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

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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

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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 *)

A144318 Shifts left when Dirichlet convolution (DC:(b,b)->a) applied 4 times.

Original entry on oeis.org

1, 1, 16, 256, 4216, 67456, 1083136, 17330176, 277344816, 4437547776, 71001776256, 1136028420096, 18176471920896, 290823550734336, 4653177071702016, 74450833163421696, 1191213334782285596, 19059413356516569536, 304950613771087329536, 4879209820337397272576
Offset: 1

Views

Author

Alois P. Heinz, Sep 17 2008

Keywords

Links

Crossrefs

4th column of A144324.

Programs

  • Maple
    k:=4: 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);

A144321 Shifts left when Dirichlet convolution (DC:(b,b)->a) applied 7 times.

Original entry on oeis.org

1, 1, 128, 16384, 2105280, 269475840, 34494988288, 4415358500864, 565166154790272, 72341267946323968, 9259682331352899584, 1185239338413171146752, 151710635321300728430592, 19418961321126493239115776, 2485627049104751885136429056
Offset: 1

Views

Author

Alois P. Heinz, Sep 17 2008

Keywords

Links

Crossrefs

7th column of A144324.

Programs

  • Maple
    k:=7: 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..25);

A144323 Shifts left when Dirichlet convolution (DC:(b,b)->a) applied 9 times.

Original entry on oeis.org

1, 1, 512, 262144, 134348544, 68786454528, 35218798673920, 18032024921047040, 9232396828183582208, 4726987176064286720000, 2420217434180064678903808, 1239151326300193115598749696, 634445479065716907074042200064
Offset: 1

Views

Author

Alois P. Heinz, Sep 17 2008

Keywords

Links

Crossrefs

9th column of A144324.

Programs

  • Maple
    k:=9: 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..20);

A144319 Shifts left when Dirichlet convolution (DC:(b,b)->a) applied 5 times.

Original entry on oeis.org

1, 1, 32, 1024, 33264, 1064448, 34094080, 1091010560, 34913358688, 1117227985920, 35751328547328, 1144042513514496, 36609361521378304, 1171499568684105728, 37487986231712710656, 1199615559415862673408
Offset: 1

Views

Author

Alois P. Heinz, Sep 17 2008

Keywords

Links

Crossrefs

5th column of A144324.

Programs

  • Maple
    k:=5: 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..25);

A144320 Shifts left when Dirichlet convolution (DC:(b,b)->a) applied 6 times.

Original entry on oeis.org

1, 1, 64, 4096, 264160, 16906240, 1082257408, 69264474112, 4432942899904, 283708353851392, 18157335711582208, 1162069485541261312, 74372447143871647744, 4759836617207785455616, 304629543505661931028480
Offset: 1

Views

Author

Alois P. Heinz, Sep 17 2008

Keywords

Links

Crossrefs

6th column of A144324.

Programs

  • Maple
    k:=6: 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..25);

A144322 Shifts left when Dirichlet convolution (DC:(b,b)->a) applied 8 times.

Original entry on oeis.org

1, 1, 256, 65536, 16809856, 4303323136, 1101667434496, 282026863230976, 72198881268083456, 18482913606768459776, 4731625884430073102336, 1211296226414098714198016, 310091833962291289108054016
Offset: 1

Views

Author

Alois P. Heinz, Sep 17 2008

Keywords

Links

Crossrefs

8th column of A144324.

Programs

  • Maple
    k:=8: 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..20);
Showing 1-9 of 9 results.

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