A158943 INVERT transform of A027656: (1, 0, 2, 0, 3, 0, 4, 0, 5, ...).

1, 1, 3, 5, 10, 19, 36, 69, 131, 250, 476, 907, 1728, 3292, 6272, 11949, 22765, 43371, 82629, 157422, 299915, 571388, 1088589, 2073943, 3951206, 7527704, 14341527, 27322992, 52054840, 99173120, 188941273, 359964521, 685792227, 1306548149

Offset: 1

Author

Gary W. Adamson, Mar 31 2009

Keywords

• nonn

Comments

Equals row sums of triangle A158945.

Number of compositions of n into odd parts where there is 1 sort of part 1, 2 sorts of part 3, 3 sorts of part 5, ..., k sorts of part 2*k-1. - Joerg Arndt, Aug 04 2014

Examples

			The INVERT transform of (1, N, ...) begins (1, (N+1), ...) so that we have (1, 1, ...) placed in ascending magnitude in the bottom row. In the top row we place an equal number of descending terms: (..., 0, 3, 0, 2, 0, 1). Take the dot product of terms in top and bottom rows, adding the result to the next term A027656: (1, 0, 2, 0, 3, ...). a(6) = 19 given: 3, 0, 2, 0, 1 1, 1, 3, 5, 10 Dot product of top row terms * bottom row terms = (1, 0, 2, 0, 3) dot (1, 1, 3, 5, 10) = (3 + 0 + 6 + 0 + 10) = 19, which is added to the next term in (1, 0, 2, 0, 3, ...); i.e., (an 0) = 19.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Jean-Luc Baril, Nathanaël Hassler, Sergey Kirgizov, and José L. Ramírez, Grand zigzag knight's paths, arXiv:2402.04851 [math.CO], 2024. See p. 18.
• Jia Huang, A coin flip game and generalizations of Fibonacci numbers, arXiv:2501.07463 [math.CO], 2025. See p. 9.
• Index entries for linear recurrences with constant coefficients, signature (1,2,0,-1).

Crossrefs

Cf. A027656, A052535, A158944, A158945.

Programs

GAP

a:=[1,1,3,5];; for n in [5..40] do a[n]:=a[n-1]+2*a[n-2]-a[n-4]; od; a; # G. C. Greubel, Jul 12 2019

Magma

I:=[1,1,3,5]; [n le 4 select I[n] else Self(n-1)+2*Self(n-2)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jul 09 2019

Maple

A027656 := proc(n) if type(n,odd) then 0; else n/2+1 ; fi; end: L := [seq(A027656(n), n=0..100)] ; read("transforms"); INVERT(L) ; # R. J. Mathar, Apr 02 2009

Mathematica

LinearRecurrence[{1, 2, 0, -1}, {1, 1, 3, 5}, 40] (* Vincenzo Librandi, Jul 09 2019 *)

PARI

my(x='x+O('x^40)); Vec(x/(1-x-2*x^2+x^4)) \\ G. C. Greubel, Jul 12 2019

Sage

a=(x/(1-x-2*x^2+x^4)).series(x, 40).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Jul 12 2019

Formula

INVERT transform of (1, 0, 2, 0, 3, 0, 4, ...); i.e., the natural numbers interleaved with zeros.

From R. J. Mathar, Apr 02 2009: (Start)

a(n) = a(n-1) + 2*a(n-2) - a(n-4).

G.f.: x/(1 - x - 2*x^2 + x^4). (End)

The sequence is the second INVERT transform of (1, -1, 3, -5, 10, -19, ...). - Gary W. Adamson, Jul 08 2019

a(n) = Sum_{k=0..floor(n/2)} binomial(2*n-3*k-1,k). - Seiichi Manyama, Jun 12 2024

Extensions

Extended by R. J. Mathar, Apr 02 2009