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A140164 Binomial transform of [1, 1, 1, 1, -1, -1, 5, -11, 19, -29, 41, ...]. 2
1, 2, 4, 8, 14, 20, 26, 32, 38, 44, 50, 56, 62, 68, 74, 80, 86, 92, 98, 104, 110, 116, 122, 128, 134, 140, 146, 152, 158, 164, 170, 176, 182, 188, 194, 200, 206, 212, 218, 224, 230, 236, 242, 248, 254, 260, 266, 272, 278, 284, 290, 296, 302, 308, 314, 320, 326 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Sum of antidiagonal terms of the following arithmetic array:

  1,  1,  1,  1,  1,  1,  1,  1, ...

  1,  2,  3,  4,  5,  6,  7,  8, ...

  1,  3,  5,  7,  9, 11, 13, 15, ...

  1,  4,  7, 10, 13, 16, 19, 22, ...

  1,  5,  9, 13, 17, 21, 25, 29, ...

  1,  6, 11, 16, 21, 26, 31, 36, ...

  1,  7, 13, 19, 25, 31, 37, 43, ...

  1,  8, 15, 22, 29, 36, 43, 50, ...

  ...

For [1, 1, 1, 1, -1, -1, 5, -11, 19, -29, 41, -55, ...], see A??????.

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (2,-1).

FORMULA

Binomial transform of [1, 1, 1, 1, -1, -1, 5, -11, 19, -29, 41, -55,...]; where A028387 = (1, 5, 11, 19, 29, 41,...), such that A028387(n) = (2*T(n) - 1).

From R. J. Mathar, May 03 2010: (Start)

G.f.: x*(1+x^2+2*x^3+2*x^4)/(1-x)^2. [G.f. amended by Georg Fischer, May 12 2019]

a(n) = A016933(n-2), n>2. (End)

a(n) = 2*(3n-5), n >= 3, if offset is 0 instead of 1. - Daniel Forgues, Jul 17 2016

EXAMPLE

a(4) = 8 = (1, 3, 3, 1) dot (1, 1, 1, 1) = (1 + 3 + 3 + 1).

a(5) = 14 = (4 + 5 + 4 + 1).

MAPLE

From R. J. Mathar, May 03 2010: (Start)

A028387 := proc(n) option remember; if n <= 2 then op(n+1, [1, 5, 11]) ; else 3*procname(n-1)-3*procname(n-2)+procname(n-3) ; end if; end proc:

read("transforms") ; L := [1, 1, 1, 1, -1, seq((-1)^(n+1)*A028387(n), n=0..60)]; BINOMIAL(L) ; (End)

MATHEMATICA

Table[If[n < 4, 2^(n - 1), 6 n - 16], {n, 60}] (* or *)

Rest@CoefficientList[Series[x*(1+x^2+2x^3+2x^4)/(1-x)^2, {x, 0, 60}], x] (* Michael De Vlieger, Jul 18 2016 *)

PROG

(PARI) a(n)=if(n<4, 2^(n-1), 6*n-16) \\ Charles R Greathouse IV, Jul 17 2016

(MAGMA) R<x>:=PowerSeriesRing(Integers(), 60); Coefficients(R!( x*(1+x^2+2*x^3+2*x^4)/(1-x)^2 )); // G. C. Greubel, May 12 2019

(Sage) (x*(1+x^2+2*x^3+2*x^4)/(1-x)^2).series(x, 60).coefficients(x, sparse=False) # G. C. Greubel, May 12 2019

(GAP) Concatenation([1, 2, 4], List([4..60], n-> 6*n-16)) # G. C. Greubel, May 12 2019

CROSSREFS

Cf. A028387.

Sequence in context: A294065 A248845 A169926 * A160730 A190402 A175300

Adjacent sequences:  A140161 A140162 A140163 * A140165 A140166 A140167

KEYWORD

nonn,easy

AUTHOR

Gary W. Adamson, May 10 2008

EXTENSIONS

More terms from R. J. Mathar, May 03 2010

STATUS

approved

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Last modified December 7 13:08 EST 2021. Contains 349581 sequences. (Running on oeis4.)