The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A006047 Number of entries in n-th row of Pascal's triangle not divisible by 3. (Formerly M0422) 27
 1, 2, 3, 2, 4, 6, 3, 6, 9, 2, 4, 6, 4, 8, 12, 6, 12, 18, 3, 6, 9, 6, 12, 18, 9, 18, 27, 2, 4, 6, 4, 8, 12, 6, 12, 18, 4, 8, 12, 8, 16, 24, 12, 24, 36, 6, 12, 18, 12, 24, 36, 18, 36, 54, 3, 6, 9, 6, 12, 18, 9, 18, 27, 6, 12, 18, 12, 24, 36, 18, 36, 54, 9, 18, 27, 18, 36, 54, 27, 54 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Fixed point of the morphism a -> a, 2a, 3a, starting from a(1) = 1. - Robert G. Wilson v, Jan 24 2006 This is a particular case of the number of entries in n-th row of Pascal's triangle not divisible by a prime p, which is given by a simple recursion using ⊗, the Kronecker (or tensor) product of vectors. Let v_0=(1,2,...,p). Then v_{n+1)=v_0 ⊗ v_n, where the vector v_n contains the values for the first p^n rows of Pascal's triangle (rows 0 through p^n-1). - William B. Everett (bill(AT)chgnet.ru), Mar 29 2008 a(n) = A206424(n) + A227428(n); number of nonzero terms in row n of triangle A083093. - Reinhard Zumkeller, Jul 11 2013 REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Reinhard Zumkeller (terms 0..1000) & Antti Karttunen, Table of n, a(n) for n = 0..19683 J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197. Michael Gilleland, Some Self-Similar Integer Sequences H. Harborth, Number of Odd Binomial Coefficients, Proc. Amer. Math. Soc. 62.1 (1977), 19-22. (Annotated scanned copy) Sam Northshield, Sums across Pascal’s triangle modulo 2, Congressus Numerantium, 200, pp. 35-52, 2010. FORMULA Write n in base 3; if the representation contains r 1's and s 2's then a(n) = 3^s * 2^r. Also a(n) = Sum_{k=0..n} (C(n, k)^2 mod 3). - Avi Peretz (njk(AT)netvision.net.il), Apr 21 2001 a(n) = b(n+1), with b(1)=1, b(2)=2, b(3n)=3b(n), b(3n+1)=b(n+1), b(3n+2)=2b(n+1). - Ralf Stephan, Sep 15 2003 G.f.: Product_{n>=0} (1+2*x^(3^n)+3*x^(2*3^n)) (Northshield). - Johannes W. Meijer, Jun 05 2011 G.f. g(x) satisfies g(x) = (1 + 2*x + 3*x^2)*g(x^3). - Robert Israel, Oct 15 2015 From Tom Edgar, Oct 15 2015: (Start) a(3^k) = 2 for k>=0; a(2*3^k) = 3 for k>=0; a(n) = Product_{b_j != 0} a(b_j*3^j) where n = Sum_{j>=0} b_j*3^j is the ternary representation of n. (End) A056239(a(n)) = A053735(n). - Antti Karttunen, Jun 03 2017 EXAMPLE 15 in base 3 is 120, here r=1 and s=1 so a(15) = 3*2 = 6. William B. Everett's comment with p=3, n=2: v_0 = (1,2,3), v_1 = (1,2,3) => v_2 = (1*1,1*2,1*3,2*1,2*2,2*3,3*1,3*2,3*3) = (1,2,3,2,4,6,3,6,9), the first 3^2 values of the present sequence. - Wolfdieter Lang, Mar 19 2014 MAPLE p:=proc(n) local ct, k: ct:=0: for k from 0 to n do if binomial(n, k) mod 3 = 0 then else ct:=ct+1 fi od: end: seq(p(n), n=0..82); # Emeric Deutsch f:= proc(n) option remember; ((n mod 3)+1)*procname(ceil((n+1)/3)-1) end proc: f(0):= 1: f(1):= 2: seq(f(i), i=0..100); # Robert Israel, Oct 15 2015 MATHEMATICA Nest[Flatten[ # /. a_Integer -> {a, 2a, 3a}] &, {1}, 4] (* Robert G. Wilson v, Jan 24 2006 *) Nest[ Join[#, 2#, 3#] &, {1}, 4] (* Robert G. Wilson v, Jul 27 2014 *) PROG (PARI) b(n)=if(n<3, n, if(n%3==0, 3*b(n/3), if(n%3==1, 1*b((n+2)/3), 2*b((n+1)/3)))) \\ Ralf Stephan (PARI) A006047(n) = b(1+n); \\ (The above PARI-program by Ralf Stephan is for offset-1-version of this sequence.) - Antti Karttunen, May 28 2017 (PARI) A006047(n) = { my(m=1, d); while(n, d = (n%3); m *= (1+d); n \= 3); m; }; \\ Antti Karttunen, May 28 2017 (PARI) a(n) = prod(i=1, #d=digits(n, 3), (1+d[i])) \\ David A. Corneth, May 28 2017 (PARI) upto(n) = my(res = , v); while(#res < n, v = concat(2*res, 3*res); res = concat(res, v)); res \\ David A. Corneth, May 29 2017 (Haskell) a006047 = sum . map signum . a083093_row -- Reinhard Zumkeller, Jul 11 2013 (Scheme) (define (A006047 n) (if (zero? n) 1 (let ((d (mod n 3))) (* (+ 1 d) (A006047 (/ (- n d) 3)))))) ;; For R6RS standard. Use modulo instead of mod in older Schemes like MIT/GNU Scheme. - Antti Karttunen, May 28 2017 (Python) from sympy.ntheory.factor_ import digits from sympy import prod def a(n):     d=digits(n, 3)     return n + 1 if n<3 else prod(1 + d[i] for i in range(1, len(d))) print([a(n) for n in range(51)]) # Indranil Ghosh, Jun 06 2017 CROSSREFS Cf. A001316, A003586, A038148, A053735, A083093, A089898, A206424, A227428, A286586, A286587, A286633. Sequence in context: A202864 A328880 A291290 * A285712 A062068 A328219 Adjacent sequences:  A006044 A006045 A006046 * A006048 A006049 A006050 KEYWORD nonn AUTHOR EXTENSIONS More terms from Ralf Stephan, Sep 15 2003 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified October 25 06:05 EDT 2020. Contains 338011 sequences. (Running on oeis4.)