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 1-4 of 4 results.

A155039 A155038-A155033.

Original entry on oeis.org

0, 1, 0, 2, 0, 0, 4, 1, 0, 0, 8, 1, 0, 0, 0, 16, 4, 1, 0, 0, 0, 32, 6, 1, 0, 0, 0, 0, 64, 14, 3, 1, 0, 0, 0, 0, 128, 27, 6, 1, 0, 0, 0, 0, 0, 256, 57, 13, 3, 1, 0, 0, 0, 0, 0, 512, 110, 24, 5, 1, 0, 0, 0, 0, 0, 0, 1024, 227, 53, 13, 3, 1, 0, 0, 0, 0, 0, 0, 2048, 447, 101, 23, 5, 1, 0, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Mats Granvik, Jan 19 2009

Keywords

Examples

			Table begins:
0,
1,0,
2,0,0,
4,1,0,0,
8,1,0,0,0,
16,4,1,0,0,0,
32,6,1,0,0,0,0,

Crossrefs

Cf. A155042.

A155038 Triangle read by rows: T(n,k) is the number of compositions of n with first part k.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 8, 4, 2, 1, 1, 16, 8, 4, 2, 1, 1, 32, 16, 8, 4, 2, 1, 1, 64, 32, 16, 8, 4, 2, 1, 1, 128, 64, 32, 16, 8, 4, 2, 1, 1, 256, 128, 64, 32, 16, 8, 4, 2, 1, 1, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1, 1, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1, 1, 2048, 1024, 512
Offset: 1

Views

Author

Mats Granvik, Jan 19 2009

Keywords

Comments

Previous name was: Matrix inverse of A154990.
Apart from first term essentially the same as A057728.
A011782 appears in the columns.
Riordan array ((1-x)/(1-2x), x). - Philippe Deléham, Jan 24 2010
Indexing the triangle from n=0 and k=0, T(n,k) is the number of binary words of length n that begin with a run of exactly k 0's. O.g.f.: 1/((1-y*x)*(1-x/(1-x))). - Geoffrey Critzer, Feb 15 2012

Examples

			T(5,2) = 4 because the compositions of 5 with first part 2 are: [2,3], [2,2,1], [2,1,2], and [2,1,1,1]. - _Emeric Deutsch_, Jan 12 2018
Table begins:
   1,
   1,  1,
   2,  1,  1,
   4,  2,  1,  1,
   8,  4,  2,  1,  1,
  16,  8,  4,  2,  1,  1,
  32, 16,  8,  4,  2,  1,  1,
  64, 32, 16,  8,  4,  2,  1,  1,
Production matrix begins:
  1, 1
  1, 0, 1
  1, 0, 0, 1
  1, 0, 0, 0, 1
  1, 0, 0, 0, 0, 1
  1, 0, 0, 0, 0, 0, 1
  1, 0, 0, 0, 0, 0, 0, 1
  1, 0, 0, 0, 0, 0, 0, 0, 1
  ... - _Philippe Deléham_, Oct 04 2014

Links

Crossrefs

Cf. A011782, A057728, A154990, A155033, A155039.

Programs

  • Haskell
    a155038 n k = a155038_tabl !! (n-1) !! (k-1)
a155038_row n = a155038_tabl !! (n-1)
a155038_tabl = iterate
   (\row -> zipWith (+) (row ++ [0]) (init row ++ [0,1])) [1]
-- Reinhard Zumkeller, Aug 08 2013
  • Maple
    T := proc(n, k) if k = n then 1 elif k < n then 2^(n-k-1) else 0 end if end proc: for n to 13 do seq(T(n, k), k = 1 .. n) end do; # yields sequence in triangular form - Emeric Deutsch, Jan 12 2018
G:= (1-2*x+t*x^2)/((1-2*x)*(1-t*x)): Gser := simplify(series(G, x = 0, 15)): for n to 14 do P[n] := coeff(Gser, x, n) end do: for n to 14 do seq(coeff(P[n], t, j), j = 1 .. n) end do; # yields sequence in triangular form - Emeric Deutsch, Jan 19 2018
  • Mathematica
    nn = 15; a = 1/(1 - y x); f[list_] := Select[list, # > 0 &];Map[f, CoefficientList[Series[ a/(1 - x/(1 - x)), {x, 0, nn}], {x, y}]] // Flatten (* Geoffrey Critzer, Feb 15 2012 *)

Formula

T(j,k) = A011782(j-k), j>=1, k>=1. - Omar E. Pol, Feb 14 2013
T(n,k) = 2^{n-k-1} if kn. - Emeric Deutsch, Jan 12 2018
G.f.: G(t,x) = (1-2*x+t*x^2)/((1-2*x)*(1-t*x)). - Emeric Deutsch, Jan 19 2018

Extensions

New name from Joerg Arndt, May 04 2014

A101173 First differences of sequence defined in A101172. Also, the Mobius transform of that sequence.

Original entry on oeis.org

1, 1, 2, 3, 7, 11, 25, 46, 94, 182, 372, 727, 1471, 2916, 5849, 11657, 23364, 46620, 93348, 186503, 373172, 745998, 1492369, 2983948, 5968679, 11935893, 23873162, 47743475, 95489895, 190973738, 381953528, 763895349, 1527802031, 3055581071, 6111185475
Offset: 1

Views

Author

Mark Hudson (mrmarkhudson(AT)hotmail.com), Dec 03 2004

Keywords

Comments

The inverse Mobius transform of this sequence is equal to the sequence A101172.
Row sums of A155033. [Mats Granvik, Jan 19 2009]

Crossrefs

Cf. A101172, A155033.

Extensions

More terms from Emeric Deutsch, Mar 08 2005
a(32) onward corrected by Sean A. Irvine, May 01 2025

A307856 a(1) = a(2) = 1; a(n) = Sum_{1 < k < n, k not dividing n} a(k).

Original entry on oeis.org

1, 1, 1, 1, 3, 4, 10, 18, 37, 71, 146, 285, 577, 1143, 2293, 4570, 9160, 18277, 36597, 73118, 146301, 292466, 585079, 1169848, 2340003, 4679431, 9359402, 18717687, 37436529, 74870685, 149743743, 299482896, 598970235, 1197931456, 2395872060, 4791725527, 9583469660, 19166902722
Offset: 1

Views

Author

Ilya Gutkovskiy, May 01 2019

Keywords

Links

Crossrefs

Cf. A003238, A045545.
Second column of A155033.

Programs

  • Maple
    a := proc(n) local j; option remember;
if n < 3 then 1;
else add(`if`(`mod`(n, j) <> 0, a(j), 0), j = 2 .. n - 1);
end if; end proc;
seq(a(n), n = 1..40); # G. C. Greubel, Mar 08 2021
  • Mathematica
    a[n_] := a[n] = Sum[Boole[Mod[n, k] != 0] a[k], {k,n-1}]; a[1] = a[2] = 1; Table[a[n], {n, 1, 38}]
terms = 38; A[] = 0; Do[A[x] = x (1 + x) + A[x]/(1 - x) - Sum[A[x^k], {k, 1, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; Rest[CoefficientList[A[x], x]]
a[n_] := a[n] = SeriesCoefficient[x (1 + x + 1/(1 - x) Sum[a[k] x^k (1 - x^(k - 1))/(1 - x^k), {k, 1, n - 1}]), {x, 0, n}]; Table[a[n], {n, 1, 38}]
  • Sage
    @CachedFunction
def a(n):
    if n<3: return 1
    else: return sum( a(j) if n%j!=0 else 0 for j in (2..n-1) )
[a(n) for n in (1..40)] # G. C. Greubel, Mar 08 2021

Formula

G.f. A(x) satisfies: A(x) = x*(1 + x) + A(x)/(1 - x) - Sum_{k>=1} A(x^k).
G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * (1 + x + (1/(1 - x)) * Sum_{n>=1} a(n)*x^n*(1 - x^(n-1))/(1 - x^n)).
a(n) ~ c * 2^n, where c = 0.0697287852138897098746368547699891689134990049613293203832908827967121295... - Vaclav Kotesovec, May 06 2019
Showing 1-4 of 4 results.

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