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%I #45 Sep 08 2022 08:44:36
%S 1,1,2,3,4,6,8,10,13,16,19,23,27,31,36,41,46,52,58,64,71,78,85,93,101,
%T 109,118,127,136,146,156,166,177,188,199,211,223,235,248,261,274,288,
%U 302,316,331,346,361,377,393,409,426,443,460,478,496,514,533,552,571
%N Expansion of (1 + x^5) / ((1-x) * (1-x^2) * (1-x^3)) in powers of x.
%H Vincenzo Librandi, <a href="/A008748/b008748.txt">Table of n, a(n) for n = 0..1000</a>
%H Peter M. Chema, <a href="/A008748/a008748.pdf">Illustration of first 26 terms as corners of a double hexagon spiral from 1</a>
%H Clark Kimberling and John E. Brown, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL7/Kimberling/kimber67.html">Partial Complements and Transposable Dispersions</a>, J. Integer Seqs., Vol. 7, 2004.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-2,1).
%F a(n) = 1 + floor( n(n+1)/6 ). - _Michael Somos_, Jun 16 1999
%F a(n) = 1 + A001840(n-1). - _Michael Somos_, Jun 16 1999
%F a(n) = 1 + a(n-1) + a(n-3) - a(n-4) if n>4; a(n) = n if n=1..4. - _Michael Somos_, Jun 16 1999
%F a(-1-n) = a(n). - _Michael Somos_, Sep 06 2013
%e G.f. = 1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 6*x^5 + 8*x^6 + 10*x^7 + 13*x^8 + ...
%p A061347 := proc(n) op(1+(n mod 3),[-2,1,1]) ; end proc:
%p A008748 := proc(n) 1/6*n^2+1/6*n+8/9+A061347(n+2)/9 ; end proc:
%p seq(A008748(n),n=0..60) ; # _R. J. Mathar_, Mar 22 2011
%t Table[Floor[((n*(n+1)+2)/2+3)/3],{n,0,60}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 26 2010 *)
%t CoefficientList[Series[(1+x^5)/((1-x)(1-x^2)(1-x^3)), {x,0,60}], x] (* _Vincenzo Librandi_, Jun 11 2013 *)
%t LinearRecurrence[{2,-1,1,-2,1}, {1,1,2,3,4}, 60] (* _Harvey P. Dale_, Apr 08 2019 *)
%o (PARI) {a(n) = (n^2 + n)\6 + 1} /* _Michael Somos_, Sep 06 2013 */
%o (Magma) [1 + Floor(n*(n+1)/6): n in [0..60]]; // _G. C. Greubel_, Aug 03 2019
%o (Sage) ((1 + x^5)/((1-x)*(1-x^2)*(1-x^3))).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Aug 03 2019
%o (GAP) List([0..60], n-> 1 + Int(n*(n+1)/6)); # _G. C. Greubel_, Aug 03 2019
%Y Cf. A001840, A008748, A061347.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_
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