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

0, 0, 0, 0, 1, 6, 22, 64, 163, 382, 848, 1816, 3797, 7814, 15914, 32192, 64839, 130238, 261156, 523128, 1047225, 2095590, 4192510, 8386560, 16774891, 33551806, 67105912, 134214424, 268431773, 536866822, 1073737298

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

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

(1 + 6x + 22x^2 + 64x^3 + ...) = (1 + 3x + 6x^2 + 10x^3 + ...) * (1 + 3x + 7x^2 + 15x^3 + ...). - Gary W. Adamson, Mar 14 2012

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

Number of binary sequences with at least four 0's. - Enrique Navarrete, Jul 23 2025

References

• 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).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• J. Eckhoff, Der Satz von Radon in konvexen Productstrukturen II, Monat. f. Math., 73 (1969), 7-30.
• R. K. Guy, Letter to N. J. A. Sloane
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• Index entries for linear recurrences with constant coefficients, signature (6,-14,16,-9,2).

Crossrefs

a(n)= A055248(n, 4). Partial sums of A002662.

Cf. A000079, A000225, A000295, A002664, A035038-A035042, A007318.

Programs

Haskell

a002663 n = a002663_list !! n
a002663_list = map (sum . drop 4) a007318_tabl
-- Reinhard Zumkeller, Jun 20 2015

Magma

[2^n - Binomial(n,0)- Binomial(n,1) - Binomial(n,2) - Binomial(n,3): n in [0..35]]; // Vincenzo Librandi, May 20 2011

Maple

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

Mathematica

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 *)
Table[Sum[ Binomial[n + 4, k + 4], {k, 0, n}], {n, -4, 26}] (* Zerinvary Lajos, Jul 08 2009 *)

PARI

a(n)=(6*2^n-n^3-5*n-6)/6 \\ Charles R Greathouse IV, Sep 24 2015

Formula

a(n) = 2^n - A000125(n).

G.f.: x^4/((1-2*x)*(1-x)^4). - Simon Plouffe in his 1992 dissertation

a(n) = Sum_{k=0..n} binomial(n,k+4) = Sum_{k=4..n} binomial(n,k). - Paul Barry, Aug 23 2004

a(n) = 2*a(n-1) + binomial(n-1,3). - Paul Barry, Aug 23 2004

a(n) = (6*2^n - n^3 - 5*n - 6)/6. - Mats Granvik, Gary W. Adamson, Feb 17 2010

From Enrique Navarrete, Jul 23 2025: (Start)

a(n) = 6*a(n-1) - 14*a(n-2) + 16*a(n-3) - 9*a(n-4) + 2*a(n-5).

E.g.f.: exp(x)*(exp(x) - 1 - x - x^2/2 - x^3/6). (End)