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.

A140944 Triangle T(n,k) read by rows, the k-th term of the n-th differences of the Jacobsthal sequence A001045.

Original entry on oeis.org

0, 1, 0, -1, 2, 0, 3, -2, 4, 0, -5, 6, -4, 8, 0, 11, -10, 12, -8, 16, 0, -21, 22, -20, 24, -16, 32, 0, 43, -42, 44, -40, 48, -32, 64, 0, -85, 86, -84, 88, -80, 96, -64, 128, 0, 171, -170, 172, -168, 176, -160, 192, -128, 256, 0, -341, 342, -340, 344, -336, 352, -320, 384, -256, 512, 0
Offset: 0

Views

Author

Paul Curtz, Jul 24 2008

Keywords

Comments

A variant of the triangle A140503, now including the diagonal.
Since the diagonal contains zeros, rows sums are those of A140503.

Examples

			Triangle begins as:
    0;
    1,   0;
   -1,   2,   0;
    3,  -2,   4,  0;
   -5,   6,  -4,  8,   0;
   11, -10,  12, -8,  16,  0;
  -21,  22, -20, 24, -16, 32,  0;

Links

Crossrefs

Cf. A000079, A001045, A016131, A045883, A052992, A083086, A084175.
Cf. A091056, A097038, A109765, A115489, A140503, A192382, A246036.

Programs

  • Magma
    [2^k*(1-(-2)^(n-k))/3: k in [0..n], n in [0..15]]; // G. C. Greubel, Feb 18 2023
  • Maple
    A001045:= n -> (2^n-(-1)^n)/3;
A140944:= proc(n,k) if n = 0 then A001045(k); else procname(n-1,k+1)-procname(n-1,k) ; fi; end:
seq(seq(A140944(n,k),k=0..n),n=0..10); # R. J. Mathar, Sep 07 2009
  • Mathematica
    T[0, 0]=0; T[1, 0]= T[0, 1]= 1; T[0, k_]:= T[0, k]= T[0, k-1] + 2*T[0, k-2]; T[n_, n_]=0; T[n_, k_]:= T[n, k] = T[n-1, k+1] - T[n-1, k]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* Jean-François Alcover, Dec 17 2014 *)
Table[2^k*(1-(-2)^(n-k))/3, {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 18 2023 *)
  • PARI
    T(n, k) = (2^k - 2^n*(-1)^(n+k))/3 \\ Jianing Song, Aug 11 2022
  • SageMath
    def A140944(n,k): return 2^k*(1 - (-2)^(n-k))/3
flatten([[A140944(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Feb 18 2023

Formula

T(n, k) = T(n-1, k+1) - T(n-1, k). T(0, k) = A001045(k).
T(n, k) = (2^k - 2^n*(-1)^(n+k))/3, for n >= k >= 0. - Jianing Song, Aug 11 2022
From G. C. Greubel, Feb 18 2023: (Start)
T(n, n-1) = A000079(n).
T(2*n, n) = (-1)^(n+1)*A192382(n+1).
T(2*n, n-1) = (-1)^n*A246036(n-1).
T(2*n, n+1) = A083086(n).
T(3*n, n) = -A115489(n).
Sum_{k=0..n} T(n, k) = A052992(n)*[n>0] + 0*[n=0].
Sum_{k=0..n} (-1)^k*T(n, k) = A045883(n).
Sum_{k=0..n} 2^k*T(n, k) = A084175(n).
Sum_{k=0..n} (-2)^k*T(n, k) = (-1)^(n+1)*A109765(n).
Sum_{k=0..n} 3^k*T(n, k) = A091056(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = (-1)^(n+1)*A097038(n).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (-1)^(n+1)*A138495(n). (End)

Extensions

Edited and extended by R. J. Mathar, Sep 07 2009

A099579 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k-1) * 3^(k-1).

Original entry on oeis.org

0, 0, 1, 1, 7, 10, 40, 70, 217, 427, 1159, 2440, 6160, 13480, 32689, 73129, 173383, 392770, 919480, 2097790, 4875913, 11169283, 25856071, 59363920, 137109280, 315201040, 727060321, 1672663441, 3855438727, 8873429050, 20444528200
Offset: 0

Views

Author

Paul Barry, Oct 23 2004

Keywords

Comments

In general a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k-1)*r^(k-1) has g.f. x^2/((1-r*x^2)*(1-x-r*x^2)) and satisfies the recurrence a(n) = a(n-1) + 2*r*a(n-2) - r*a(n-3) - r^2*a(n-4).

Links

Crossrefs

Cf. A097038, A099580, A140167.

Programs

  • Magma
    [n le 4 select Floor((n-1)/2) else Self(n-1) +6*Self(n-2) -3*Self(n-3) -9*Self(n-4): n in [1..41]]; // G. C. Greubel, Jul 24 2022
  • Mathematica
    LinearRecurrence[{1,6,-3,-9}, {0,0,1,1}, 50] (* G. C. Greubel, Jul 24 2022 *)
  • SageMath
    @CachedFunction
def a(n): # a = A099579
    if (n<4): return (n//2)
    else: return a(n-1) +6*a(n-2) -3*a(n-3) -9*a(n-4)
[a(n) for n in (0..40)] # G. C. Greubel, Jul 24 2022

Formula

G.f.: x^2/((1-3*x^2)*(1-x-3*x^2)).
a(n) = a(n-1) + 6*a(n-2) - 3*a(n-3) - 9*a(n-4).
From G. C. Greubel, Jul 24 2022: (Start)
a(n) = (i*sqrt(3))^(n-1)*ChebyshevU(n-1, -i/(2*sqrt(3))) - 3^((n-1)/2)*(1 - (-1)^n)/2.
E.g.f.: (1/sqrt(39))*( 2*sqrt(3)*exp(x/2)*sinh(sqrt(13)*x/2) - sqrt(13)*sinh(sqrt(3)*x) ). (End)

A099580 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k-1) * 4^(k-1).

Original entry on oeis.org

0, 0, 1, 1, 9, 13, 65, 117, 441, 909, 2929, 6565, 19305, 45565, 126881, 309141, 833049, 2069613, 5467345, 13745797, 35877321, 90860509, 235418369, 598860405, 1544728185, 3940169805, 10135859761, 25896538981, 66507086889, 170093242813
Offset: 0

Views

Author

Paul Barry, Oct 23 2004

Keywords

Comments

In general a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k-1) * r^(k-1) has g.f. x^2/((1-r*x^2)*(1-x-r*x^2)) and satisfies the recurrence a(n) = a(n-1) + 2*r*a(n-2) - r*a(n-3) - r^2*a(n-4).

Links

Crossrefs

Cf. A006131, A097038, A099579.

Programs

  • Magma
    [n le 4 select Floor((n-1)/2) else Self(n-1) +8*Self(n-2) -4*Self(n-3) -16*Self(n-4): n in [1..41]]; // G. C. Greubel, Jul 24 2022
  • Mathematica
    LinearRecurrence[{1,8,-4,-16}, {0,0,1,1}, 51] (* G. C. Greubel, Jul 24 2022 *)
  • SageMath
    @CachedFunction
def a(n): # a = A099580
    if (n<4): return (n//2)
    else: return a(n-1) +8*a(n-2) -4*a(n-3) -16*a(n-4)
[a(n) for n in (0..40)] # G. C. Greubel, Jul 24 2022

Formula

G.f.: x^2/((1-4*x^2) * (1-x-4*x^2)).
a(n) = a(n-1) + 8*a(n-2) - 4*a(n-3) - 16*a(n-4).
From G. C. Greubel, Jul 24 2022: (Start)
a(n) = (4*(2/i)^(n-1)*ChebyshevU(n-1, i/4) - 2^n*(1-(-1)^n))/4.
E.g.f.: ( 4*exp(x/2)*sinh(sqrt(17)*x/2) - sqrt(17)*sinh(2*x) )/(2*sqrt(17)). (End)
Showing 1-3 of 3 results.

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