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A037453 Positive numbers n such that the base 5 representation of n contains no 3 or 4. 8
1, 2, 5, 6, 7, 10, 11, 12, 25, 26, 27, 30, 31, 32, 35, 36, 37, 50, 51, 52, 55, 56, 57, 60, 61, 62, 125, 126, 127, 130, 131, 132, 135, 136, 137, 150, 151, 152, 155, 156, 157, 160, 161, 162, 175, 176, 177, 180, 181, 182, 185, 186, 187 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

5 divides neither C(2s-1,s) = A001700[ s ] (nor C(2s,s) = A000984[ s ], central column of Pascal's Triangle) if and only if s is one of the terms in this sequence.

n such that the last decimal digit of C(2n,n) is not zero. - Benoit Cloitre, Aug 18 2002

Let us recall the plan of Apery's irrationality proof. Consider the recurrence (n + 1)^3 * u_(n+1) = (34n^3 + 51n^2 + 27n + 5)u_n - n^3 * u_(n-1). The solution with starting values u_0 = 1; u_1 = 5 has the peculiar property that it has integral terms, despite the fact that at every recursion step we divide by (n + 1)^3. The n-th term is given by f(n) = sum(i=0,n,binomial(n+i,i)^2*binomial(n,i)^2) (see Beukers link) and m such that if Mod(f(m),5) <>0 is same as 2*a(n). - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 08 2004

LINKS

Clark Kimberling, Table of n, a(n) for n = 1..1000

Frits Beukers Consequences of Apery's work on zeta(3)

W. Shur, The last digit of C(2*n,n) and sigma C(n,i)*C(2*n-2*i,n-i), The Electronic Journal of Combinatorics, R16, Volume 4, Issue 2 (1997).

FORMULA

a(3n)=5a(n), a(3n+1)=5a(n)+1, a(3n+2)=5a(n)+2, where by definition a(0)=0. - Emeric Deutsch, Mar 23 2004

G.f. satisfies g(x) = 5*(1+x+x^2)*g(x^3) + (x + 2*x^2)/(1-x^3). - Robert Israel, Sep 02 2014

MAPLE

a:= proc(t) option remember; 5*procname(floor(t/3))+ (t mod 3) end proc:

a(0):= 0:

seq(a(n), n=1..100); # Robert Israel, Sep 02 2014

MATHEMATICA

Table[FromDigits[IntegerDigits[k, 3], 5], {k, 60}] - T. D. Noe, Apr 18 2007

Rest[FromDigits[#, 5]&/@Tuples[{0, 1, 2}, 4]] (* Harvey P. Dale, Aug 31 2016 *)

PROG

(PARI) f(n)=sum(i=0, n, binomial(n+i, i)^2*binomial(n, i)^2); for (i=1, 1000, if(Mod(f(i), 5)<>0, print1(i/2, ", ")))

CROSSREFS

Cf. A050607, A005836.

Sequence in context: A047217 A219650 A039015 * A014528 A286751 A293278

Adjacent sequences:  A037450 A037451 A037452 * A037454 A037455 A037456

KEYWORD

nonn,base

AUTHOR

Clark Kimberling

EXTENSIONS

Better definition from T. D. Noe, Apr 18 2007

STATUS

approved

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Last modified March 24 06:56 EDT 2019. Contains 321444 sequences. (Running on oeis4.)