A261132 Number of ways to write n as the sum u+v+w of three palindromes (from A002113) with 0 <= u <= v <= w.

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

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 graph has a kind of self-similarity: looking at the first 100 values, there is a Gaussian-shaped peak centered at the first local maximum a(15) = 18. Looking at the first 10000 values, one sees just one Gaussian-shaped peak centered around the record and local maximum a(1453) = 766, but to both sides of this value there are smaller peaks, roughly at distances which are multiples of 10. In the range [1..10^6], one sees a Gaussian-shaped peak centered around the record a(164445) = 57714. In the range [1..3*10^7], there is a similar peak of height ~ 4.3*10^6 at 1.65*10^7, with smaller neighbor peaks at distances which are multiples of 10^6, etc. - M. F. Hasler, Sep 09 2018

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 [math.NT], 2016-2017.

Erich Friedman, Problem of the Month (June 1999)

James Grime and Brady Haran, Every Number is the Sum of Three Palindromes, Numberphile video (2018)

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

Formula

a(n) = Sum_{k=0..3} A319453(n,k). - Alois P. Heinz, Sep 19 2018

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]

Prog

(PARI) A261132(n)=n||return(1); my(c=0, i=inv_A002113(n)); A2113[i] > n && i--; until( A2113[i--]*3 < n, j = inv_A002113(D = n-A2113[i]); if( j>i, j=i, A2113[j] > D && j--); while( j >= k = inv_A002113(D - A2113[j]), A2113[k] == D - A2113[j] && c++; j--||break)); c \\ For efficiency, this uses an array A2113 precomputed at least up to n. - M. F. Hasler, Sep 10 2018

Crossrefs

Cf. A002113, A035137, A260254, A261131, A319453.

See A261422 for another version.

Sequence in context: A034155 A306403 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