefs

3rd column of A144324, 7th column of A144823.

Programs

  • Maple
    k:=3: with (numtheory): dc:= proc(b,c) proc(n) option remember; add (b(d) *c(n/d), d=`if`(n<0,{}, divisors(n))) end end: a:='a': b[1]:= dc(a,a): for t from 2 to k do b[t]:= dc(b[t-1], b[t-1]) od: a:= n-> `if`(n=1, 1, b[k](n-1)): seq (a(n), n=1..30);
  • Mathematica
    dc[b_, c_] := Module[{proc}, proc[n_] := proc[n] = Sum[b[d]*c[n/d], {d, If[n < 0, {}, Divisors[n]]}]; proc];
A[n_, k_] := Module[{a, b, t}, b[1] = dc[a, a]; For[t = 2, t <= k, t++, b[t] = dc[b[t-1], b[t-1]]]; a = Function[m, If[m == 1, 1, b[k][m - 1]]]; a[n]];
a[n_] := A[n, 3];
Array[a, 30] (* Jean-François Alcover, Jun 11 2018, after Alois P. Heinz *)

A144817 Shifts left when Dirichlet convolution with a (DC:(b,a)->c) applied twice.

Original entry on oeis.org

1, 1, 3, 9, 30, 90, 288, 864, 2647, 7968, 24084, 72252, 217467, 652401, 1958931, 5877333, 17637453, 52912359, 158754606, 476263818, 1428840972, 4286528100, 12859728804, 38579186412, 115738013592, 347214043476, 1041643435230, 3124930353363, 9374794990911
Offset: 1

Views

Author

Alois P. Heinz, Sep 21 2008

Keywords

Links

Crossrefs

2nd column of A144823.

Programs

  • Maple
    with (numtheory): dc:= proc(b,c) proc(n) option remember; add (b(d) *c(n/d), d=`if`(n<0,{},divisors(n))) end end: A:= proc(n, k) local a, b, t; b[1]:= dc(a,a); for t from 2 to k do b[t]:= dc(b[t-1],a) od: a:= n-> `if`(n=1, 1, b[k](n-1)); a(n) end: a:= n-> A(n,2): seq (a(n), n=1..30);
  • Mathematica
    dc[b_, c_] := Module[{f}, f[n_] := f[n] = Sum[b[d] c[n/d], {d, If[n<0, {}, Divisors[n]]}]; f];
A[n_, k_] := Module[{a, b, t}, b[1] = dc[a, a]; For[t = 2, t <= k, t++, b[t] = dc[b[t-1], a]]; a = Function[m, If[m==1, 1, b[k][m-1]]]; a[n]];
a[n_] := A[n, 2];
Array[a, 30] (* Jean-François Alcover, Dec 18 2020, after Maple *)

Formula

a(n) ~ c * 3^n, where c = 0.1365983596534181021630692308337960543393478528568767041107748567859... . - Vaclav Kotesovec, Sep 03 2014

A144818 Shifts left when Dirichlet convolution with a (DC:(b,a)->c) applied 4 times.

Original entry on oeis.org

1, 1, 5, 25, 135, 675, 3475, 17375, 87385, 437175, 2188575, 10942875, 54730525, 273652625, 1368332625, 6841676625, 34208737630, 171043688150, 855220256700, 4276101283500, 21380515232550, 106902576510250, 534512926322750
Offset: 1

Views

Author

Alois P. Heinz, Sep 21 2008

Keywords

Links

Crossrefs

4th column of A144823.

Programs

  • Maple
    with(numtheory): dc:= proc(b,c) proc(n) option remember; add(b(d) *c(n/d), d=`if`(n<0,{},divisors(n))) end end: A:= proc(n, k) local a, b, t; b[1]:= dc(a,a); for t from 2 to k do b[t]:= dc(b[t-1],a) od: a:= n-> `if`(n=1, 1, b[k](n-1)); a(n) end: a:= n-> A(n,4): seq(a(n), n=1..30);
  • Mathematica
    dc[b_, c_] := Module[{f}, f[n_] := f[n] = Sum[b[d] c[n/d], {d, If[n<0, {}, Divisors[n]]}]; f];
A[n_, k_] := Module[{a, b, t}, b[1] = dc[a, a]; For[t = 2, t <= k, t++, b[t] = dc[b[t-1], a]]; a = Function[m, If[m= 1, 1, b[k][m-1]]]; a[n]];
a[n_] := A[n, 4];
Array[a, 30] (* Jean-François Alcover, Dec 18 2020, after Maple *)

A144819 Shifts left when Dirichlet convolution with a (DC:(b,a)->c) applied 5 times.

Original entry on oeis.org

1, 1, 6, 36, 231, 1386, 8496, 50976, 306956, 1842276, 11060586, 66363516, 398229516, 2389377096, 14336517456, 86019146316, 516116428791, 3096698572746, 18580200896796, 111481205380776, 668887287816276
Offset: 1

Views

Author

Alois P. Heinz, Sep 21 2008

Keywords

Links

Crossrefs

5th column of A144823.

Programs

  • Maple
    with (numtheory): dc:= proc(b,c) proc(n) option remember; add (b(d) *c(n/d), d=`if`(n<0,{},divisors(n))) end end: A:= proc(n, k) local a, b, t; b[1]:= dc(a,a); for t from 2 to k do b[t]:= dc(b[t-1],a) od: a:= n-> `if`(n=1, 1, b[k](n-1)); a(n) end: a:= n-> A(n,5): seq (a(n), n=1..30);
  • Mathematica
    dc[b_, c_] := Module[{f}, f[n_] := f[n] = Sum[b[d] c[n/d], {d, If[n<0, {}, Divisors[n]]}]; f];
A[n_, k_] := Module[{a, b, t}, b[1] = dc[a, a]; For[t = 2, t <= k, t++, b[t] = dc[b[t-1], a]]; a = Function[m, If[m==1, 1, b[k][m-1]]]; a[n]];
a[n_] := A[n, 5];
Array[a, 30] (* Jean-François Alcover, Dec 18 2020, after Maple *)

A144820 Shifts left when Dirichlet convolution with a (DC:(b,a)->c) applied 6 times.

Original entry on oeis.org

1, 1, 7, 49, 364, 2548, 18130, 126910, 890463, 6234270, 43655178, 305586246, 2139225879, 14974581153, 104822829531, 733759913733, 5136324781952, 35954273473664, 251679952469351, 1761759667285457, 12332317933624871
Offset: 1

Views

Author

Alois P. Heinz, Sep 21 2008

Keywords

Links

Crossrefs

6th column of A144823.

Programs

  • Maple
    with(numtheory): dc:= proc(b,c) proc(n) option remember; add(b(d) *c(n/d), d=`if`(n<0,{},divisors(n))) end end: A:= proc(n, k) local a, b, t; b[1]:= dc(a,a); for t from 2 to k do b[t]:= dc(b[t-1],a) od: a:= n-> `if`(n=1, 1, b[k](n-1)); a(n) end: a:= n-> A(n,6): seq(a(n), n=1..30);

A144821 Shifts left when Dirichlet convolution with a (DC:(b,a)->c) applied 8 times.

Original entry on oeis.org

1, 1, 9, 81, 765, 6885, 62613, 563517, 5077569, 45701037, 411364413, 3702279717, 33321067929, 299889611361, 2699011010385, 24291099589185, 218619937132623, 1967579434193607, 17708215277809323, 159373937500283907
Offset: 1

Views

Author

Alois P. Heinz, Sep 21 2008

Keywords

Links

Crossrefs

8th column of A144823.

Programs

  • Maple
    with(numtheory): dc:= proc(b,c) proc(n) option remember; add(b(d) *c(n/d), d=`if`(n<0,{},divisors(n))) end end: A:= proc(n, k) local a, b, t; b[1]:= dc(a,a); for t from 2 to k do b[t]:= dc(b[t-1],a) od: a:= n-> `if`(n=1, 1, b[k](n-1)); a(n) end: a:= n-> A(n,8): seq(a(n), n=1..30);

A144822 Shifts left when Dirichlet convolution with a (DC:(b,a)->c) applied 9 times.

Original entry on oeis.org

1, 1, 10, 100, 1045, 10450, 105400, 1054000, 10549120, 105495700, 1055051050, 10550510500, 105506139100, 1055061391000, 10550623396000, 105506234900500, 1055062444351210, 10550624443512100, 105506245393982800
Offset: 1

Views

Author

Alois P. Heinz, Sep 21 2008

Keywords

Links

Crossrefs

9th column of A144823.

Programs

  • Maple
    with (numtheory): dc:= proc(b,c) proc(n) option remember; add (b(d) *c(n/d), d=`if`(n<0,{},divisors(n))) end end: A:= proc(n, k) local a, b, t; b[1]:= dc(a,a); for t from 2 to k do b[t]:= dc(b[t-1],a) od: a:= n-> `if`(n=1, 1, b[k](n-1)); a(n) end: a:= n-> A(n,9): seq (a(n), n=1..30);
Showing 1-9 of 9 results.

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