A216182 -id:A216182 - cp's OEIS frontend

A073717 a(n) = T(2n+1), where T(n) are the tribonacci numbers A000073.

0, 1, 4, 13, 44, 149, 504, 1705, 5768, 19513, 66012, 223317, 755476, 2555757, 8646064, 29249425, 98950096, 334745777, 1132436852, 3831006429, 12960201916, 43844049029, 148323355432, 501774317241, 1697490356184, 5742568741225

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Aug 05 2002

In general, the bisection of a third-order linear recurrence with signature (x,y,z) will result in a third-order recurrence with signature (x^2 + 2*y, 2*z*x - y^2, z^2). - Gary Detlefs, May 29 2024

G. C. Greubel, Table of n, a(n) for n = 0..1000
Meng-Han Wu, Henryk A. Witek, Rafał Podeszwa, Clar Covers and Zhang-Zhang Polynomials of Zigzag and Armchair Carbon Nanotubes, MATCH Commun. Math. Comput. Chem. (2025) Vol. 93, 415-462. See p. 437.
Index entries for linear recurrences with constant coefficients, signature (3,1,1).

Cf. A000073, A099463, A113300.

Row sums of A216182.

[n le 3 select (n-1)^2 else 3*Self(n-1) +Self(n-2) +Self(n-3): n in [1..31]]; // G. C. Greubel, Nov 19 2021

CoefficientList[Series[(x+x^2)/(1-3x-x^2-x^3), {x, 0, 30}], x]
LinearRecurrence[{3,1,1},{0,1,4},30] (* Harvey P. Dale, Sep 07 2015 *)

def A073717_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1+x)/(1-3*x-x^2-x^3) ).list()
A073717_list(30) # G. C. Greubel, Nov 19 2021

a(n) = 3*a(n-1) + a(n-2) + a(n-3), a(0)=0, a(1)=1, a(2)=4.
G.f.: x*(1+x)/(1-3*x-x^2-x^3).
a(n+1) = Sum_{k=0..n} A216182(n,k). - Philippe Deléham, Mar 11 2013
a(n) = A113300(n-1) + A113300(n). - R. J. Mathar, Jul 04 2019

A112743 An aerated Delannoy triangle.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 1, 0, 5, 0, 1, 0, 5, 0, 7, 0, 1, 1, 0, 13, 0, 9, 0, 1, 0, 7, 0, 25, 0, 11, 0, 1, 1, 0, 25, 0, 41, 0, 13, 0, 1, 0, 9, 0, 63, 0, 61, 0, 15, 0, 1, 1, 0, 41, 0, 129, 0, 85, 0, 17, 0, 1, 0, 11, 0, 129, 0, 231, 0, 113, 0, 19, 0, 1, 1, 0, 61, 0, 321, 0, 377, 0, 145, 0, 21, 0, 1

Offset: 0

Paul Barry, Sep 17 2005

Diagonal sums are aerated Pell numbers.

			Rows begin
  1;
  0,  1;
  1,  0,  1;
  0,  3,  0,  1;
  1,  0,  5,  0,  1;
  0,  5,  0,  7,  0,  1;
  1,  0, 13,  0,  9,  0,  1;
  0,  7,  0, 25,  0, 11,  0,  1;
  1,  0, 25,  0, 41,  0, 13,  0,  1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000073, A008288, A114123, A216182.

function T(n,k)
  if k lt 0 or k gt n then return 0;
  elif k eq n then return 1;
  elif k eq 0 then return (1+(-1)^n)/2;
  else return T(n-1,k-1) + T(n-2,k) + T(n-3,k-1);
  end if;
  return T;
end function;
[T(n,k): k in [0..n], n in [0..14]]; // G. C. Greubel, Nov 20 2021

A008288[n_, k_]:= Hypergeometric2F1[-n, -k, 1, 2];
T[n_, k_]:= T[n, k]= (1+(-1)^(n-k))*A008288[(n-k)/2, k]/2;
Table[T[n, k], {n,0,14}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 20 2021 *)

def A008288(n, k): return simplify( hypergeometric([-n, -k], [1], 2) )
def A112743(n, k): return (1 + (-1)^(n-k))*A008288((n-k)/2, k)/2
flatten([[A112743(n,k) for k in (0..n)] for n in (0..14)]) # G. C. Greubel, Nov 20 2021

Riordan array (1/(1-x^2), x*(1+x^2)/(1-x^2)).
T(n,k) = Sum_{j=0..k} (1+(-1)^(n-k))*binomial(k,j)*binomial((n-k)/2,j)*2^(j-1).
Sum_{k=0..n} T(n, k) = A000073(n).
T(n,k) = T(n-1,k-1) + T(n-2,k) + T(n-3,k-1). - Philippe Deléham, Mar 11 2013

cp's OEIS Frontend

A073717 a(n) = T(2n+1), where T(n) are the tribonacci numbers A000073.

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