

A261132


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


16



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
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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 selfsimilarity: looking at the first 100 values, there is a Gaussianshaped peak centered at the first local maximum a(15) = 18. Looking at the first 10000 values, one sees just one Gaussianshaped 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 Gaussianshaped 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], 20162017.
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 := nxy ;
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)=nreturn(1); my(c=0, i=inv_A002113(n)); A2113[i] > n && i; until( A2113[i]*3 < n, j = inv_A002113(D = nA2113[i]); if( j>i, j=i, A2113[j] > D && j); while( j >= k = inv_A002113(D  A2113[j]), A2113[k] == D  A2113[j] && c++; jbreak)); 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



