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.

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A155144 a(n) = 9*a(n-1) + 9*a(n-2), n>2; a(0)=1, a(1)=8, a(2)=80.

Original entry on oeis.org

1, 8, 80, 792, 7848, 77760, 770472, 7634088, 75641040, 749476152, 7426054728, 73579777920, 729052493832, 7223690445768, 71574686456400, 709185392119512, 7026840707183208, 69624234893724480, 689859680408169192
Offset: 0

Views

Author

Philippe Deléham, Jan 21 2009

Keywords

Links

Crossrefs

Sequences of the form a(n) = m*(a(n-1) + a(n-2)) with a(0)=1, a(1) = m-1, a(2) = m^2 -1: A155020 (m=2), A155116 (m=3), A155117 (m=4), A155119 (m=5), A155127 (m=6), A155130 (m=7), A155132 (m=8), this sequence (m=9), A155157 (m=10).

Programs

  • Magma
    [1] cat [n le 2 select 8*(9*n-8) else 9*(Self(n-1) + Self(n-2)): n in [1..30]]; // G. C. Greubel, Mar 24 2021
  • Maple
    m:=30; S:=series( (1-x-x^2)/(1-9*x-9*x^2), x, m+1):
seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Mar 24 2021
  • Mathematica
    Join[{1},LinearRecurrence[{9,9},{8,80},20]] (* or *) CoefficientList[ Series[ (1-x-x^2)/(1-9x-9x^2),{x,0,20}],x] (* Harvey P. Dale, Jun 19 2011 *)
  • Sage
    [1]+[-8*(3*i)^(n-2)*chebyshev_U(n, -3*i/2) for n in (1..30)] # G. C. Greubel, Mar 24 2021

Formula

G.f.: (1-x-x^2)/(1-9*x-9*x^2).
From G. C. Greubel, Mar 24 2021: (Start)
a(n) = (1/9)*[n=0] - 8*3^(n-2)*ChebyshevU(n, -3*i/2).
a(n) = (1/9)*[n=0] + 8*3^(n-2)*Fibonacci(n+1, 3). (End)

A155157 a(n) = 10*a(n-1) + 10*a(n-2), with a(0)=1, a(1)=9, a(2)=99.

Original entry on oeis.org

1, 9, 99, 1080, 11790, 128700, 1404900, 15336000, 167409000, 1827450000, 19948590000, 217760400000, 2377089900000, 25948503000000, 283255929000000, 3092044320000000, 33753002490000000, 368450468100000000
Offset: 0

Views

Author

Philippe Deléham, Jan 21 2009

Keywords

Links

Crossrefs

Sequences of the form a(n) = m*(a(n-1) + a(n-2)) with a(0)=1, a(1) = m-1, a(2) = m^2 -1: A155020 (m=2), A155116 (m=3), A155117 (m=4), A155119 (m=5), A155127 (m=6), A155130 (m=7), A155132 (m=8), A155144 (m=9), this sequence (m=10).
Cf. A057093.

Programs

  • Magma
    [1]cat[n le 2 select 9*(10*n-9) else 10*(Self(n-1) + Self(n-2)): n in [1..30]]; // G. C. Greubel, Mar 20 2021
  • Maple
    1,seq( simplify(9*(-I*sqrt(10))^n*ChebyshevU(n, I*sqrt(10)/2)/10), n=1..30); # G. C. Greubel, Mar 20 2021
  • Mathematica
    LinearRecurrence[{10,10},{1,9,99},20] (* Harvey P. Dale, Jan 27 2016 *)
  • Sage
    [1]+[(9/10)*(-i*sqrt(10))^n*chebyshev_U(n, i*sqrt(10)/2) for n in (1..30)] # G. C. Greubel, Mar 20 2021

Formula

G.f.: (1-x-x^2)/(1-10*x-10*x^2).
From G. C. Greubel, Mar 20 2021: (Start)
a(n) = ([n=0] + 9*A057093(n))/10.
a(n) = (1/10)*([n=0] + 9*(-i*sqrt(10))^n*ChebyshevU(n, i*sqrt(10)/2)). (End)

A155112 Triangle T(n,k), 0<=k<=n, read by rows given by [0,2,-1/2,-1/2,0,0,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 4, 1, 0, 5, 10, 6, 1, 0, 8, 22, 21, 8, 1, 0, 13, 45, 59, 36, 10, 1, 0, 21, 88, 147, 124, 55, 12, 1, 0, 34, 167, 339, 366, 225, 78, 14, 1, 0, 55, 310, 741, 976, 770, 370, 105, 16, 1, 0, 89, 566, 1557, 2422, 2337, 1443, 567, 136, 18, 1, 0, 144, 1020, 3174, 5696, 6505, 4920, 2485, 824, 171, 20, 1
Offset: 0

Views

Author

Philippe Deléham, Jan 20 2009

Keywords

Comments

A Fibonacci convolution triangle; Riordan array (1, x*(1+x)/(1-x-x^2)).

Examples

			Triangle begins:
  1;
  0,  1;
  0,  2,  1;
  0,  3,  4,  1;
  0,  5, 10,  6,  1;
  0,  8, 22, 21,  8,  1;
  0, 13, 45, 59, 36, 10, 1;
  ...

Links

Crossrefs

Cf. A000007, A155020, A155116, A155117, A155119, A155127, A155130, A155132, A155144, A155157.
Cf. A000012, A155020, A154964, A154968, A154996, A154997, A154999, A155000, A155001, A155017.
Cf. A000045, A154929.
T(2n,n) gives A262910.
Cf. A291385.

Programs

  • Magma
    T:= func< n,k | n eq 0 select 1 else (&+[ Binomial(n-j,j)*Binomial(n-j,k)*k/(n-j): j in [0..Floor(n/2)]]) >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 26 2021
  • Maple
    # Uses function PMatrix from A357368.
PMatrix(10, n -> combinat:-fibonacci(n+1)); # Peter Luschny, Oct 19 2022
  • Mathematica
    T[n_, k_]:= If[n==0, 1, Sum[Binomial[n-j, j]*Binomial[n-j, k]*k/(n-j), {j, 0, Floor[n/2]}]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 26 2021 *)
  • Sage
    def T(n,k): return 1 if n==0 else sum( binomial(n-j,j)*binomial(n-j,k)*k/(n-j) for j in (0..n//2) )
flatten([[T(n,k) for k in [0..n]] for n in [0..12]]) # G. C. Greubel, Mar 26 2021

Formula

Recurrence: T(n+2,k+1) = T(n+1,k+1) + T(n+1,k) + T(n,k+1) + T(n,k).
Explicit formula: T(n,k) = Sum_{i=0..floor(n/2)} binomial(n-i, i)*binomial(n-i, k)*k/(n-i), for n > 0.
G.f.: (1-x-x^2)/(1-(1+y)*x-(1+y)*x^2). - Philippe Deléham, Feb 21 2012
Sum_{k=0..n} T(n,k)*x^(n-k) = A000012(n), A155020(n), A154964(n), A154968(n), A154996(n), A154997(n), A154999(n), A155000(n), A155001(n), A155017(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, respectively.
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A155020(n), A155116(n), A155117(n), A155119(n), A155127(n), A155130(n), A155132(n), A155144(n), A155157(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, respectively. - Philippe Deléham, Feb 21 2012
Sum_{k=0..n} T(n, k)*(m-1)^k = (1/m)*[n=0] - (m-1)*(i*sqrt(m))^(n-2)*ChebyshevU(n, -i*sqrt(m)/2). - G. C. Greubel, Mar 26 2021
Sum_{k=0..n} k * T(n,k) = A291385(n-1) for n>=1. - Alois P. Heinz, Sep 29 2022

Extensions

Typos in two terms corrected by Alois P. Heinz, Aug 08 2015

A370173 Riordan array (1-x-x^2, x*(1+x)).

Original entry on oeis.org

1, -1, 1, -1, 0, 1, 0, -2, 1, 1, 0, -1, -2, 2, 1, 0, 0, -3, -1, 3, 1, 0, 0, -1, -5, 1, 4, 1, 0, 0, 0, -4, -6, 4, 5, 1, 0, 0, 0, -1, -9, -5, 8, 6, 1, 0, 0, 0, 0, -5, -15, -1, 13, 7, 1, 0, 0, 0, 0, -1, -14, -20, 7, 19, 8, 1, 0, 0, 0, 0, 0, -6, -29, -21, 20, 26, 9, 1
Offset: 0

Views

Author

Philippe Deléham, Feb 27 2024

Keywords

Comments

Triangle T(n,k) read by rows : matrix product of A155112*A130595.
Triangle T(n,k), read by rows, given by [-1, 2, -1/2, -1/2, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

Examples

			Triangle T(n,k) begins:
  1;
 -1,  1;
 -1,  0,  1;
  0, -2,  1,  1;
  0, -1, -2,  2, 1;
  0,  0, -3, -1, 3, 1;
...

Crossrefs

Cf. A000007, A155020, A155116, A155117, A155119, A155127, A155130, A155132, A155144, A155157.
Cf. A084938, A130595, A155112.

Programs

  • Python
    from functools import cache
@cache
def T(n, k):
    if k > n: return 0
    if n == 0: return 1
    if k == 0: return -1 if n == 1 or n == 2 else 0
    return T(n-1, k-1) + T(n-2, k-1)
for n in range(9):
    print([T(n, k) for k in range(n+1)])  # Peter Luschny, Feb 28 2024

Formula

T(n,k) = T(n-1,k-1) + T(n-2,k-1), T(0,0) = 1, T(1,0) = T(2,0) = -1, T(n,0) = 0 for n>2, T(n,k) = 0 if k>n.
T(n,k) = Sum_{j = k..n} A155112(n,j)*A130595(j,k).
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A155020(n), A155116(n), A155117(n), A155119(n), A155127(n), A155130(n), A155132(n), A155144(n), A155157(n) for x = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 respectively.
Previous Showing 11-14 of 14 results.

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