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.

A164544 a(n) = 2*a(n-1) + 7*a(n-2) for n > 1; a(0) = 1, a(1) = 7.

Original entry on oeis.org

1, 7, 21, 91, 329, 1295, 4893, 18851, 71953, 275863, 1055397, 4041835, 15471449, 59235743, 226771629, 868193459, 3323788321, 12724930855, 48716379957, 186507275899, 714029211497, 2733609354287, 10465423189053, 40066111858115
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Aug 15 2009

Keywords

Comments

Binomial transform of A164640. Inverse binomial transform of A164545.

Links

Crossrefs

Cf. A164545, A164640.

Programs

  • Magma
    Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((2+3*r)*(1+2*r)^n+(2-3*r)*(1-2*r)^n)/4: n in [0..23] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 19 2009
  • Mathematica
    LinearRecurrence[{2,7},{1,7},40] (* Harvey P. Dale, Jul 15 2012 *)
  • Sage
    [(i*sqrt(7))^(n-1)*(i*sqrt(7)*chebyshev_U(n, -i/sqrt(7)) + 5*chebyshev_U(n-1, -i/sqrt(7))) for n in (0..40)] # G. C. Greubel, Jul 18 2021

Formula

a(n) = 2*a(n-1) + 7*a(n-2) for n > 1; a(0) = 1, a(1) = 7.
a(n) = ((2+3*sqrt(2))*(1+2*sqrt(2))^n + (2-3*sqrt(2))*(1-2*sqrt(2))^n)/4.
G.f.: (1+5*x)/(1-2*x-7*x^2).
a(n)/a(n-1) ~ 1 + 2*sqrt(2). - Kyle MacLean Smith, Dec 15 2019
E.g.f.: exp(x)*cosh(2*sqrt(2)*x) + 3*exp(x)*sinh(2*sqrt(2)*x)/sqrt(2). - Stefano Spezia, Dec 16 2019
From G. C. Greubel, Jul 18 2021: (Start)
a(n) = (i*sqrt(7))^(n-1)*(i*sqrt(7)*ChebyshevU(n, -i/sqrt(7)) + 5*ChebyshevU(n-1, -i/sqrt(7))).
a(n) = Sum_{j=0..floor(n/2)} binomial(n-k, k)*((7*n -12*k)/(n-k))*7^k*2^(n-2*k-1). (End)

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus, Aug 19 2009

A154346 a(n) = 12*a(n-1) - 28*a(n-2) for n > 1, with a(0)=1, a(1)=12.

Original entry on oeis.org

1, 12, 116, 1056, 9424, 83520, 738368, 6521856, 57587968, 508443648, 4488860672, 39629905920, 349870772224, 3088811900928, 27269361188864, 240745601040384, 2125405099196416, 18763984361226240, 165656469557215232
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Jan 07 2009

Keywords

Comments

Binomial transform of A164547, second binomial transform of A164546, third binomial transform of A038761, fourth binomial transform of A164545, fifth binomial transform of A164544, sixth binomial transform of A164640.
Lim_{n -> infinity} a(n)/a(n-1) = 6 + 2*sqrt(2) = 8.8284271247....

Links

Crossrefs

Cf. A002193 (decimal expansion of sqrt(2)), A164547, A164546, A038761, A164545, A164544, A164640.

Programs

  • Magma
    Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((6+2*r)^n-(6-2*r)^n)/(4*r): n in [1..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jan 12 2009
  • Mathematica
    Join[{a=1,b=12},Table[c=12*b-28*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 31 2011 *)
LinearRecurrence[{12,-28},{1,12},20] (* Harvey P. Dale, May 23 2012 *)
Rest@ CoefficientList[Series[x/(1 - 12 x + 28 x^2), {x, 0, 19}], x] (* Michael De Vlieger, Sep 13 2016 *)
  • PARI
    a(n)=([0,1; -28,12]^(n-1)*[1;12])[1,1] \\ Charles R Greathouse IV, Sep 13 2016
  • Sage
    [lucas_number1(n,12,28) for n in range(1, 20)] # Zerinvary Lajos, Apr 27 2009

Formula

a(n) = 12*a(n-1) - 28*a(n-2) for n > 1. - Philippe Deléham, Jan 12 2009
a(n) = ( (6 + 2*sqrt(2))^n - (6 - 2*sqrt(2))^n )/(4*sqrt(2)).
G.f.: x/(1 - 12*x + 28*x^2). - Klaus Brockhaus, Jan 12 2009, corrected Oct 08 2009
E.g.f.: (1/(2*sqrt(2)))*exp(6*x)*sinh(2*sqrt(2)*x). - G. C. Greubel, Sep 13 2016
a(n) =2^(n-1)*A081179(n). - R. J. Mathar, Feb 04 2021

Extensions

Extended beyond a(7) by Klaus Brockhaus, Jan 12 2009
Edited by Klaus Brockhaus, Oct 08 2009
Offset corrected. - R. J. Mathar, Jun 19 2021

A238160 A skewed version of triangular array A029653.

Original entry on oeis.org

1, 0, 2, 0, 1, 2, 0, 0, 3, 2, 0, 0, 1, 5, 2, 0, 0, 0, 4, 7, 2, 0, 0, 0, 1, 9, 9, 2, 0, 0, 0, 0, 5, 16, 11, 2, 0, 0, 0, 0, 1, 14, 25, 13, 2, 0, 0, 0, 0, 0, 6, 30, 36, 15, 2, 0, 0, 0, 0, 0, 1, 20, 55, 49, 17, 2, 0, 0, 0, 0, 0, 0, 7, 50, 91, 64, 19, 2, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Philippe Deléham, Feb 18 2014

Keywords

Comments

Triangle T(n,k), 0<=k<=n, read by rows, given by (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Row sums are Fib(n+2).
Column sums are A003945(k).
Diagonal sums are (-1)^(n+1)*A109266(n+1).
T(3*n,2*n) = A029651(n).

Examples

			Triangle begins:
1;
0, 2;
0, 1, 2;
0, 0, 3, 2;
0, 0, 1, 5, 2;
0, 0, 0, 4, 7, 2;
0, 0, 0, 1, 9, 9, 2;
0, 0, 0, 0, 5, 16, 11, 2;
0, 0, 0, 0, 1, 14, 25, 13, 2;
0, 0, 0, 0, 0, 6, 30, 36, 15, 2;
0, 0, 0, 0, 0, 1, 20, 55, 49, 17, 2;
0, 0, 0, 0, 0, 0, 7, 50, 91, 64, 19, 2;
...

Crossrefs

Cf. A029635, A029653, A178524.

Formula

G.f.: (1+x*y)/(1-x*y-x^2*y).
T(n,k) = T(n-1,k-1) + T(n-2,k-1), T(0,0) = 1, T(1,0) = 0, T(1,1) = 2, T(n,k) = 0 if k<0 or if k>n.
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A000045(n+2), A026150(n+1), A108306(n), A164545(n), A188168(n+1) for x = 0, 1, 2, 3, 4, 5 respectively.
Showing 1-3 of 3 results.

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

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