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.

A087130 a(n) = 5*a(n-1)+a(n-2) for n>1, a(0)=2, a(1)=5.

Original entry on oeis.org

2, 5, 27, 140, 727, 3775, 19602, 101785, 528527, 2744420, 14250627, 73997555, 384238402, 1995189565, 10360186227, 53796120700, 279340789727, 1450500069335, 7531841136402, 39109705751345, 203080369893127
Offset: 0

Views

Author

Paul Barry, Aug 16 2003

Keywords

Comments

Sequence is related to the fifth metallic mean [5;5,5,5,5,...] (see A098318).
The solution to the general recurrence b(n) = (2*k+1)*b(n-1)+b(n-2) with b(0)=2, b(1) = 2*k+1 is b(n) = ((2*k+1)+sqrt(4*k^2+4*k+5))^n+(2*k+1)-sqrt(4*k^2+4*k+5))^n)/2; b(n) = 2^(1-n)*Sum_{j=0..n} C(n, 2*j)*(4*k^2+4*k+5)^j*(2*k+1)^(n-2*j); b(n) = 2*T(n, (2*k+1)*x/2)(-1)^i with T(n, x) Chebyshev's polynomials of the first kind (see A053120) and i^2=-1. - Paul Barry, Nov 15 2003
Primes in this sequence include a(0) = 2; a(1) = 5; a(4) = 727; a(8) = 528527 (3) semiprimes in this sequence include a(7) = 101785; a(13) = 1995189565; a(16) = 279340789727; a(19) = 39109705751345; a(20) = 203080369893127 - Jonathan Vos Post, Feb 09 2005
a(n)^2 - 29*A052918(n-1)^2 = 4*(-1)^n, with n>0 - Gary W. Adamson, Oct 07 2008
For more information about this type of recurrence follow the Khovanova link and see A054413 and A086902. - Johannes W. Meijer, Jun 12 2010
Binomial transform of A072263. - Johannes W. Meijer, Aug 01 2010

Links

Crossrefs

Cf. A006497, A014448, A085447.
Cf. A086902, A000032, A052918.

Programs

  • Magma
    I:=[2,5]; [n le 2 select I[n] else 5*Self(n-1)+Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 19 2016
  • Mathematica
    RecurrenceTable[{a[0] == 2, a[1] == 5, a[n] == 5 a[n-1] + a[n-2]}, a, {n, 30}] (* Vincenzo Librandi, Sep 19 2016 *)
  • PARI
    {a(n) = if( n<0, (-1)^n * a(-n), polsym(x^2 - 5*x -1, n) [n + 1])} /* Michael Somos, Nov 04 2008 */
  • Sage
    [lucas_number2(n,5,-1) for n in range(0, 21)] # Zerinvary Lajos, May 14 2009

Formula

a(n) = ((5+sqrt(29))/2)^n+((5-sqrt(29))/2)^n.
a(n) = A100236(n) + 1.
E.g.f. : 2*exp(5*x/2)*cosh(sqrt(29)*x/2); a(n) = 2^(1-n)*Sum_{k=0..floor(n/2)} C(n, 2k)*29^k*5^(n-2*k). a(n) = 2T(n, 5i/2)(-i)^n with T(n, x) Chebyshev's polynomials of the first kind (see A053120) and i^2=-1. - Paul Barry, Nov 15 2003
O.g.f.: (-2+5*x)/(-1+5*x+x^2). - R. J. Mathar, Dec 02 2007
a(-n) = (-1)^n * a(n). - Michael Somos, Nov 01 2008
A090248(n) = a(2*n). 5 * A097834(n) = a(2*n + 1). - Michael Somos, Nov 01 2008
Limit(a(n+k)/a(k), k=infinity) = (A087130(n) + A052918(n-1)*sqrt(29))/2. Limit(A087130(n)/A052918(n-1), n= infinity) = sqrt(29). - Johannes W. Meijer, Jun 12 2010
a(3n+1) = A041046(5n), a(3n+2) = A041046(5n+3) and a(3n+3) = 2*A041046 (5n+4). - Johannes W. Meijer, Jun 12 2010
a(n) = 2*A052918(n) - 5*A052918(n-1). - R. J. Mathar, Oct 02 2020
From Peter Bala, Jul 09 2025 : (Start)
The following series telescope (Cf. A000032):
For k >= 1, Sum_{n >= 1} (-1)^((k+1)*(n+1)) * a(2*n*k)/(a((2*n-1)*k)*a((2*n+1)*k)) = 1/a(k)^2.
For positive even k, Sum_{n >= 1} 1/(a(k*n) - (a(k) + 2)/a(k*n)) = 1/(a(k) - 2) and
Sum_{n >= 1} (-1)^(n+1)/(a(k*n) + (a(k) - 2)/a(k*n)) = 1/(a(k) + 2).
For positive odd k, Sum_{n >= 1} 1/(a(k*n) - (-1)^n*(a(2*k) + 2)/a(k*n)) = (a(k) + 2)/(2*(a(2*k) - 2)) and
Sum_{n >= 1} (-1)^(n+1)/(a(k*n) - (-1)^n*(a(2*k) + 2)/a(k*n)) = (a(k) - 2)/(2*(a(2*k) - 2)). (End)

A100235 Triangle, read by rows, of the coefficients of [x^k] in G100234(x)^n such that the row sums are 6^n-1 for n>0, where G100234(x) is the g.f. of A100234.

Original entry on oeis.org

1, 1, 4, 1, 8, 26, 1, 12, 63, 139, 1, 16, 116, 436, 726, 1, 20, 185, 965, 2830, 3774, 1, 24, 270, 1790, 7335, 17634, 19601, 1, 28, 371, 2975, 15505, 52444, 106827, 101784, 1, 32, 488, 4584, 28860, 124424, 358748, 633952, 528526, 1, 36, 621, 6681, 49176, 256194
Offset: 0

Views

Author

Paul D. Hanna, Nov 29 2004

Keywords

Comments

The main diagonal forms A100236. Secondary diagonal is: T(n+1,n) = (n+1)*A100237(n). More generally, if g.f. F(x) satisfies: m^n-b^n = Sum_{k=0..n} [x^k]F(x)^n, then F(x) also satisfies: (m+z)^n - (b+z)^n + z^n = Sum_{k=0..n} [x^k](F(x)+z*x)^n for all z and F(x)=(1+(m-1)*x+sqrt(1+2*(m-2*b-1)*x+(m^2-2*m+4*b+1)*x^2))/2; the triangle formed from powers of F(x) will have the g.f.: G(x,y)=(1-2*x*y+m*x^2*y^2)/((1-x*y)*(1-(m-1)*x*y-x^2*y^2-x*(1-x*y))).

Examples

			Rows begin:
[1],
[1,4],
[1,8,26],
[1,12,63,139],
[1,16,116,436,726],
[1,20,185,965,2830,3774],
[1,24,270,1790,7335,17634,19601],
[1,28,371,2975,15505,52444,106827,101784],
[1,32,488,4584,28860,124424,358748,633952,528526],...
where row sums form 6^n-1 for n>0:
6^1-1 = 1+4 = 5
6^2-1 = 1+8+26 = 35
6^3-1 = 1+12+63+139 = 215
6^4-1 = 1+16+116+436+726 = 1295
6^5-1 = 1+20+185+965+2830+3774 = 7775.
The main diagonal forms A100236 = [1,4,26,139,726,3774,...],
where Sum_{n>=1} A100236(n)/n*x^n = log((1-x)/(1-5*x-x^2)).

Crossrefs

Cf. A100234, A100236, A100237, A100232.

Programs

  • Mathematica
    row[n_] := CoefficientList[ Series[ (1 + 5*x + Sqrt[1 + 6*x + 29*x^2])^n/2^n, {x, 0, n}], x]; Flatten[ Table[ row[n], {n, 0, 9}]](* Jean-François Alcover, May 11 2012, after PARI *)
  • PARI
    T(n,k,m=6)=if(n

Formula

G.f.: A(x, y)=(1-2*x*y+6*x^2*y^2)/((1-x*y)*(1-5*x*y-x^2*y^2-x*(1-x*y))).

A100237 Secondary diagonal of triangle A100235 divided by row number: a(n) = A100235(n+1,n)/(n+1) for n >= 0.

Original entry on oeis.org

1, 4, 21, 109, 566, 2939, 15261, 79244, 411481, 2136649, 11094726, 57610279, 299146121, 1553340884, 8065850541, 41882593589, 217478818486, 1129276686019, 5863862248581, 30448587928924, 158106801893201, 820982597394929, 4263019788867846, 22136081541734159
Offset: 0

Views

Author

Paul D. Hanna, Nov 30 2004

Keywords

Comments

G.f. equals the ratio of the g.f.s of any two adjacent diagonals of triangle A100235.
a(n) is the number of compositions of n when there are 4 types of 1 and 5 types of other natural numbers. - Milan Janjic, Aug 13 2010

Links

Crossrefs

Cf. A100234, A100235, A100236.
First differences of A052918.
Cf. A052918.

Programs

  • Maple
    a[0]:=1: a[1]:=4: for n from 2 to 26 do a[n]:=5*a[n-1]+a[n-2] od: seq(a[n], n=0..21); # Zerinvary Lajos, Jul 26 2006
  • PARI
    a(n)=polcoeff((1-x)/(1-5*x-x^2)+x*O(x^n),n)
  • PARI
    Vec((1-x)/(1-5*x-x^2) + O(x^40)) \\ Colin Barker, Oct 13 2015

Formula

a(n) = 5*a(n-1) + a(n-2) for n>1, with a(0)=1, a(1)=4.
G.f.: (1-x)/(1-5*x-x^2).
Numerators in continued fraction [1, 4, 5, 5, 5, ...]. Continued fraction [1, 4, 5, 5, 5, ...] = 0.807417596433..., the inradius of a right triangle with legs 2 and 5. n-th convergent (n>0) to [1, 4, 5, 5, 5, ...] = A100237(n)/A052918(n), the first few being 1/1, 4/5, 21/26, 109/135, ... - Gary W. Adamson, Dec 21 2007
If p[1]=4, p[i]=5, (i>1), and if A is Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n >= 1, a(n)=det A. - Milan Janjic, Apr 29 2010
a(n) = (2^(-1-n)*((5-sqrt(29))^n*(-3+sqrt(29)) + (3+sqrt(29))*(5+sqrt(29))^n))/sqrt(29). - Colin Barker, Oct 13 2015

A100234 G.f. A(x) satisfies: 6^n - 1 = Sum_{k=0..n} [x^k]A(x)^n and also satisfies: (6+z)^n - (1+z)^n + z^n = Sum_{k=0..n} [x^k](A(x)+z*x)^n for all z, where [x^k]A(x)^n denotes the coefficient of x^k in A(x)^n.

Original entry on oeis.org

1, 4, 5, -15, 20, 90, -695, 1785, 3895, -53985, 196255, 121635, -4907130, 23332140, -13181145, -470127465, 2866898820, -4455872910, -44776087145, 356263904235, -873534120380, -3988869806010, 44179467566755, -147200296896765, -293052319462105, 5409366658571715
Offset: 0

Views

Author

Paul D. Hanna, Nov 29 2004

Keywords

Comments

More generally, if g.f. A(x) satisfies: m^n-b^n = Sum_{k=0..n} [x^k]A(x)^n, then A(x) also satisfies: (m+z)^n - (b+z)^n + z^n = Sum_{k=0..n} [x^k](A(x)+z*x)^n for all z and A(x)=(1+(m-1)*x+sqrt(1+2*(m-2*b-1)*x+(m^2-2*m+4*b+1)*x^2))/2.

Examples

			From the table of powers of A(x) (A100235), we see that
6^n-1 = Sum of coefficients [x^0] through [x^n] in A(x)^n:
A^1=[1,4],5,-15,20,90,-695,1785,...
A^2=[1,8,26],10,-55,190,-245,-1690,...
A^3=[1,12,63,139],15,-120,635,-2130,...
A^4=[1,16,116,436,726],20,-210,1480,...
A^5=[1,20,185,965,2830,3774],25,-325,...
A^6=[1,24,270,1790,7335,17634,19601],30,...

Crossrefs

Cf. A100235, A100236, A100231.

Programs

  • PARI
    a(n)=if(n==0,1,(6^n-1-sum(k=0,n,polcoeff(sum(j=0,min(k,n-1),a(j)*x^j)^n+x*O(x^k),k)))/n)
  • PARI
    a(n)=if(n==0,1,if(n==1,4,if(n==2,5,-(3*(2*n-3)*a(n-1)+29*(n-3)*a(n-2))/n)))
  • PARI
    a(n)=polcoeff((1+5*x+sqrt(1+6*x+29*x^2+x^2*O(x^n)))/2,n)

Formula

a(n)=-(3*(2*n-3)*a(n-1)+29*(n-3)*a(n-2))/n for n>2, with a(0)=1, a(1)=4, a(2)=5. G.f.: A(x) = (1+5*x+sqrt(1+6*x+29*x^2))/2.
Showing 1-4 of 4 results.

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