

A178482


Phiantipalindromic numbers.


8



1, 3, 4, 7, 8, 10, 11, 18, 19, 21, 22, 25, 26, 28, 29, 47, 48, 50, 51, 54, 55, 57, 58, 65, 66, 68, 69, 72, 73, 75, 76, 123, 124, 126, 127, 130, 131, 133, 134, 141, 142, 144, 145, 148, 149, 151, 152, 170, 171, 173, 174
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OFFSET

1,2


COMMENTS

We call m a phiantipalindromic number if for the vector (a,...,b) (a<...<b) of exponents of its basephi expansion, we have (a,...,b)=(b,...,a). For n>=2, either a(n)+1 or a(n)1 is in the sequence; also either a(n)+3 or a(n)3 is in the sequence.
Conjecture: this is the sequence of numbers k for which f(k) is an integer, where f(x) is the changeofbase function defined at A214969 using b=phi and c=b^2.  Clark Kimberling, Oct 17 2012


LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..3071 from R. J. Mathar)


FORMULA

For k>=1, a(2^k)=A005248(k); if 2^k<n<2^(k+1), then a(n)=a(2^k)+a(n2^k).


EXAMPLE

The vectors of exponents of 4 and 5 are (2,0,2) and (4,1,3) correspondingly (cf.A104605). Therefore by definition 4 is a phiantipalindromic number, while 5 is not. Let n=38. Then k=5. Thus a(38)=A005248(5)+a(6)=123+10=133. The vector of exponents of phi in the basephi expansion of 133 is (10,4,2,2,4,10).


MATHEMATICA

phiAPQ[1] = True; phiAPQ[n_] := Module[{d = RealDigits[n, GoldenRatio, 2*Ceiling[Log[GoldenRatio, n]]]}, e = d[[2]]  Flatten @ Position[d[[1]], 1]; Reverse[e] == e]; Select[Range[200], phiAPQ] (* Amiram Eldar, Apr 23 2020 *)


CROSSREFS

Cf. A005248, A055778, A104605, A104626, A104627, A104628.
For bisections see A171070, A171071.
Sequence in context: A184823 A242921 A091934 * A284658 A286340 A024515
Adjacent sequences: A178479 A178480 A178481 * A178483 A178484 A178485


KEYWORD

nonn,base


AUTHOR

Vladimir Shevelev, May 28 2010, May 29 2010


STATUS

approved



