A078012 - OEIS

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A078012

Expansion of (1 - x) / (1 - x - x^3) in powers of x.

1, 0, 0, 1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60, 88, 129, 189, 277, 406, 595, 872, 1278, 1873, 2745, 4023, 5896, 8641, 12664, 18560, 27201, 39865, 58425, 85626, 125491, 183916, 269542, 395033, 578949, 848491, 1243524, 1822473, 2670964, 3914488, 5736961, 8407925 (list; graph; refs; listen; history; internal format)

	OFFSET	`0,7`
	COMMENTS	`For n>=1, a(n) = number of compositions of n in which each part is >=3. [From Milan R. Janjic (agnus(AT)blic.net), Jun 28 2010]` `From Adi Dani, May 22 2011: (Start)` `Number of compositions of number n into parts of the form 3*k+1, k>=0.` `For example, a(10)=19 and all compositions of 10 in parts 1,4,7 or 10 are` `(1,1,1,1,1,1,1,1,1,1), (1,1,1,1,1,1,4), (1,1,1,1,1,4,1), (1,1,1,1,4,1,1), (1,1,1,4,1,1,1), (1,1,4,1,1,1,1), (1,4,1,1,1,1,1), (4,1,1,1,1,1,1), (1,1,4,4), (1,4,1,4), (1,4,4,1), (4,1,1,4),(4,1,4,1), (4,4,1,1), (1,1,1,7), (1,1,7,1), (1,7,1,1), (7,1,1,1), (10). (End)`
	REFERENCES	`C. K. Fan, Structure of a Hecke algebra quotient. J. Amer. Math. Soc. 10 (1997), no. 1, 139-167. [Page 156, f_n.]` `Taylor L. Booth, Sequential Machines and Automata Theory, John Wiley and Sons, Inc., 1967, page 331ff.`
	LINKS	`Index to sequences with linear recurrences with constant coefficients, signature (1,0,1).`
	FORMULA	`a(n) = sum(binomial(n-3-2*i, i), i=0..(n-3)/3), n>=1, a(0) = 1.` `From Michael Somos, May 03 2011: (Start)` `Euler transform of A065417.` `G.f.: (1 - x) / (1 - x - x^3). a(n) = a(n-1) + a(n-3).` `a(-n) = A077961(n). a(n+3) = A000930(n). a(n+5) = A068921(n). (End)` `a(n+1) = A013979(n-3) + A135851(n) + A107458(n), n>=3.` `a(n) = a(n-1) + a(n-3) for n >= 4. - Jaroslav Krizek, May 07 2011.`
	EXAMPLE	`1 + x^3 + x^4 + x^5 + 2x^6 + 3x^7 + 4x^8 + 6x^9 + 9x^10 + 13x^11 + ...`
	MAPLE	`A078012 := proc(n): if n=0 then 1 else add(binomial(n-3-2*i, i), i=0..(n-3)/3) fi: end: seq(A078012(n), n=0..46); # [Johannes W. Meijer, Aug 11 2011]`
	MATHEMATICA	`CoefficientList[ Series[(1 - x)/(1 - x - x^3), {x, 0, 46}], x] (* Robert G. Wilson v, May 25 2011 *)`
	PROG	`(PARI) {a(n) = if( n<0, n = -n; polcoeff( 1 / (1 + x^2 - x^3) + x * O(x^n), n) , polcoeff( (1 - x) / (1 - x - x^3) + x * O(x^n), n))} /* Michael Somos May 03 2011 */`
	CROSSREFS	`Cf. A000930, A065417, A068921, A077961.` `Sequence in context: A068921 A000930 * A135851 A199804 A101913 A121653` `Adjacent sequences: A078009 A078010 A078011 * A078013 A078014 A078015`
	KEYWORD	`nonn,easy`
	AUTHOR	`N. J. A. Sloane (njas(AT)research.att.com), Nov 17 2002, Mar 08 2008`

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Last modified February 16 14:07 EST 2012. Contains 205930 sequences.