A254667 The nonnegative numbers with 2 instead of 1.

0, 2, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62

Offset: 0

Author

Paul Curtz, Feb 04 2015

Keywords

• nonn
• easy

Comments

An autosequence of the first kind is a sequence whose main diagonal in the difference table is A000004 = 0's.

This is the case for a(n).

Difference table of a(n):

0, 2, 2, 3, 4, 5, ...

2, 0, 1, 1, 1, 1, ...

-2, 1, 0, 0, 0, 0, ...

3, -1, 0, 0, 0, 0, ...

-4, 1, 0, 0, 0, 0, ...

5, -1, 0, 0, 0, 0, ...

etc.

The inverse binomial transform of a(n) is (-1)^(n+1)*a(n).

0 followed by A000012(n) is not in the OEIS. See A054977.

What is the meaning of a(n)?

Among many others, A015441 is an autosequence of the first kind.

General form for such autosequence.

Starting from the first upper diagonal s0, s1, s2, s3, s4, ...,

the autosequence is

0, s0, s0, s0 + s1, s0 + 2*s1, s0 + 3*s1 + s2, s0 + 4*s1 + 3*s2, ... .

After 0, the corresponding coefficients are A011973(n).

Examples

			G.f. = 2*x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 6*x^6 + 7*x^7 + 8*x^8 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2500
• Index entries for linear recurrences with constant coefficients, signature (2,-1).

Crossrefs

Cf. A000004, A001477, A011973, A015441, A027641, A027642, A054977, A063524, A164555, A164558, A233583.

Programs

Magma

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((2*x-2*x^2+x^3)/(1-x)^2)); // G. C. Greubel, Aug 03 2018

Mathematica

CoefficientList[Series[(2*x-2*x^2+x^3)/(1-x)^2, {x, 0, 60}], x] (* G. C. Greubel, Aug 03 2018 *)
a[ n_] := n + Boole[n == 1]; (* Michael Somos, Aug 19 2018 *)
Join[{0,2},Range[2,70]] (* Harvey P. Dale, Oct 10 2024 *)

PARI

{a(n) = n + (n==1)}; /* Michael Somos, Feb 09 2015 */

Formula

a(n) = (A164558(n) + (-1)^(n+1)*A164555(n))/A027642(n).

a(n) = A063524(n) + A001477(n). - David A. Corneth, Aug 03 2018

G.f.: (2*x - 2*x^2 + x^3) / (1 - x)^2. - Michael Somos, Feb 09 2015