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A001972
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Expansion of 1/((1-x)^2*(1-x^4)) = 1/( (1+x)*(1+x^2)*(1-x)^3 ) .
(Formerly M0551 N0199)
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11
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1, 2, 3, 4, 6, 8, 10, 12, 15, 18, 21, 24, 28, 32, 36, 40, 45, 50, 55, 60, 66, 72, 78, 84, 91, 98, 105, 112, 120, 128, 136, 144, 153, 162, 171, 180, 190, 200, 210, 220, 231, 242, 253, 264, 276, 288, 300, 312, 325, 338, 351, 364, 378, 392, 406, 420, 435, 450, 465
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| First differences are A008621 - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 26 2004
a(n) = least k>a(n-1) such that k+a(n-1)+a(n-2)+a(n-3) is triangular. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 26 2004
Column sums of the following array:
1 2 3 4 5 6 7 8 9...
1 2 3 4 5...
1...
......................
--------------------
1 2 3 4 6 8 10 12 15
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REFERENCES
| A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, p. 276-281.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..10000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 208
Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.
Brian OSullivan and Thomas Busch, Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas, arXiv 0810.0231v1 [quant-ph], 2008. [Eq 8a, lambda=4]
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
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FORMULA
| a(n) = a(n-1)+a(n-4)-a(n-5)+1. a(n)=floor((n+3)^2/8) - Michael Somos, Apr 21 2000.
a(n)=sum{k=0..n, floor((k+4)/4)}=n+1+sum{k=0..n, floor(k/4)}. - Paul Barry, Aug 19 2003
a(n)=a(n-4)+n+1. - Paul Barry, Jul 14 2004
a(n) = sum(floor(j/4), {j,0,n+4}), a(n-4) = (1/2)floor(n/4)*(2n-2-4*floor(n/4)) [From Mitch Harris, Sep 08 2008]
A002620(n+1)=a(2*n-1)/2. A000217(n+1)=a(2*n).
a(n)+a(n+1)+a(n+2)+a(n+3) = (n+4)*(n+5)/2. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 26 2004
a(n) = n^2/8+3*n/4+15/16+(-1)^n/16+A056594(n+3)/4. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 26 2004
a(n)=A130519(n+4). - Franklin T. Adams-Watters, Jul 10 2009
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MAPLE
| A001972:=-(2-z+z**3-2*z**4+z**5)/(z+1)/(z**2+1)/(z-1)**3; [Conjectured by S. Plouffe in his 1992 dissertation. Gives sequence except for the initial 1.]
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MATHEMATICA
| CoefficientList[Series[1/((1-x)^2(1-x^4)), {x, 0, 80}], x] (* From Harvey P. Dale, Mar 27 2011 *)
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PROG
| (PARI) a(n)=(n+3)^2\8
(MAGMA) [Floor((n+3)^2/8): n in [0..60]]; // Vincenzo Librandi, Aug 15 2011
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CROSSREFS
| Bisections are A000217 and A007590. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 26 2004
Sequence in context: A054041 A019293 A130519 * A005705 A139542 A093717
Adjacent sequences: A001969 A001970 A001971 * A001973 A001974 A001975
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Partially edited by R. J. Mathar, Jul 11 2009
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