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.

Previous Showing 31-36 of 36 results.

A123971 Triangle T(n,k), read by rows, defined by T(n,k)=3*T(n-1,k)-T(n-1,k-1)-T(n-2,k), T(0,0)=1, T(1,0)=2, T(1,1)=-1, T(n,k)=0 if k<0 or if k>n.

Original entry on oeis.org

1, 2, -1, 5, -5, 1, 13, -19, 8, -1, 34, -65, 42, -11, 1, 89, -210, 183, -74, 14, -1, 233, -654, 717, -394, 115, -17, 1, 610, -1985, 2622, -1825, 725, -165, 20, -1, 1597, -5911, 9134, -7703, 3885, -1203, 224, -23, 1, 4181, -17345, 30691, -30418, 18633, -7329
Offset: 0

Views

Author

Gary W. Adamson and Roger L. Bagula, Oct 30 2006

Keywords

Comments

This entry is the result of merging two sequences, this one and a later submission by Philippe Deléham, Nov 29 2013 (with edits from Ralf Stephan, Dec 12 2013). Most of the present version is the work of Philippe Deléham, the only things remaining from the original entry are the sequence data and the Mathematica program. - N. J. A. Sloane, May 31 2014
Subtriangle of the triangle given by (0, 2, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Apart from signs, equals A126124.
Row sums = 1.
Sum_{k=0..n} T(n,k)*(-x)^k = A001519(n+1), A079935(n+1), A004253(n+1), A001653(n+1), A049685(n), A070997(n), A070998(n), A072256(n+1), A078922(n+1), A077417(n), A085260(n+1), A001570(n+1) for x=0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 respectively.

Examples

			Triangle begins:
  1
  2, -1
  5, -5, 1
  13, -19, 8, -1
  34, -65, 42, -11, 1
  89, -210, 183, -74, 14, -1
  233, -654, 717, -394, 115, -17, 1
Triangle (0, 2, 1/2, 1/2, 0, 0, ...) DELTA (1, -2, 0, 0, ...) begins:
  1
  0, 1
  0, 2, -1
  0, 5, -5, 1
  0, 13, -19, 8, -1
  0, 34, -65, 42, -11, 1
  0, 89, -210, 183, -74, 14, -1
  0, 233, -654, 717, -394, 115, -17, 1

Crossrefs

Cf. A094954, A098495, A123971, A126124, A152063, A001519, A079935, A004253, A001653, A049685, A070997, A070998, A072256, A078922, A077417, A085260, A001570, A001870, A126124.

Programs

  • Mathematica
    Mathematica ( general k th center) Clear[M, T, d, a, x, k] k = 3 T[n_, m_, d_] := If[ n == m && n < d && m < d, k, If[n == m - 1 || n == m + 1, -1, If[n == m == d, k - 1, 0]]] M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}] Table[M[d], {d, 1, 10}] Table[Det[M[d]], {d, 1, 10}] Table[Det[M[d] - x*IdentityMatrix[d]], {d, 1, 10}] a = Join[{M[1]}, Table[CoefficientList[ Det[M[d] - x*IdentityMatrix[d]], x], {d, 1, 10}]] Flatten[a] MatrixForm[a] Table[NSolve[Det[M[d] - x*IdentityMatrix[d]] == 0, x], {d, 1, 10}] Table[x /. NSolve[Det[M[d] - x*IdentityMatrix[d]] == 0, x][[d]], {d, 1, 10}]
  • PARI
    T(n,k)=polcoeff(polcoeff(Ser((1-x)/(1+(y-3)*x+x^2)),n,x),n-k,y) \\ Ralf Stephan, Dec 12 2013
  • Sage
    @CachedFunction
def A123971(n,k): # With T(0,0) = 1!
    if n< 0: return 0
    if n==0: return 1 if k == 0 else 0
    h = 2*A123971(n-1,k) if n==1 else 3*A123971(n-1,k)
    return A123971(n-1,k-1) - A123971(n-2,k) - h
for n in (0..9): [A123971(n,k) for k in (0..n)] # Peter Luschny, Nov 20 2012

Formula

T(n,k) = (-1)^n*A126124(n+1,k+1).
T(n,k) = (-1)^k*Sum_{m=k..n} binomial(m,k)*binomial(m+n,2*m). - Wadim Zudilin, Jan 11 2012
G.f.: (1-x)/(1+(y-3)*x+x^2).
T(n,0) = A001519(n+1) = A000045(2*n+1).
T(n+1,1) = -A001870(n).

Extensions

Edited by N. J. A. Sloane, May 31 2014

A352181 a(n) = A200993(n)/2.

Original entry on oeis.org

0, 5, 495, 48510, 4753490, 465793515, 45643010985, 4472549283020, 438264186724980, 42945417749765025, 4208212675290247475, 412361896760694487530, 40407257669872769530470, 3959498889750770719498535, 387990483937905657741325965, 38019107927025003687930446040
Offset: 0

Views

Author

N. J. A. Sloane, Mar 08 2022

Keywords

Comments

Halves of triangular numbers which are also thirds of triangular numbers.

Links

Crossrefs

Cf. A200993, A200994, A278620.

Formula

a(n) = A200994(n)/3.
From Chai Wah Wu, Apr 22 2024: (Start)
a(n) = 99*a(n-1) - 99*a(n-2) + a(n-3) for n > 2.
G.f.: -5*x/((x - 1)*(x^2 - 98*x + 1)). (End)
a(n) = 5*A278620(n). - Hugo Pfoertner, Apr 22 2024

A095685 Expansion of (1+x)^4/(1-11*x+11*x^2-x^3).

Original entry on oeis.org

1, 15, 160, 1600, 15856, 156976, 1553920, 15382240, 152268496, 1507302736, 14920758880, 147700286080, 1462082101936, 14473120733296, 143269125231040, 1418218131577120, 14038912190540176, 138970903773824656, 1375670125547706400, 13617730351703239360
Offset: 0

Views

Author

N. J. A. Sloane, Jul 04 2004

Keywords

Links

Crossrefs

Cf. A072256.

Programs

  • Mathematica
    CoefficientList[Series[(1 + x)^4/(1 - 11*x + 11*x^2 - x^3), {x, 0, 20}], x] (* Wesley Ivan Hurt, Dec 27 2023 *)
  • PARI
    Vec((1+x)^4/(1-11*x+11*x^2-x^3) + O(x^25)) \\ Colin Barker, Mar 05 2016

Formula

a(n) = 18*A072256(n)-2, n>1. - R. J. Mathar, Jan 11 2009
a(n) = -2+3*(5-2*sqrt(6))^n*(3+sqrt(6))-3*(-3+sqrt(6))*(5+2*sqrt(6))^n for n>1. - Colin Barker, Mar 05 2016

A352182 Twice A200994.

Original entry on oeis.org

0, 30, 2970, 291060, 28520940, 2794761090, 273858065910, 26835295698120, 2629585120349880, 257672506498590150, 25249276051741484850, 2474171380564166925180, 242443546019236617182820, 23756993338504624316991210, 2327942903627433946447955790, 228114647562150022127582676240
Offset: 0

Views

Author

N. J. A. Sloane, Mar 08 2022

Keywords

Comments

Also 3 times A200993 and 6 times A352181.
Numbers that both doubles and triples of triangular numbers.

Links

Crossrefs

Cf. A200993, A200994, A278620, A352181.

Formula

From Chai Wah Wu, Apr 22 2024: (Start)
a(n) = 99*a(n-1) - 99*a(n-2) + a(n-3) for n > 2.
G.f.: -30*x/((x - 1)*(x^2 - 98*x + 1)). (End)
a(n) = 30*A278620. - Hugo Pfoertner, Apr 22 2024

A269028 a(n) = 40*a(n - 1) - a(n - 2) for n>1, a(0) = 1, a(1) = 1.

Original entry on oeis.org

1, 1, 39, 1559, 62321, 2491281, 99588919, 3981065479, 159143030241, 6361740144161, 254310462736199, 10166056769303799, 406387960309415761, 16245352355607326641, 649407706263983649879, 25960062898203738668519, 1037753108221885563090881
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 18 2016

Keywords

Comments

In general, the ordinary generating function for the recurrence relation b(n) = k*b(n - 1) - b(n - 2) with n>1 and b(0)=1, b(1)=1, is (1 - (k - 1)*x)/(1 - k*x +x^2). This recurrence gives the closed form b(n) = (2^( -n - 1)*((k - 2)*(k - sqrt(k^2 - 4))^n + sqrt(k^2 - 4)*(k - sqrt(k^2 - 4))^n - (k - 2)*(sqrt(k^2 - 4) + k)^n + sqrt(k^2 - 4)*(sqrt(k^2 - 4) + k)^n))/sqrt(k^2 - 4).

Links

Crossrefs

Cf. A001519, A001835, A001653, A049685, A070997, A070998, A072256, A078922, A160682, A007805, A075839, A157014, A159664, A159668, A157877, A238379, A097315.

Programs

  • Magma
    [n le 2 select 1 else 40*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Feb 19 2016
  • Mathematica
    Table[Cosh[n Log[20 + Sqrt[399]]] - Sqrt[19/21] Sinh[n Log[20 + Sqrt[399]]], {n, 0, 17}]
Table[(2^(-n - 2) (38 (40 - 2 Sqrt[399])^n + 2 Sqrt[399] (40 - 2 Sqrt[399])^n - 38 (40 + 2 Sqrt[399])^n + 2 Sqrt[399] (40 + 2 Sqrt[399])^n))/Sqrt[399], {n, 0, 17}]
LinearRecurrence[{40, -1}, {1, 1}, 17]

Formula

G.f.: (1 - 39*x)/(1 - 40*x + x^2).
a(n) = cosh(n*log(20 + sqrt(399))) - sqrt(19/21)*sinh(n*log(20 + sqrt(399))).
a(n) = (2^(-n - 2)*(38*(40 - 2*sqrt(399))^n + 2*sqrt(399)*(40 - 2*sqrt(399))^n - 38*(40 + 2*sqrt(399))^n + 2*sqrt(399)*(40 + 2*sqrt(399))^n))/sqrt(399).
Sum_{n>=0} 1/a(n) = 2.0262989201139499769986...

A290284 Number of pairs of integers (x,y) satisfying the Diophantine equation x^2 - A000037(n)*y^2 = m such that x/y gives a convergent series towards sqrt(A000037(n)).

Original entry on oeis.org

3, 3, 5, 4, 5, 4, 7, 6, 5, 15, 8, 5, 9, 7, 12, 6, 10, 12, 9, 6, 11, 9, 12, 21, 7, 17, 9, 10, 11, 7, 13, 10, 9, 9, 19, 8, 20, 15, 13, 24, 12, 8, 15, 12, 16, 27, 16, 13, 9, 14, 27, 17, 12
Offset: 1

Views

Author

A.H.M. Smeets, Jul 25 2017

Keywords

Comments

If (x(0),y(0)) and (x(1),y(1)) are solutions of the Diophantine equation x^2 - A000037(n)*y^2 = m, then (x(i),y(i)) with x(i) = A*x(i-1) - x(i-2) and y(i) = A*y(i-1) - y(i-2) are also solutions for i > 1. The sequence represents the number of different integer pair sequences where in all cases A = 2*A033313(A000037(n)). Each contributing sequences has to satisfy the condition that for all x < x(i) and all y < y(i), |x/y - sqrt(A000037(n))| > |x(i)/y(i) - sqrt(A000037(n))|.
a(A000037(n)) is not equal to the number of all sequences of pairs (x(i),y(i)) that are solutions of a Diophantine equation x^2 - D*y^2 = m, with -D <= m < D and D = A000037(n). For example for D = 5 we obtain two other sequences from Fibonacci sequence: (first) x(i) = 2*Fib(6i)-Fib(6i-1) and y(i) = Fib(6i-1) satisfy x^2 - D*y^2 = -4 and (second) x(i) = 2*Fib(6i+3) - Fib(6i+2) and y(i) = Fib(6i+2) satisfy x^2 - D*y^2 = 4; but neither of these satisfy the restriction that, for all x < x(i) and all y < y(i), |x/y - sqrt(D)| > |x(i)/y(i) - sqrt(D)|.
A good approximation for the order of magnitude of a(n) is given by 2*log(2*A033313(n)).
For a lower bound, all values m satisfying either m = -D + k^2 for k^2 < D or m = 1, D = A000037(n), contribute with a sequence to the convergent series of sqrt(D), so a(n) > floor(sqrt(D)) + 1.

Examples

			For A000037(4) = 6, a(4) = 4 we have the following sequences of pairs (x,y):
m = 1: x(0) = 1, x(1) = 5, x(i) = 10*x(i-1) - x(i-2) as in A001079(i) and y(0) = 0, y(1) = 2, y(i) = 10*y(i-1) - y(i-2) as in A001078(i);
m = -6: x(0) = 0, x(1) = 12, x(i) = 10*x(i-1) - x(i-2) as in A004291(i) (for i > 0) and y(0) = 1, y(1) = 5, y(i) = 10*y(i-1) - y(i-2) as in A001079(i);
m = -5: x(0) = 1, x(1) = 17, x(i) = 10*x(i-1) - x(i-2) and y(0) = 1, y(1) = 7, y(i) = 10*y(i-1) - y(i-2);
m = -2: x(0) = 2, x(1) = 22, x(i) = 10*x(i-1) - x(i-2) and y(0) = 1, y(1) = 9, y(i) = 10*y(i-1) - y(i-2) as in A072256(i+1).
In some cases a combination of A000037(n) and m has more than one integer pair sequence, for example A000037(5) = 7 and m = -3 has two integer pair sequences:
x(0) = 2, x(1) = 37, x(i) = 16*x(i-1) - x(i-2) and y(0) = 1, y(1) = 14, y(i) = 16*y(i-1) - y(i-2);
x(0) = -2, x(1) = 5, x(i) = 16*x(i-1) - x(i-2) and y(0) = 1, y(1) = 2, y(i) = 16*y(i-1) - y(i-2).
For A000037(4) = 6, the sequence observed from x^2 - 6y^2 = 3 is not in the convergent series of sqrt(6) due to for example x1/y1 = 2643/1079 = sqrt(6) + 5.259842e-7 while the smaller x,y pair, x2/y2 = 2158/881 from x^2 - 6y^2 = -2 is a fraction closer to sqrt(5), 2158/881 = sqrt(6) - 5.259841e-7.

Crossrefs

Cf. A001078, A001079, A004291, A033313, A072256.

Programs

  • Python
    from fractions import Fraction
def FracSqrt(p):
    a = Fraction(p/1)
    b = Fraction(1/1)
    e = Fraction(10**(-200))
    while a-b > e:
        a = (a+b)/2
        b = p/a
    return a
print("number: ")
pp = int(input())
p = FracSqrt(pp)
n = 0
while n >= 0:
    n = n+1
    q = p.limit_denominator(n)
    if (n == 1) or (q != q0):
        t = q*n
        m = t*t-pp*n*n
        print(n,q,m)
    q0 = q
Previous Showing 31-36 of 36 results.

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

Content is available under The OEIS End-User License Agreement.