

A165136


a(n) is the number of patterns for ndigit papaya numbers.


3



1, 2, 4, 10, 21, 50, 99, 250, 454, 1242, 2223, 6394, 11389, 35002, 62034, 202010, 359483, 1233518, 2203507, 7944100, 14249715, 53810836, 96911168, 382258438, 691048071, 2840120987, 5152403569, 22010733048, 40059670261, 177444599715
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OFFSET

1,2


COMMENTS

Papaya numbers are concatenations of two palindromes or palindromes themselves (think of "papaya" as the concatenation of the palindromes "pap" and "aya").
The actual number of ndigit papaya numbers is A165135. If the pattern is "aa", for example, inserting digits 1 to 9 for "a" gives 9 positive 2digit numbers, 11, 22, ...,99.The pattern "ab" inserting a<>b gives 10, 12,..,98, that is 9*9 = 81 positive 2digit numbers.(9 different choices for "a" because leading 0's are not allowed, and for each "a" 9 different choices of "b".)So the a(2) =2 two different patterns represent 9+81= A165135(2) different 2digit numbers.
The first 19 terms of this sequence are the same as in A165137. Then the sequences start to differ, because the number of patterns in an infinite alphabet can be larger than patterns in the 10digitsalphabet of ordinary numbers: A165137(20) = a(20)+10. [From Tanya Khovanova, Oct 01 2009] (Since at most 2 symbols in a papaya number can be present only once, to require 11 symbols takes a length of 2 + (112)*2 = 20. The 10 strings for A165137(20) not counted here are abcdefghijkjihgfedbc, abacdefghijkjihgfedc, abcbadefghijkjihgfed, ..., abcdefghijihgfedcbak.  Franklin T. AdamsWatters, Apr 10 2011.


LINKS

Table of n, a(n) for n=1..30.
Tanya Khovanova, Papaya Words and Numbers


FORMULA

a(n) = R(n)  Sum_{dn,d<n} phi(n/d)*a(d) where R(2*k)=k*(b(k)+b(k+1)), R(2*k+1)=(2*k+1)*b(k+1), b(k)=Sum_{j=1..10} stirling2(k,j).  Andrew Howroyd, Mar 29 2016


EXAMPLE

There are two types of twodigit papaya numbers: aa, or ab. Hence a(2) = 2.
There are four types of threedigit papaya numbers: aaa, aab, aba, abb. Hence a(3) = 4.
There is no pattern of the form "abcdefghijkl" contributing to a(12), because this requires 12 different letters in the alphabet, and the standard numbers alphabet provides only ten different digits 09.


CROSSREFS

Cf. A165135, A165137, A165610, A165611.
Sequence in context: A018003 A255711 A204804 * A165137 A065023 A301700
Adjacent sequences: A165133 A165134 A165135 * A165137 A165138 A165139


KEYWORD

base,nonn


AUTHOR

Sergei Bernstein and Tanya Khovanova, Sep 04 2009


EXTENSIONS

Three more terms from R. J. Mathar, Sep 25 2009
Keyword:base added, comment expanded  R. J. Mathar, Aug 29 2010
a(10)a(14) from Franklin T. AdamsWatters, Apr 10 2011
a(15)a(30) from Andrew Howroyd, Mar 29 2016


STATUS

approved



