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.

A190958 a(n) = 2*a(n-1) - 10*a(n-2), with a(0) = 0, a(1) = 1.

Original entry on oeis.org

0, 1, 2, -6, -32, -4, 312, 664, -1792, -10224, -2528, 97184, 219648, -532544, -3261568, -1197696, 30220288, 72417536, -157367808, -1038910976, -504143872, 9380822016, 23803082752, -46202054656, -330434936832, -198849327104, 2906650714112, 7801794699264
Offset: 0

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Author

Vladimir Joseph Stephan Orlovsky, May 24 2011

Keywords

Comments

For the difference equation a(n) = c*a(n-1) - d*a(n-2), with a(0) = 0, a(1) = 1, the solution is a(n) = d^((n-1)/2) * ChebyshevU(n-1, c/(2*sqrt(d))) and has the alternate form a(n) = ( ((c + sqrt(c^2 - 4*d))/2)^n - ((c - sqrt(c^2 - 4*d))/2)^n )/sqrt(c^2 - 4*d). In the case c^2 = 4*d then the solution is a(n) = n*d^((n-1)/2). The generating function is x/(1 - c*x + d^2) and the exponential generating function takes the form (2/sqrt(c^2 - 4*d))*exp(c*x/2)*sinh(sqrt(c^2 - 4*d)*x/2) for c^2 > 4*d, (2/sqrt(4*d - c^2))*exp(c*x/2)*sin(sqrt(4*d - c^2)*x/2) for 4*d > c^2, and x*exp(sqrt(d)*x) if c^2 = 4*d. - G. C. Greubel, Jun 10 2022

Links

Crossrefs

Sequences of the form a(n) = c*a(n-1) - d*a(n-2), with a(0)=0, a(1)=1:
c/d...1.......2.......3.......4.......5.......6.......7.......8.......9......10
1..A128834,A107920,A106852,A106853,A106854,A145934,A145976,A145978,A146078,A146080
2..A000027,A009545,A088137,A088138,A045873,A088139,A089181,A090591,A127357,A190958
3..A001906,A000225,A057083,A049072,A190959,A190960,A190961,A190962,A190963,A190964
4..A001353,A007070,A003462,A001787,A099456,A190965,A168175,A190966,A190967,A190968
5..A004254,A107839,A116415,A002450,A030191,A001047,A099450,A190969,A190970,A190971
6..A001109,A154244,A138395,A084326,A003463,A030192,A081179,A006516,A027471,A106392
7..A004187,A186446,A190972,A190973,A190974,A003464,A030240,A152268,A099459,A016127
8..A001090,A190975,A190976,A099156,A190977,A190978,A023000,A057084,A154245,A154235
9..A018913,A190979,A190980,A190981,A190982,A190983,A190984,A023001,A057085,A178869
10.A004189,A190869,A190985,A190986,A190987,A190988,A190989,A190990,A002452,A057086

Programs

  • Magma
    I:=[0,1]; [n le 2 select I[n] else 2*Self(n-1)-10*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 17 2011
  • Mathematica
    LinearRecurrence[{2,-10}, {0,1}, 50]
  • PARI
    a(n)=([0,1; -10,2]^n*[0;1])[1,1] \\ Charles R Greathouse IV, Apr 08 2016
  • SageMath
    [lucas_number1(n,2,10) for n in (0..50)] # G. C. Greubel, Jun 10 2022

Formula

G.f.: x / ( 1 - 2*x + 10*x^2 ). - R. J. Mathar, Jun 01 2011
E.g.f.: (1/3)*exp(x)*sin(3*x). - Franck Maminirina Ramaharo, Nov 13 2018
a(n) = 10^((n-1)/2) * ChebyshevU(n-1, 1/sqrt(10)). - G. C. Greubel, Jun 10 2022
a(n) = (1/3)*10^(n/2)*sin(n*arctan(3)) = Sum_{k=0..floor(n/2)} (-1)^k*3^(2*k)*binomial(n,2*k+1). - Gerry Martens, Oct 15 2022

A254601 Numbers of n-length words on alphabet {0,1,...,6} with no subwords ii, where i is from {0,1,2}.

Original entry on oeis.org

1, 7, 46, 304, 2008, 13264, 87616, 578752, 3822976, 25252864, 166809088, 1101865984, 7278432256, 48078057472, 317582073856, 2097804673024, 13857156333568, 91534156693504, 604633565495296, 3993938019745792, 26382162380455936, 174268726361718784
Offset: 0

Views

Author

Milan Janjic, Feb 02 2015

Keywords

Links

Crossrefs

Cf. A055099, A126473, A126501, A126528, A135032, A190976 (shifted bin. trans).

Programs

  • Magma
    [n le 1 select 7^n else 6*Self(n)+4*Self(n-1): n in [0..25]]; // Bruno Berselli, Feb 03 2015
  • Mathematica
    RecurrenceTable[{a[0] == 1, a[1] == 7, a[n] == 6 a[n - 1] + 4 a[n - 2]}, a[n], {n, 0, 25}]
LinearRecurrence[{6,4},{1,7},30] (* Harvey P. Dale, Oct 10 2017 *)
  • PARI
    Vec((1 + x)/(1 - 6*x - 4*x^2) + O(x^30)) \\ Colin Barker, Jan 22 2017

Formula

G.f.: (1 + x)/(1 - 6*x - 4*x^2).
a(n) = 6*a(n-1) + 4*a(n-2) with n>1, a(0) = 1, a(1) = 7.
a(n) = ((3-r)^n*(-4+r) + (3+r)^n*(4+r)) / (2*r), where r=sqrt(13). - Colin Barker, Jan 22 2017
a(n) = A135032(n-1)+A135032(n). - R. J. Mathar, Apr 07 2022

A332936 Number of blue nodes in n-th power graph W exponentiation of a cycle graph with 7 blue nodes and 1 green node.

Original entry on oeis.org

7, 51, 387, 2943, 22383, 170235, 1294731, 9847143, 74892951, 569602179, 4332138579, 32948302095, 250590001023, 1905875101899, 14495230812123, 110244221191287, 838468077093927, 6377011953177555, 48500691394138659, 368874495293576607, 2805493888166196879, 21337327619448845211
Offset: 0

Views

Author

George Strand Vajagich, Mar 02 2020

Keywords

Comments

The series of green nodes in n-th power W exponentiation for all n<6 n blue 1 green, 2 edge per node graphs already corresponds with an existing OEIS sequence (empirical). For example the number of blue nodes in n-th power W exponentiation of a square containing 3 blue nodes and 1 green node corresponds to A163063.

Examples

			For n = 2 take g(1)=15 and b(1)=51. Multiply b(1) by 7 to get 357 add 30 to get 387.
For n = 3 take g(2)=117 and b(2)=387. Multiply b(2) by 7 to get 774 add 234 to get 2943.

Links

Crossrefs

Cf. A331211.

Programs

  • PARI
    Vec((1 + 43*x - 18*x^2) / (1 - 8*x + 3*x^2) + O(x^40)) \\ Colin Barker, Mar 03 2020
  • Python
    g=1
b=7
sg=0
sb=0
bl=[]
gl=[]
for int in range(1,20):
  sg=g*1+b*2
  sb=b*7+g*2
  g=sg
  b=sb
  gl.append(g)
  bl.append(b)
print(bl)

Formula

g(n) = g(n-1) + 2*a(n-1), a(n) = 2*g(n-1) + 7*a(n-1) with g(0) = 1 and b(0) = 7, where g(n) = A332211(n).
From Colin Barker, Mar 03 2020: (Start)
G.f.: (1 + 43*x - 18*x^2) / (1 - 8*x + 3*x^2).
a(n) = 8*a(n-1) - 3*a(n-2) for n > 1.
(End)
From Stefano Spezia, Mar 03 2020: (Start)
a(n) = ((4 - sqrt(13))^n*(-23 + 7*sqrt(13)) + (4 + sqrt(13))^n*(23 + 7*sqrt(13)))/(2*sqrt(13)).
E.g.f.: exp(4*x)*(91*cosh(sqrt(13)*x) + 23*sqrt(13)*sinh(sqrt(13)*x))/13.
(End)
a(n) = 7*A190976(n+1) -5*A190976(n). - R. J. Mathar, Apr 30 2020
Showing 1-3 of 3 results.

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