A118182 Antidiagonal sums of triangle A118180: a(n) = Sum_{k=0..[n/2]} (3^k)^(n-2*k) for n>=0.

1, 1, 2, 4, 11, 37, 164, 1000, 8021, 81001, 1076006, 19683244, 473632031, 14349084877, 571833704648, 31381448626000, 2265367321680041, 205893684435186001, 24615565942378859210, 4052605390737766057684

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..125

Formula

G.f.: A(x) = Sum_{n>=0} x^n/(1-3^n*x^2).

a(2*n) = Sum_{k=0..n} (3^k)^(2*(n-k)).

a(2*n+1) = Sum_{k=0..n} (3^k)^(2*(n-k) +1).

Example

A(x) = 1/(1-x^2) + x/(1-3x^2) + x^2/(1-9x^2) + x^3/(1-27x^2) +...
= 1 + x + 2*x^2 + 4*x^3 + 11*x^4 + 37*x^5 + 164*x^6 + 1000*x^7 +...

Mathematica

Table[Sum[3^(k*(n-2*k)), {k, 0, Floor[n/2]}], {n, 0, 30}] (* G. C. Greubel, Jun 29 2021 *)

Prog

(PARI) a(n)=sum(k=0, n\2, (3^k)^(n-2*k) );
(Magma) [(&+[3^(k*(n-2*k)): k in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, Jun 29 2021
(Sage) [sum(3^(k*(n-2*k)) for k in (0..n//2)) for n in (0..30)] # G. C. Greubel, Jun 29 2021

Crossrefs

Cf. A118180 (triangle), A118181 (row sums), A118183, A118184.

Sequence in context: A367495 A173939 A328433 * A179327 A107107 A243565

Adjacent sequences: A118179 A118180 A118181 * A118183 A118184 A118185

Keyword

nonn

Author

Paul D. Hanna, Apr 15 2006

Status

approved