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A261132 Number of ways to write n as the sum u+v+w of three palindromes (from A002113) with 0 <= u <= v <= w. 14
1, 1, 2, 3, 4, 5, 7, 8, 10, 12, 13, 15, 16, 17, 17, 18, 17, 17, 16, 15, 13, 12, 11, 10, 9, 8, 7, 7, 6, 6, 6, 6, 5, 7, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 8, 7, 7, 7, 7, 7, 7, 7, 7, 7, 5, 9, 7, 7, 7, 7, 7, 7, 7, 7, 7, 5, 11, 8, 8, 8, 8, 8, 8, 8, 8, 8, 5, 12, 8, 8, 8 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

It is known that a(n) > 0 for every n, i.e., every number can be written as the sum of 3 palindromes.

The conjecture has been checked up to 1.2*10^14. - Hugo Pfoertner, Sep 24 2015

LINKS

Giovanni Resta, Table of n, a(n) for n = 0..10000

Javier Cilleruelo and Florian Luca, Every positive integer is a sum of three palindromes, arXiv: 1602.06208, 2016.

Erich Friedman, Problem of the Month (June 1999)

Hugo Pfoertner, Plot of first 10^6 terms

Hugo Pfoertner, Plot of first 3*10^7 terms

Hugo Pfoertner, Plot of low values in range 7*10^6 ... 10^7

EXAMPLE

a(0)=1 because 0 = 0+0+0;

a(1)=1 because 1 = 0+0+1;

a(2)=2 because 2 = 0+1+1 = 0+0+2;

a(3)=3 because 3 = 1+1+1 = 0+1+2 = 0+0+3.

a(28) = 6 since 28 can be expressed in 6 ways as the sum of 3 palindromes, namely, 28 = 0+6+22 = 1+5+22 = 2+4+22 = 3+3+22 = 6+11+11 = 8+9+11.

MAPLE

A261132 := proc(n)

    local xi, yi, x, y, z, a ;

    a := 0 ;

    for xi from 1 do

        x := A002113(xi) ;

        if 3*x > n then

            return a;

        end if;

        for yi from xi do

            y := A002113(yi) ;

            if x+2*y > n then

                break;

            else

                z := n-x-y ;

                if z >= y and isA002113(z) then

                    a := a+1 ;

                end if;

            end if;

        end do:

    end do:

    return a;

end proc:

seq(A261132(n), n=0..80) ; # R. J. Mathar, Sep 09 2015

MATHEMATICA

pal=Select[Range[0, 1000], (d = IntegerDigits@ #; d == Reverse@ d)&]; a[n_] := Length@ IntegerPartitions[n, {3}, pal]; a /@ Range[0, 1000]

CROSSREFS

Cf. A002113, A035137, A260254, A261131.

See A261422 for another version.

Sequence in context: A103740 A034155 A129590 * A262525 A184105 A062469

Adjacent sequences:  A261129 A261130 A261131 * A261133 A261134 A261135

KEYWORD

nonn,base,look

AUTHOR

Giovanni Resta, Aug 10 2015

EXTENSIONS

Examples revised and plots for large n added by Hugo Pfoertner, Aug 11 2015

STATUS

approved

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Last modified February 23 11:22 EST 2018. Contains 299564 sequences. (Running on oeis4.)