login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A165136 a(n) is the number of patterns for n-digit 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 (list; graph; refs; listen; history; text; internal format)
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 n-digit papaya numbers is A165135. If the pattern is "aa", for example, inserting digits 1 to 9 for "a" gives 9 positive 2-digit numbers, 11, 22, ...,99.The pattern "ab" inserting a<>b gives 10, 12,..,98, that is 9*9 = 81 positive 2-digit 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 2-digit 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 10-digits-alphabet 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 + (11-2)*2 = 20. The 10 strings for A165137(20) not counted here are abcdefghijkjihgfedbc, abacdefghijkjihgfedc, abcbadefghijkjihgfed, ..., abcdefghijihgfedcbak. - Franklin T. Adams-Watters, 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_{d|n,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 two-digit papaya numbers: aa, or ab. Hence a(2) = 2.

There are four types of three-digit 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 0-9.

CROSSREFS

Cf. A165135, A165137, A165610, A165611.

Sequence in context: A018003 A255711 A204804 * A165137 A065023 A123445

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. Adams-Watters, Apr 10 2011

a(15)-a(30) from Andrew Howroyd, Mar 29 2016

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 | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 12 15:01 EST 2017. Contains 295939 sequences.