A129687 -id:A129687 - cp's OEIS frontend

A129686 Triangle read by rows: row n is 0^(n-3), 1, 0, 1.

1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1

Offset: 1

Gary W. Adamson, Apr 28 2007

Alternate term operator, sums.

Let A129686 = matrix M, with V any sequence as a vector. Then M*V is the alternate term sum operator. Given V = [1,2,3,...], M*V = [1, 2, 4, 6, 8, 10, 12, 14, ...]. The analogous operation using A097807, (the pairwise operator), gives [1, 3, 5, 7, 9, 11, 13, 15, ...]. Binomial transform of A129686 = A124725. A129686 * A007318 = A129687. Row sums of A129686 = (1, 1, 2, 2, 2, ...).

			First few rows of the triangle:
  1;
  0, 1;
  1, 0, 1;
  0, 1, 0, 1
  0, 0, 1, 0, 1;
  0, 0, 0, 1, 0, 1;
  ...

Cf. A124725, A129687.

T[n_, k_] := If[k == n || k == n-2, 1, 0];
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 10 2019 *)

tabl(nn) = {t129686 = matrix(nn, nn, n, k, (k<=n)*((k==n) || (k==(n-2)))); for (n = 1, nn, for (k = 1, n, print1(t129686[n, k], ", ");););} \\ Michel Marcus, Feb 12 2014

As an infinite lower triangular matrix, (1,1,1,...) in the main diagonal, (0,0,0,...) in the subdiagonal and (1,1,1,...) in the subsubdiagonal; with the rest zeros. (1, 0, 1, 0, 0, 0, ...) in every column.

More terms from Michel Marcus, Feb 12 2014

A097809 a(n) = 52^n - 2n - 4.

Original entry on oeis.org

1, 4, 12, 30, 68, 146, 304, 622, 1260, 2538, 5096, 10214, 20452, 40930, 81888, 163806, 327644, 655322, 1310680, 2621398, 5242836, 10485714, 20971472, 41942990, 83886028, 167772106, 335544264, 671088582, 1342177220, 2684354498

Offset: 0

Paul Barry, Aug 25 2004

Rows sums of the infinite triangle defined by T(n,n) = 1, T(n,0) = n*(n+1) + 1 for n=0, 1, 2, ... and interior terms defined by the Pascal-type recurrence T(n,k) = T(n-1,k-1) +T(n-1,k): Sum_{k=0..n} T(n,k) = a(n). T is apparently obtained by deleting the first two columns of A129687. - J. M. Bergot, Feb 23 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Tamas Lengyel, On p-adic properties of the Stirling numbers of the first kind, Journal of Number Theory, 148 (2015) 73-94.
Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A079583, A097810.

[5*2^n-2*n-4: n in [0..30]]; // Vincenzo Librandi, Feb 24 2013

LinearRecurrence[{4,-5,2},{1,4,12},30] (* Harvey P. Dale, Oct 11 2018 *)

[5*2^n -2*(n+2) for n in (0..30)] # G. C. Greubel, Dec 30 2021

G.f.: (1+x^2)/((1-x)^2*(1-2*x)).
a(n) = 2*a(n-1) + 2*n, n>0.
a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3), with a(0)=1, a(1)=4, a(2)=12.
E.g.f.: 5*exp(2*x) - 2*(2+x)*exp(x). - G. C. Greubel, Dec 30 2021

cp's OEIS Frontend

A129686 Triangle read by rows: row n is 0^(n-3), 1, 0, 1.

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A097809 a(n) = 52^n - 2n - 4.

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cp's OEIS Frontend

A129686 Triangle read by rows: row n is 0^(n-3), 1, 0, 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A097809 a(n) = 5*2^n - 2*n - 4.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

A097809 a(n) = 52^n - 2n - 4.