A155734 Binomial transform of A154879.

3, 1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907, 43046721, 129140163, 387420489, 1162261467, 3486784401, 10460353203, 31381059609, 94143178827, 282429536481, 847288609443, 2541865828329

Offset: 0

Author

Paul Curtz, Jan 26 2009

Keywords

• nonn
• easy
• less

Comments

Binomial transform of the third differences of A001045.

The binomial transform of the first differences of A001045 is in A133494.

The binomial transform of the 2nd differences of A001045 is in A133494, with the sign of A133494(0) flipped.

The binomial transform of the p-th differences of A001045 is the number A077925(p-1) followed by A000244.

Links

• Index entries for linear recurrences with constant coefficients, signature (3).

Crossrefs

Cf. A154879, A078008. Essentially the same as A140429 and A000244.

Programs

Maple

read("transforms") ; A001045 := proc(n) option remember ; if n <= 1 then n; else procname(n-1)+2*procname(n-2) ; fi; end:
a001045 := [seq(A001045(n),n=0..80) ] ; a154879 := DIFF(DIFF(DIFF(a001045))) ; BINOMIAL(a154879) ; # R. J. Mathar, Jul 23 2009

Formula

From Colin Barker, Apr 05 2012: (Start)

a(n) = 3*a(n-1) for n > 1.

G.f.: (3-8*x)/(1-3*x). (End)

G.f.: (1 - 2/G(0))/x where G(k) = 1 + 2^k/(1 - 2*x/(2*x + 2^k/G(k+1))); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 06 2012

Extensions

Edited and extended by R. J. Mathar, Jul 23 2009