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A131708 A024494 prefixed by a 0. 18
0, 1, 2, 3, 5, 10, 21, 43, 86, 171, 341, 682, 1365, 2731, 5462, 10923, 21845, 43690, 87381, 174763, 349526, 699051, 1398101, 2796202, 5592405, 11184811, 22369622, 44739243, 89478485, 178956970, 357913941, 715827883, 1431655766, 2863311531, 5726623061, 11453246122 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Sequence is identical to its 3rd differences. a(n)=3a(n-1)-3a(n-2)+2a(n-2), n=3,4.. Also binomial transform of 0, 1, 0. Also A024495 = first differences. Recurrence: a(n+1)-2a(n)= 1, 0, -1, -1, 0, 1, 1 .

{A024493, A131708, A024495} is the difference analog of the hyperbolic functions {h_1(x), h_2(x), h_3(x)} of order 3. For the definitions of {h_i(x)} and the difference analog {H_i(n)} see [Erdelyi] and the Shevelev link respectively. - Vladimir Shevelev, Aug 01 2017

REFERENCES

A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, Chapter XVIII.

LINKS

Table of n, a(n) for n=0..35.

Vladimir Shevelev, Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n, arXiv:1706.01454 [math.CO], 2017.

Index entries for linear recurrences with constant coefficients, signature (3,-3,2).

FORMULA

G.f.: x*(-1+x)/(2*x-1)/(x^2-x+1). - R. J. Mathar, Nov 14 2007

Recurrences: a(n) = k*a(n - 1) + (6 - 3k)*a(n - 2) + (3k - 7)*a(n - 3) + (6 - 2k)*a(n - 4);k = 0: a(n) = 6a(n - 2) - 7a(n - 3) + 6a(n - 4), k = 1: a(n) = a(n - 1) + 3a(n - 2) - 4a(n - 3) + 4a(n - 4), k = 2: a(n) = 2a(n - 1) - a(n - 3) + 2a(n - 4), cf. A113405, A135350, k = 3: a(n) = 3a(n - 1) - 3a(n - 2) + 2a(n - 3), here and many other sequences, k = 4: a(n) = 4a(n - 1) - 6a(n - 2) + 5a(n - 3) - 2a(n - 4), k = 5: a(n) = 5a(n - 1) - 9a(n - 2) + 8a(n - 3) - 4a(n - 4). For k sum of coefficients = 5 - k. Of the family k=3 gives the best recurrence.

a(n+m) = a(n)*A024493(m) + A024493(n)*a(m) + A024495(n)*A024495(m). - Vladimir Shevelev, Aug 01 2017

MATHEMATICA

LinearRecurrence[{3, -3, 2}, {0, 1, 2}, 40] (* Harvey P. Dale, Nov 27 2013 *)

PROG

(PARI) v=vector(99, i, i); for(i=4, #v, v[i]=3*v[i-1]-3*v[i-2]+2*v[i-3]); v \\ Charles R Greathouse IV, Jun 01 2011

CROSSREFS

Sequence in context: A014626 A132418 A024494 * A002991 A218532 A022861

Adjacent sequences:  A131705 A131706 A131707 * A131709 A131710 A131711

KEYWORD

nonn,easy

AUTHOR

Paul Curtz, Sep 14 2007, Mar 01 2008

STATUS

approved

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Last modified August 21 00:36 EDT 2017. Contains 290855 sequences.