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.

A193361 a(0)=0, a(1)=0; for n>1, a(n) = a(n-1) + (n-3)*a(n-2) + 1.

Original entry on oeis.org

0, 0, 1, 2, 4, 9, 22, 59, 170, 525, 1716, 5917, 21362, 80533, 315516, 1281913, 5383622, 23330405, 104084736, 477371217, 2246811730, 10839493637, 53528916508, 270318789249, 1394426035918, 7341439399397, 39413238225512, 215607783811041, 1200938739448842
Offset: 0

Views

Author

Vincenzo Librandi, Dec 25 2012

Keywords

Links

Crossrefs

Cf. A185108, A185109, A185308, A185309, A186738, A186739, A213720.

Programs

  • Magma
    [n le 2 select 0 else Self(n-1)+(n-4)*Self(n-2) + 1: n in [1..30]];
  • Mathematica
    RecurrenceTable[{a[1]==0,a[2]==0,a[n]==a[n-1]+(n-4) a[n-2]+1},a,{n,30}]

Formula

a(0)=a(1)=0, a(2)=1, a(n) = 2*a(n-1)+(n-4)*a(n-2)-(n-4)*a(n-3).
a(n) ~ (sqrt(Pi)+sqrt(2))/2 * n^(n/2-1)*exp(sqrt(n)-n/2-1/4) * (1-17/(24*sqrt(n))). - Vaclav Kotesovec, Dec 27 2012
a(n) = A187044(n-2). - Vaclav Kotesovec, Feb 14 2014

A187830 a(n)=2*a(n-1)+(n+3)*a(n-2)-(n+3)*a(n-3), a(0)=0, a(1)=0, a(2)=1.

Original entry on oeis.org

0, 0, 1, 2, 11, 30, 141, 472, 2165, 8302, 38613, 163144, 780953, 3554402, 17611557, 85145196, 437376337, 2225425454, 11847704869, 63032490312, 347377407169, 1923189664970, 10955002251365, 62881123205556, 369621186243777, 2193173759204902, 13281809346518213
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 27 2012

Keywords

Comments

This is case k=3. In general case, recurrence a(n)=2*a(n-1)+(n+k)*(a(n-2)-a(n-3)) is asymptotic to a(n) ~ c * n^(n/2+k/2+1)*exp(sqrt(n)-n/2-1/4) * (1+(12*k+31)/(24*sqrt(n))), where c is constant dependent only on k.
EGF is solution of the equation DSolve[{(3+k)*f[x] + (x-3-k)*f'[x] - (x+2)*f''[x] + f'''[x]==0, f[0]==0, f'[0]==0, f''[0]==1}, f, x]

Links

Crossrefs

Cf. A220700 (k=2), A213720 (k=1), A185309 (k=0), A185308 (k=-1), A186738 (k=-2), A186739 (k=-3), A193361 (k=-4), A220699 (k=-5).

Programs

  • Mathematica
    RecurrenceTable[{(3+n)*a[-3+n]+(-3-n)*a[-2+n]-2*a[-1+n]+a[n]==0,a[0]==0,a[1]==0,a[2]==1},a,{n,20}]
FullSimplify[CoefficientList[Series[1/30*E^(-(x^2/2))*((8*Sqrt[2*E*Pi]*Erf[1/Sqrt[2]]-27)*E^(x^2+x)*(x+1)*(x*(x+2)*(x*(x+2)+12)+26)+Sqrt[2*Pi]*E^(x^2+x)*(x+1)*(x*(x+2)*(x*(x+2)+12)+26)*(15*Erf[x/Sqrt[2]]-8*Sqrt[E]*Erf[(x+1)/Sqrt[2]])-16*E^(x^2/2)*(x*(x+2)+2)*(x*(x+2)+9)+30*E^(1/2*x*(x+2))*(x*(x*(x*(x+5)+19)+35)+33)), {x, 0, 20}], x]* Range[0, 20]!]

Formula

E.g.f.: 1/30*exp(-(x^2/2))*((8*sqrt(2*exp(1)*Pi)*erf(1/sqrt(2))-27)*exp(x^2+x)*(x+1)*(x*(x+2)*(x*(x+2)+12)+26)+sqrt(2*Pi)*exp(x^2+x)*(x+1)*(x*(x+2)*(x*(x+2)+12)+26)*(15*erf(x/sqrt(2))-8*sqrt(exp(1))*erf((x+1)/sqrt(2)))-16*exp(x^2/2)*(x*(x+2)+2)*(x*(x+2)+9)+30*exp(1/2*x*(x+2))*(x*(x*(x*(x+5)+19)+35)+33))
a(n) ~ (1/2*sqrt(Pi)-9/(10*sqrt(2))+4/15*sqrt(Pi)*exp(1/2)*(erf(1/sqrt(2))-1)) * n^(n/2+5/2)*exp(sqrt(n)-n/2-1/4) * (1+(67/(24*sqrt(n))))

A220699 a(0)=0, a(1)=0; for n>1, a(n) = a(n-1) + (n-4)*a(n-2) + 1.

Original entry on oeis.org

0, 0, 1, 2, 3, 6, 13, 32, 85, 246, 757, 2480, 8537, 30858, 116229, 455668, 1850417, 7774102, 33679941, 150291472, 689170529, 3244125554, 15649195077, 77287580604, 390271482145, 2013310674830, 10599283282021
Offset: 0

Views

Author

Vincenzo Librandi, Dec 25 2012

Keywords

Links

Crossrefs

Cf. A185108, A185109, A185308, A185309, A186738, A186739, A213720, A187830.

Programs

  • Magma
    [n le 2 select 0 else Self(n-1)+(n-5)*Self(n-2) + 1: n in [1..30]];
  • Mathematica
    RecurrenceTable[{a[1]==0, a[2]==0, a[n]==a[n-1] + (n-5) a[n-2] + 1}, a, {n, 40}]

Formula

a(0)=a(1)=0, a(2)=1, a(n) = 2*a(n-1)+(n-5)*a(n-2)-(n-5)*a(n-3).
a(n) ~ (sqrt(Pi)/2+sqrt(2)) * n^(n/2-3/2)*exp(sqrt(n)-n/2-1/4) * (1-29/(24*sqrt(n))). - Vaclav Kotesovec, Dec 27 2012

A220700 a(0)=0, a(1)=0; for n>1, a(n) = a(n-1) + (n+3)*a(n-2) + 1.

Original entry on oeis.org

0, 0, 1, 2, 10, 27, 118, 389, 1688, 6357, 28302, 117301, 541832, 2418649, 11629794, 55165477, 276131564, 1379441105, 7178203950, 37525908261, 202624599112, 1103246397377, 6168861375178, 34853267706981, 201412524836788, 1177304020632257, 7018267240899110
Offset: 0

Views

Author

Vincenzo Librandi, Dec 25 2012

Keywords

Links

Crossrefs

Cf. A185108, A185109, A185308, A185309, A186738, A186739, A213720.

Programs

  • Magma
    [n le 2 select 0 else Self(n-1)+(n+2)*Self(n-2) + 1: n in [1..30]];
  • Mathematica
    RecurrenceTable[{a[0] == 0, a[1] == 0, a[n] == a[n-1] + (n+3) a[n-2] + 1}, a, {n, 0, 40}] (* corrected by Georg Fischer, Dec 05 2019 *)
FullSimplify[CoefficientList[Series[1/8*E^(-(x^2/2))*(E^(x^2/2)*(3*Sqrt[2*Pi]*Erf[1/Sqrt[2]]*E^(1/2*(x+1)^2)*(x*(x+2)*(x*(x+2)+8)+10)-6*(x+1)*(x*(x+2)+6)-6*E^(1/2*x*(x+2))*(x*(x+2)*(x*(x+2)+8)+10)+8*E^x*(x*(x*(x+4)+11)+12))+Sqrt[2*Pi]*E^(x^2+x)*(x*(x+2)*(x*(x+2)+8)+10)*(4*Erf[x/Sqrt[2]]-3*Sqrt[E]*Erf[(x+1)/Sqrt[2]])), {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Dec 27 2012 *)
nxt[{n_,a_,b_}]:={n+1,b,b+a(n+4)+1}; NestList[nxt,{1,0,0},30][[All,2]] (* Harvey P. Dale, Mar 01 2020 *)

Formula

a(0)=a(1)=0, a(2)=1, a(n) = 2*a(n-1)+(n+2)*a(n-2)-(n+2)*a(n-3).
E.g.f.: 1/8*exp(-(x^2/2))*(exp(x^2/2)*(3*sqrt(2*Pi)*erf(1/sqrt(2))*exp(1/2*(x+1)^2)*(x*(x+2)*(x*(x+2)+8)+10)-6*(x+1)*(x*(x+2)+6)-6*exp(1/2*x*(x+2))*(x*(x+2)*(x*(x+2)+8)+10)+8*exp(x)*(x*(x*(x+4)+11)+12))+sqrt(2*Pi)*exp(x^2+x)*(x*(x+2)*(x*(x+2)+8)+10)*(4*erf(x/sqrt(2))-3*sqrt(exp(1))*erf((x+1)/sqrt(2)))). - Vaclav Kotesovec, Dec 27 2012
a(n) ~ (1/2*sqrt(Pi)-3/(4*sqrt(2))+3/8*sqrt(Pi)*exp(1/2)*(erf(1/sqrt(2))-1)) * n^(n/2+2)*exp(sqrt(n)-n/2-1/4) * (1+55/(24*sqrt(n))). - Vaclav Kotesovec, Dec 27 2012
Showing 1-4 of 4 results.

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