cp's OEIS Frontend

A002663 a(n) = 2^n - C(n,0) - C(n,1) - C(n,2) - C(n,3).

%I A002663 M4152 N1725 #74 Jul 28 2025 18:01:43
%S A002663 0,0,0,0,1,6,22,64,163,382,848,1816,3797,7814,15914,32192,64839,
%T A002663 130238,261156,523128,1047225,2095590,4192510,8386560,16774891,
%U A002663 33551806,67105912,134214424,268431773,536866822,1073737298
%N A002663 a(n) = 2^n - C(n,0) - C(n,1) - C(n,2) - C(n,3).
%C A002663 Starting with "1" = eigensequence of a triangle with bin(n,4), A000332 as the left border: (1, 5, 15, 35, 70, ...) and the rest 1's. - _Gary W. Adamson_, Jul 24 2010
%C A002663 The Kn25 sums, see A180662, of triangle A065941 equal the terms (doubled) of this sequence minus the four leading zeros. - _Johannes W. Meijer_, Aug 14 2011
%C A002663 (1 + 6x + 22x^2 + 64x^3 + ...) = (1 + 3x + 6x^2 + 10x^3 + ...) * (1 + 3x + 7x^2 + 15x^3 + ...). - _Gary W. Adamson_, Mar 14 2012
%C A002663 The sequence starting (1, 6, 22, ...) is the binomial transform of A171418 and starting (0, 0, 0, 1, 6, 22, ...) is the binomial transform of (0, 0, 0, 1, 2, 2, 2, 2, 2, ...). - _Gary W. Adamson_, Jul 27 2015
%C A002663 Number of binary sequences with at least four 0's. - _Enrique Navarrete_, Jul 23 2025
%D A002663 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002663 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002663 Vincenzo Librandi, <a href="/A002663/b002663.txt">Table of n, a(n) for n = 0..1000</a>
%H A002663 J. Eckhoff, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002476398&amp;IDDOC=247621">Der Satz von Radon in konvexen Productstrukturen II</a>, Monat. f. Math., 73 (1969), 7-30.
%H A002663 R. K. Guy, <a href="/A000346/a000346.pdf">Letter to N. J. A. Sloane</a>
%H A002663 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A002663 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A002663 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (6,-14,16,-9,2).
%F A002663 a(n) = 2^n - A000125(n).
%F A002663 G.f.: x^4/((1-2*x)*(1-x)^4). - _Simon Plouffe_ in his 1992 dissertation
%F A002663 a(n) = Sum_{k=0..n} binomial(n,k+4) = Sum_{k=4..n} binomial(n,k). - _Paul Barry_, Aug 23 2004
%F A002663 a(n) = 2*a(n-1) + binomial(n-1,3). - _Paul Barry_, Aug 23 2004
%F A002663 a(n) = (6*2^n - n^3 - 5*n - 6)/6. - _Mats Granvik_, _Gary W. Adamson_, Feb 17 2010
%F A002663 From _Enrique Navarrete_, Jul 23 2025: (Start)
%F A002663 a(n) = 6*a(n-1) - 14*a(n-2) + 16*a(n-3) - 9*a(n-4) + 2*a(n-5).
%F A002663 E.g.f.: exp(x)*(exp(x) - 1 - x - x^2/2 - x^3/6). (End)
%p A002663 A002663 := proc(n): 2^n - add(binomial(n,k),k=0..3) end: seq(A002663(n), n=0..30); # _Johannes W. Meijer_, Aug 14 2011
%t A002663 a=1;lst={};s1=s2=s3=s4=0;Do[s1+=a;s2+=s1;s3+=s2;s4+=s3;AppendTo[lst,s4];a=a*2,{n,5!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Jan 10 2009 *)
%t A002663 Table[Sum[ Binomial[n + 4, k + 4], {k, 0, n}], {n, -4, 26}] (* _Zerinvary Lajos_, Jul 08 2009 *)
%o A002663 (Magma) [2^n - Binomial(n,0)- Binomial(n,1) - Binomial(n,2) - Binomial(n,3): n in [0..35]]; // _Vincenzo Librandi_, May 20 2011
%o A002663 (Haskell)
%o A002663 a002663 n = a002663_list !! n
%o A002663 a002663_list = map (sum . drop 4) a007318_tabl
%o A002663 -- _Reinhard Zumkeller_, Jun 20 2015
%o A002663 (PARI) a(n)=(6*2^n-n^3-5*n-6)/6 \\ _Charles R Greathouse IV_, Sep 24 2015
%Y A002663 a(n)= A055248(n, 4). Partial sums of A002662.
%Y A002663 Cf. A000079, A000225, A000295, A002664, A035038-A035042, A007318.
%K A002663 nonn,easy
%O A002663 0,6
%A A002663 _N. J. A. Sloane_