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 1-3 of 3 results.

A256272 G.f.: Sum_{n>=1} Pell(n+1) * x^n / (1 - x^n), where Pell(n) = A000129(n).

Original entry on oeis.org

2, 7, 14, 36, 72, 188, 410, 1021, 2392, 5818, 13862, 33678, 80784, 195440, 470916, 1137710, 2744212, 6627675, 15994430, 38619812, 93222780, 225072548, 543339722, 1311772784, 3166816034, 7645450834, 18457558444, 44560677618, 107578520352, 259717999680, 627013566050, 1513745792655, 3654502889812
Offset: 1

Views

Author

Paul D. Hanna, Jun 01 2015

Keywords

Examples

			G.f.: A(x) = 2*x + 7*x^2 + 14*x^3 + 36*x^4 + 72*x^5 + 188*x^6 +...
where by definition
A(x) = 2*x/(1-x) + 5*x^2/(1-x^2) + 12*x^3/(1-x^3) + 29*x^4/(1-x^4) + 70*x^5/(1-x^5) + 169*x^6/(1-x^6) + 408*x^7/(1-x^7) + 985*x^8/(1-x^8) + 2378*x^9/(1-x^9) + 5741*x^10/(1-x^10) +...+ Pell(n+1)*x^n/(1-x^n) +...
The g.f. is also given by the series identity:
A(x) = x*(2+x)/(1-2*x-x^2) + x^2*(2+x^2)/(1-2*x^2-x^4) + x^3*(2+x^3)/(1-2*x^3-x^6) + x^4*(2+x^4)/(1-2*x^4-x^8) + x^5*(2+x^5)/(1-2*x^5-x^10) + x^6*(2+x^6)/(1-2*x^6-x^12) + x^7*(2+x^7)/(1-2*x^7-x^14) +...+ x^n*(1+x^n)/(1-x^n-x^(2*n)) +...
And also we have the series:
A(x) = x*(2 + x) + x^2*((2+x)^2 + (2+x^2)) + x^3*((2+x)^3 + (2+x^3))
+ x^4*((2+x)^4 + (2+x^2)^2 + (2+x^4)) + x^5*((2+x)^5 + (2+x^5))
+ x^6*((2+x)^6 + (2+x^2)^3 + (2+x^3)^2 + (2+x^6))
+ x^7*((2+x)^7 + (2+x^7))
+ x^8*((2+x)^8 + (2+x^2)^4 + (2+x^4)^2 + (2+x^8))
+ x^9*((2+x)^9 + (2+x^3)^3 + (2+x^9))
+ x^10*((2+x)^10 + (2+x^2)^5 + (2+x^5)^2 + (2+x^10))
+ x^11*((2+x)^11 + (2+x^11))
+ x^12*((2+x)^12 + (2+x^2)^6 + (2+x^3)^4 + (2+x^4)^3 + (2+x^6)^2 + (2+x^12))
+...+ x^n * Sum_{d|n} (2 + x^d)^(n/d) +...
or, more explicitly,
A(x) = x*(2 + x) + x^2*(6 + 4*x + 2*x^2)
+ x^3*(10 + 12*x + 6*x^2 + 2*x^3)
+ x^4*(22 + 32*x + 28*x^2 + 8*x^3 + 3*x^4)
+ x^5*(34 + 80*x + 80*x^2 + 40*x^3 + 10*x^4 + 2*x^5)
+ x^6*(78 + 192*x + 252*x^2 + 164*x^3 + 66*x^4 + 12*x^5 + 4*x^6)
+ x^7*(130 + 448*x + 672*x^2 + 560*x^3 + 280*x^4 + 84*x^5 + 14*x^6 + 2*x^7)
+ x^8*(278 + 1024*x + 1824*x^2 + 1792*x^3 + 1148*x^4 + 448*x^5 + 120*x^6 + 16*x^7 + 4*x^8)
+ x^9*(522 + 2304*x + 4608*x^2 + 5388*x^3 + 4032*x^4 + 2016*x^5 + 678*x^6 + 144*x^7 + 18*x^8 + 3*x^9) +...

Links

Crossrefs

Cf. A256281, A245282, A000129.

Programs

  • Mathematica
    a[n_] := SeriesCoefficient[Sum[(x^k*(2+x^k))/(1-2*x^k-x^(2*k)), {k, 1, n}], {x, 0, n}]; Array[a, 40] (* Jean-François Alcover, Dec 19 2015 *)
  • PARI
    {Pell(n)=polcoeff(x/(1-2*x-x^2 +x*O(x^n)),n)}
{a(n)=polcoeff(sum(m=1, n, Pell(m+1)*x^m/(1-x^m +x*O(x^n))), n)}
for(n=1, 40, print1(a(n), ", "))
  • PARI
    {a(n)=polcoeff(sum(m=1, n, x^m*(2+x^m)/(1-2*x^m-x^(2*m) +x*O(x^n)) ), n)}
for(n=1, 40, print1(a(n), ", "))
  • PARI
    {a(n)=local(A=x+x^2); A=sum(m=1, n, x^m*sumdiv(m, d, (2 + x^(m/d) +x*O(x^n))^d) ); polcoeff(A, n)}
for(n=1, 40, print1(a(n), ", "))

Formula

G.f.: Sum_{n>=1} x^n * (2 + x^n) / (1 - 2*x^n - x^(2*n)).
G.f.: Sum_{n>=1} x^n * Sum_{d|n} (2 + x^d)^(n/d).
a(2*n^2) == 1 (mod 2), with a(n) == 0 (mod 2) elsewhere.
a(n) ~ (1+sqrt(2))^(n+1) / (2*sqrt(2)). - Vaclav Kotesovec, Jun 02 2015

A133726 Möbius transform of the Pell numbers (A000129).

Original entry on oeis.org

1, 1, 4, 10, 28, 64, 168, 396, 980, 2348, 5740, 13780, 33460, 80612, 194992, 470424, 1136688, 2743160, 6625108, 15992040, 38613792, 93216616, 225058680, 543325464, 1311738092, 3166782500, 7645369060, 18457475260, 44560482148, 107578322912, 259717522848
Offset: 1

Views

Author

Gary W. Adamson, Sep 21 2007

Keywords

Examples

			a(4) = 10 = (0, -1, 0, 1) dot (1, 2, 5, 12) = (0, -2, 0, 12).

Links

Crossrefs

Cf. A000129, A054525, A256281.

Programs

  • Maple
    with(numtheory):
a:= n-> add(mobius(n/d)*(<<0|1>, <1|2>>^d. <<0,1>>)[1,1], d=divisors(n)):
seq(a(n), n=1..40);  # Alois P. Heinz, Sep 26 2011
  • Mathematica
    a[n_] := Sum[MoebiusMu[n/d]*(MatrixPower[{{0, 1}, {1, 2}}, d]. {0, 1})[[1]], {d, Divisors[n]}];
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Apr 19 2025, after Alois P. Heinz *)

Formula

a(n) = Sum_{d|n} A008683(n/d)*A000129(d) = Sum_{k=1..n} A054525(n,k) * A000129(k).
G.f.: Sum_{k>=1} mu(k) * x^k / (1 - 2*x^k - x^(2*k)). - Ilya Gutkovskiy, Feb 06 2020

A309729 Expansion of Sum_{k>=1} x^k/(1 - x^k - 2*x^(2*k)).

Original entry on oeis.org

1, 2, 4, 7, 12, 26, 44, 92, 175, 354, 684, 1396, 2732, 5506, 10938, 21937, 43692, 87578, 174764, 349884, 699098, 1398786, 2796204, 5593886, 11184823, 22372354, 44739418, 89483996, 178956972, 357925242, 715827884, 1431677702, 2863312218, 5726666754, 11453246178, 22906581193
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 14 2019

Keywords

Comments

Inverse Moebius transform of Jacobsthal numbers (A001045).

Crossrefs

Cf. A001045, A007435, A055895, A100107, A104723, A256281.

Programs

  • Maple
    seq(add(2^d-(-1)^d, d=numtheory:-divisors(n))/3, n=1..50); # Robert Israel, Aug 14 2019
  • Mathematica
    nmax = 36; CoefficientList[Series[Sum[x^k/(1 - x^k - 2 x^(2 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[(1/3) Sum[(2^d - (-1)^d), {d, Divisors[n]}], {n, 1, 36}]
  • PARI
    a(n)={sumdiv(n, d, 2^d - (-1)^d)/3} \\ Andrew Howroyd, Aug 14 2019
  • Python
    n = 1
while n <= 36:
    s, d = 0, 1
    while d <= n:
        if n%d == 0:
            s = s+2**d-(-1)**d
        d = d+1
    print(n,s//3)
n = n+1 # A.H.M. Smeets, Aug 14 2019

Formula

G.f.: Sum_{k>=1} A001045(k) * x^k/(1 - x^k).
a(n) = (1/3) * Sum_{d|n} (2^d - (-1)^d).
Showing 1-3 of 3 results.

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