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 1-6 of 6 results.

A227953 Smallest m such that A070965(m) = n.

Original entry on oeis.org

3, 1, 30, 63, 187, 232, 348, 375, 912, 620, 692, 463, 864, 2303, 1615, 2806, 4844, 1239, 2239, 1383, 1835, 2236, 3716, 6575, 3447, 3670, 1384, 6894, 13422, 7486, 9131, 6462, 6480, 13703, 13039, 4958, 9908, 6368, 9912, 5534, 2767, 12734, 9719, 16598, 11060
Offset: 0

Views

Author

Reinhard Zumkeller, Aug 01 2013

Keywords

Crossrefs

Cf. A070965, A227954.

Programs

  • Haskell
    import Data.List (elemIndex); import Data.Maybe (fromJust)
a227953 = (+ 1) . fromJust . (`elemIndex` a070965_list)

Formula

A070965(a(n)) = n and A070965(m) <> n for m < a(n).
A070965(A227954(n)) = -n.

A227954 Smallest m such that A070965(m) = -n.

Original entry on oeis.org

3, 15, 78, 115, 155, 310, 231, 911, 619, 691, 464, 863, 1294, 1616, 3240, 2235, 1240, 2240, 3472, 3183, 4479, 2766, 4148, 9212, 6588, 4567, 2587, 2588, 4846, 6479, 3231, 7763, 3744, 3743, 2479, 2480, 6367, 7440, 12936, 2768, 9914, 9395, 13415, 11059, 9792
Offset: 0

Views

Author

Reinhard Zumkeller, Aug 01 2013

Keywords

Crossrefs

Cf. A070965, A227953.

Programs

  • Haskell
    import Data.List (elemIndex); import Data.Maybe (fromJust)
a227954 = (+ 1) . fromJust . (`elemIndex` a070965_list) . negate

Formula

A070965(a(n)) = -n and A070965(m) <> -n for m < a(n);
A070965(A227953(n)) = n.

A073776 a(n) = Sum_{k=1..n} -mu(k+1) * a(n-k), with a(0)=1.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 17, 28, 50, 83, 147, 249, 435, 742, 1288, 2207, 3819, 6561, 11333, 19497, 33640, 57915, 99874, 172020, 296550, 510886, 880580, 1517226, 2614889, 4505745, 7765094, 13380640, 23059193, 39735969, 68476885, 118001888
Offset: 0

Views

Author

Paul D. Hanna, Aug 10 2002

Keywords

Comments

Recurrence relation involves the Moebius function.
Radius of convergence of A(x) is r=0.5802946238073267...
Related limits are
lim_{n->infinity} a(n) r^n = 0.6303632342... and
lim_{n->infinity} a(n+1)/a(n) = 1.723262561763844...
From Gary W. Adamson, Aug 11 2016: (Start)
The definition in the heading follows from the INVERTi transform of (1, 2, 3, 6, 9, 17, ...) equals -mu(n) for n >= 2 (cf. A157658).
Then for example, a(6) = 17 = (1, 1, 0, 1, -1, 1) dot (9, 6, 3, 2, 1, 1) = (9 + 6 + 0 + 2 - 1 + 1); in agreement with the first example. (End)

Examples

			a(6) = -mu(2)a(5) - mu(3)a(4) - mu(4)a(3) - mu(5)a(2) - mu(6)a(1) - mu(7)a(0) = 9 + 6 + 0 + 2 - 1 + 1 = 17.
G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 9*x^5 + 17*x^6 + 28*x^7 + 50*x^8 + 83*x^9 + 147*x^10 + 249*x^11 + 435*x^12 + ...
where
1/A(x) = 1 - x - x^2 - x^4 + x^5 - x^6 + x^9 - x^10 - x^12 + x^13 + x^14 - x^16 - x^18 + x^20 + x^21 - x^22 + x^25 - x^28 - x^29 - x^30 + ... + mu(n)*x^n +...
Also, g.f. A(x) satisfies:
x*A(x) = x*A(x)/A(x*A(x)) + x^2*A(x)^2/A(x^2*A(x)^2) + x^3*A(x)^3/A(x^3*A(x)^3) + x^4*A(x)^4/A(x^4*A(x)^4) + x^5*A(x)^5/A(x^5*A(x)^5) + ...

Links

Crossrefs

Cf. A073777, A068341, A070965, A008683, A185694, A157658.

Programs

  • Haskell
    a073776 n = a073776_list !! (n-1)
a073776_list = 1 : f [1] where
   f xs = y : f (y : xs) where y = sum $ zipWith (*) xs ms
   ms = map negate $ tail a008683_list
-- Reinhard Zumkeller, Nov 03 2015
  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[-MoebiusMu[k + 1]*a[n - k], {k, 1, n}]; Array[a,35,0] (* Jean-François Alcover, Apr 11 2011 *)
  • PARI
    {a(n) = my(A=[1,1],F); for(i=1,n, A=concat(A,0); F=Ser(A); A = Vec(sum(m=1,#A, subst(x/F, x, x^m*F^m))) ); A[n+1]}
for(n=0,50, print1(a(n),", ")) \\ Paul D. Hanna, Apr 19 2016

Formula

G.f.: A(x) = x / (Sum_{n>=1} mu(n)*x^n), A(0)=1, where mu(n) = Moebius function of n.
G.f. A(x) satisfies: x*A(x) = Sum_{n>=1} x^n*A(x)^n / A( x^n*A(x)^n ). - Paul D. Hanna, Apr 19 2016

A007557 Shifts left when inverse Moebius transform applied twice.

Original entry on oeis.org

1, 1, 3, 5, 10, 12, 24, 26, 43, 52, 78, 80, 133, 135, 189, 219, 295, 297, 428, 430, 584, 642, 804, 806, 1100, 1123, 1395, 1494, 1856, 1858, 2428, 2430, 2977, 3143, 3739, 3811, 4790, 4792, 5654, 5930, 7072, 7074, 8656
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Equals eigensequence of triangle A127170 (the square of the inverse Mobius transform). - Gary W. Adamson, Apr 27 2009

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000005, A007554, A038045, A038046, A003238, A070965.
Cf. A127170. - Gary W. Adamson, Apr 27 2009

Programs

  • Mathematica
    a[n_] := a[n] = Sum[ DivisorSigma[0, (n - 1)/d]*a[d], {d, Divisors[n - 1]}]; a[1] = 1; Table[a[n], {n, 1, 43}] (* Jean-François Alcover, Dec 12 2011, after Vladeta Jovovic *)

Formula

a(n+1) = Sum_{d divides n} tau(n/d)*a(d). - Vladeta Jovovic, Jan 24 2003
From Ilya Gutkovskiy, Apr 30 2019: (Start)
G.f. A(x) satisfies: A(x) = x * (1 + Sum_{i>=1} Sum_{j>=1} A(x^(i*j))).
G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * (1 + Sum_{i>=1} Sum_{j>=1} a(i)*x^(i*j)/(1 - x^(i*j))). (End)

Extensions

More terms from Vladeta Jovovic, Jan 24 2003

A073777 a(n) = Sum_{k=1..n} -A068341(k+1)*a(n-k), a(0)=1.

Original entry on oeis.org

1, 2, 5, 10, 22, 42, 85, 162, 314, 588, 1113, 2066, 3847, 7080, 13036, 23824, 43504, 79048, 143441, 259376, 468313, 843352, 1516515, 2721470, 4877165, 8726118, 15593224, 27826634, 49602226, 88316198, 157089101, 279137436, 495566701, 879034448, 1557979289
Offset: 0

Views

Author

Paul D. Hanna, Aug 10 2002

Keywords

Comments

Recurrence relation involves the convolution of the Moebius function (A068341).
Radius of convergence of A(x) is r=0.5802946238073267...
Related limits are limit_{n->infinity} a(n) r^n/n = 0.406...(?) and limit_{n->infinity} a(n+1)/a(n) = 1.723262561763844...
This sequence is the self-convolution of A073776.

Examples

			a(4) = -A068341(2)*a(3) -A068341(3)*a(2) -A068341(4)*a(1) -A068341(5)*a(0) = 2*10 +1*5 -2*2 +1*1 = 22. A068341 begins {1,-2,-1,2,-1,4,-2,0,3,...}.

Links

Crossrefs

Cf. A073776, A068341, A070965, A008683.

Programs

  • Haskell
    a073777 n = a073777_list !! (n-1)
a073777_list = 1 : f [1] where
   f xs = y : f (y : xs) where y = sum $ zipWith (*) xs ms'
   ms' = map negate $ tail a068341_list
-- Reinhard Zumkeller, Nov 03 2015
  • Mathematica
    A068341[n_] := A068341[n] = Sum[MoebiusMu[k]*MoebiusMu[n + 1 - k], {k, 1, n}]; a[0] = 1; a[n_] := a[n] = Sum[-A068341[k + 1]*a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 33}] (* Jean-François Alcover, Oct 10 2011 *)

Formula

G.f.: A(x)= x/(Sum_{n=1..infinity} mu(n)*x^n)^2, A(0)=1, where mu(n)=Moebius function.

Extensions

Corrected by Jean-François Alcover, Oct 10 2011

A307837 a(1) = 1; a(n+1) = Sum_{d|n} lambda(d)*a(d), where lambda = Liouville function (A008836).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, -1, -1, -1, 2, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 2, 1, 0, 1, 0, 2, -1, -2, 3, -3, -2, 1, 1, -1, 2, 3, 3, 2, 3, 3, -2, -3, 4, 4, -3, -3, -3, 4, -3, 4, 4, -3, 4, -5, 6, 6, -6, 8, 9, -9, 10, -8, -6, -7, 8, 7, 6, 5, 6, 7, -6, -8, -7, 6, 7, 9, 9, 5, -4, 2, -1
Offset: 1

Views

Author

Ilya Gutkovskiy, May 01 2019

Keywords

Crossrefs

Cf. A001222, A008836, A061020, A070965, A206369.

Programs

  • Mathematica
    a[n_] := a[n] = Sum[LiouvilleLambda[d] a[d], {d, Divisors[n - 1]}]; a[1] = 1; Table[a[n], {n, 1, 100}]
a[n_] := a[n] = SeriesCoefficient[x (1 + Sum[LiouvilleLambda[k] a[k] x^k/(1 - x^k), {k, 1, n - 1}]), {x, 0, n}]; Table[a[n], {n, 1, 100}]

Formula

G.f.: x * (1 + Sum_{n>=1} lambda(n)*a(n)*x^n/(1 - x^n)).
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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