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.

A238379 Expansion of (1 - x)/(1 - 36*x + x^2).

Original entry on oeis.org

1, 35, 1259, 45289, 1629145, 58603931, 2108112371, 75833441425, 2727895778929, 98128414600019, 3529895029821755, 126978092658983161, 4567681440693572041, 164309553772309610315, 5910576254362452399299, 212616435603275976764449
Offset: 0

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Author

Bruno Berselli, Feb 25 2014

Keywords

Comments

First bisection of A041611.

Links

Crossrefs

Cf. similar sequences with g.f. (1-x)/(1-k*x+x^2): A122367 (k=3), A079935 (k=4), A004253 (k=5), A001653 (k=6), A049685 (k=7), A070997 (k=8), A070998 (k=9), A138288 (k=10), A078922 (k=11), A077417 (k=12), A085260 (k=13), A001570 (k=14), A160682 (k=15), A157456 (k=16), A161595 (k=17). From 18 to 38, even k only, except k=27 and k=31: A007805 (k=18), A075839 (k=20), A157014 (k=22), A159664 (k=24), A153111 (k=26), A097835 (k=27), A159668 (k=28), A157877 (k=30), A111216 (k=31), A159674 (k=32), A077420 (k=34), this sequence (k=36), A097315 (k=38).

Programs

  • Magma
    [n le 2 select 35^(n-1) else 36*Self(n-1)-Self(n-2): n in [1..20]];
  • Magma
    R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1 - x)/(1 - 36*x + x^2))); // Marius A. Burtea, Jan 14 2020
  • Mathematica
    CoefficientList[Series[(1 - x)/(1 - 36 x + x^2), {x, 0, 20}], x] (* or *) LinearRecurrence[{36, -1}, {1, 35}, 20]
  • PARI
    a(n)=([0,1; -1,36]^n*[1;35])[1,1] \\ Charles R Greathouse IV, May 10 2016
  • Sage
    m = 20; L. = PowerSeriesRing(ZZ, m); f = (1-x)/(1-36*x+x^2)
print(f.coefficients())

Formula

G.f.: (1 - x)/(1 - 36*x + x^2).
a(n) = a(-n-1) = 36*a(n-1) - a(n-2).
a(n) = ((19-sqrt(323))/38)*(1+(18+sqrt(323))^(2*n+1))/(18+sqrt(323))^n.
a(n+1) - a(n) = 34*A144128(n+1).
323*a(n+1)^2 - ((a(n+2)-a(n))/2)^2 = 34.
Sum_{n>0} 1/(a(n) - 1/a(n)) = 1/34.
See also Tanya Khovanova in Links field:
a(n) = 35*a(n-1) + 34*Sum_{i=0..n-2} a(i).
a(n+2)*a(n) - a(n+1)^2 = 36-2 = 34 = 34*1,
a(n+3)*a(n) - a(n+1)*a(n+2) = 36*(36-2) = 1224 = 34*36.
Generalizing:
a(n+4)*a(n) - a(n+1)*a(n+3) = 44030 = 34*1295,
a(n+5)*a(n) - a(n+1)*a(n+4) = 1583856 = 34*46584,
a(n+6)*a(n) - a(n+1)*a(n+5) = 56974786 = 34*1675729, etc.,
where 1, 36, 1295, 46584, 1675729, ... is the sequence A144128, which is the second bisection of A041611.
a(n)^2 - 36*a(n)*a(n+1) + a(n+1)^2 + 34 = 0 (see comments by Colin Barker in similar sequences).

A159675 Expansion of x*(1 + x)/(1 - 32*x + x^2).

Original entry on oeis.org

1, 33, 1055, 33727, 1078209, 34468961, 1101928543, 35227244415, 1126169892737, 36002209323169, 1150944528448671, 36794222701034303, 1176264181904649025, 37603659598247734497, 1202140842962022854879, 38430903315186483621631, 1228586765243005453037313
Offset: 1

Views

Author

Paul Weisenhorn, Apr 19 2009

Keywords

Comments

Previous name was: The general form of the recurrences are the a(j), b(j) and n(j) solutions of the 2 equations problem: 15*n(j) + 1 = a(j)*a(j) and 17*n(j) + 1 = b(j)*b(j) with positive integer numbers.

Links

Crossrefs

Cf. A157456, A159674, A159677.

Programs

  • Magma
    [n le 2 select (33)^(n-1) else 32*Self(n-1) -Self(n-2): n in [1..30]]; // G. C. Greubel, Sep 25 2022
  • Maple
    for a from 1 by 2 to 100000 do b:=sqrt((17*a*a-2)/15): if (trunc(b)=b) then
n:=(a*a-1)/15: La:=[op(La),a]:Lb:=[op(Lb),b]:Ln:=[op(Ln),n]: endif: enddo:
  • Mathematica
    LinearRecurrence[{32,-1},{1,33},20] (* or *)
CoefficientList[Series[(1+x)/(1-32 x+x^2),{x,0,20}], x] (* Harvey P. Dale, Apr 22 2011 *)
  • PARI
    Vec(x*(1+x)/(1-32*x+x^2) + O(x^20)) \\ Colin Barker, Feb 24 2014
  • PARI
    a(n) = round((16+sqrt(255))^(-n)*(-15-sqrt(255)+(-15+sqrt(255))*(16+sqrt(255))^(2*n))/30) \\ Colin Barker, Jul 25 2016
  • SageMath
    def A159675(n): return chebyshev_U(n-1, 16) + chebyshev_U(n-2, 16)
[A159675(n) for n in range(1,30)] # G. C. Greubel, Sep 25 2022

Formula

The a(j) recurrence is a(1)=1; a(2)=31; a(t+2)=32*a(t+1)-a(t) resulting in terms 1, 31, 991, 31681... (A159674).
The b(j) recurrence is b(1)=1; b(2)=33; b(t+2)=32*b(t+1)-b(t) resulting in terms 1, 33, 1055, 33727... (this sequence).
The n(j) recurrence is n(0)=n(1)=0; n(2)=64; n(t+3)=1023*(n(t+2)-n(t+1))+n(t) resulting in terms 0, 0, 64, 65472, 66912384... (A159677).
G.f.: x*(1 + x)/(1 - 32*x + x^2). - Harvey P. Dale, Apr 22 2011
a(n) = (16+sqrt(255))^(-n)*(-15 - sqrt(255) + (-15 + sqrt(255))*(16 + sqrt(255))^(2*n))/30. - Colin Barker, Jul 25 2016
a(n) = ChebyshevU(n-1, 16) + ChebyshevU(n-2, 16). - G. C. Greubel, Sep 25 2022

Extensions

More terms from Harvey P. Dale, Apr 22 2011
New name from Colin Barker, Feb 24 2014

A159677 Expansion of 64*x^2/(1 - 1023*x + 1023*x^2 - x^3).

Original entry on oeis.org

0, 0, 64, 65472, 66912384, 68384391040, 69888780730560, 71426265522241344, 72997573474949923072, 74603448665133299138304, 76244651538192756769423680, 77921959268584332285051862720, 79636166127841649402566234276224, 81388083860694897105090406378438272
Offset: 0

Views

Author

Paul Weisenhorn, Apr 19 2009

Keywords

Comments

Previous name was: The general form of the recurrences are the a(j), b(j) and n(j) solutions of the 2 equations problem: 15*n(j) + 1 = a(j)*a(j) and 17*n(j) + 1 = b(j)*b(j) with positive integer numbers.

Links

Crossrefs

Cf. A157456, A159674, A159675.

Programs

  • Magma
    I:=[0,0,64]; [n le 3 select I[n] else 1023*Self(n-1) - 1023*Self(n-2) +Self(n-3): n in [1..30]]; // G. C. Greubel, Jun 03 2018
  • Maple
    for a from 1 by 2 to 100000 do b:=sqrt((17*a*a-2)/15): if (trunc(b)=b) then
n:=(a*a-1)/15: La:=[op(La),a]:Lb:=[op(Lb),b]:Ln:=[op(Ln),n]: endif: enddo:
  • Mathematica
    RecurrenceTable[{a[0]==a[1]==0,a[2]==64,a[n]==1023(a[n-1]-a[n-2])+ a[n-3]}, a,{n,20}] (* Harvey P. Dale, Jan 01 2014 *)
LinearRecurrence[{1023,-1023,1},{0,0,64},20] (* Harvey P. Dale, Jan 01 2014 *)
  • PARI
    concat([0, 0], Vec(64/(-x^3+1023*x^2-1023*x+1) + O(x^20))) \\ Colin Barker, Mar 04 2014
  • PARI
    a(n) = round(-((511+32*sqrt(255))^(-n)*(-1+(511+32*sqrt(255))^n)*(16+sqrt(255)+(-16+sqrt(255))*(511+32*sqrt(255))^n))/510) \\ Colin Barker, Jul 25 2016
  • SageMath
    def A159677(n): return (16/255)*(-1 +chebyshev_U(n, 511) -1021*chebyshev_U(n-1, 511))
[A159677(n) for n in range(31)] # G. C. Greubel, Sep 25 2022

Formula

The a(j) recurrence is a(0)=1, a(1)=31, a(t+2) = 32*a(t+1) - a(t) resulting in terms 1, 31, 991, 31681, ... (A159674).
The b(j) recurrence is b(0)=1, b(1)=33, b(t+2) = 32*b(t+1) - b(t) resulting in terms 1, 33, 1055, 33727, ... (A159675).
The n(j) recurrence is n(-1) = n(0) = 0, n(1) = 64, n(t+3) = 1023*(n(t+2) -n(t+1)) + n(t) resulting in terms 0, 0, 64, 65472, 66912384, ... (this sequence).
a(n) = -((511+32*sqrt(255))^(-n)*(-1+(511+32*sqrt(255))^n)*(16+sqrt(255)+(-16+sqrt(255))*(511+32*sqrt(255))^n))/510. - Colin Barker, Jul 25 2016
a(n) = (16/255)*(-1 + ChebyshevU(n, 511) - 1021*ChebyshevU(n-1, 511)). - G. C. Greubel, Sep 25 2022

Extensions

More terms from Harvey P. Dale, Jan 01 2014
New name from Colin Barker, Feb 24 2014
Offset changed to 0 by Colin Barker, Mar 04 2014
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