A261131 -id:A261131 - cp's OEIS frontend

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

Giovanni Resta, Aug 10 2015

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

			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.

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

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

See A261422 for another version.

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

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

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

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

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

A319453 Number T(n,k) of partitions of n into exactly k nonzero decimal palindromes; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 2, 2, 1, 1, 0, 1, 3, 3, 2, 1, 1, 0, 1, 3, 4, 3, 2, 1, 1, 0, 1, 4, 5, 5, 3, 2, 1, 1, 0, 1, 4, 7, 6, 5, 3, 2, 1, 1, 0, 0, 5, 8, 9, 7, 5, 3, 2, 1, 1, 0, 1, 4, 10, 11, 10, 7, 5, 3, 2, 1, 1, 0, 0, 5, 11, 15, 13, 11, 7, 5, 3, 2, 1, 1

Offset: 0

Alois P. Heinz, Sep 19 2018

Differs from A008284 and from A072233 first at T(10,1) = 0.

			Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1, 1;
  0, 1, 1,  1;
  0, 1, 2,  1,  1;
  0, 1, 2,  2,  1,  1;
  0, 1, 3,  3,  2,  1,  1;
  0, 1, 3,  4,  3,  2,  1, 1;
  0, 1, 4,  5,  5,  3,  2, 1, 1;
  0, 1, 4,  7,  6,  5,  3, 2, 1, 1;
  0, 0, 5,  8,  9,  7,  5, 3, 2, 1, 1;
  0, 1, 4, 10, 11, 10,  7, 5, 3, 2, 1, 1;
  0, 0, 5, 11, 15, 13, 11, 7, 5, 3, 2, 1, 1;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Columns k=0-10 give: A000007, A136522 (for n>0), A319468, A261131, A319469, A319470, A319471, A319472, A319473, A319474, A319475.
Row sums give A091580.
T(2n,n) gives A319454.
Cf. A002113, A008284, A072233, A261132.

p:= proc(n) option remember; local i, s; s:= ""||n;
      for i to iquo(length(s), 2) do if
        s[i]<>s[-i] then return false fi od; true
    end:
h:= proc(n) option remember; `if`(n<1, 0,
     `if`(p(n), n, h(n-1)))
    end:
b:= proc(n, i) option remember; `if`(n=0 or i=1, x^n,
      b(n, h(i-1))+expand(x*b(n-i, h(min(n-i, i)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, h(n))):
seq(T(n), n=0..14);

T(n,k) = [x^n y^k] 1/Product_{j>=2} (1-y*x^A002113(j)).

Sum_{k=0..3} T(n,k) = A261132(n).

A341156 Number of partitions of n into 3 distinct nonzero decimal palindromes.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 7, 7, 9, 9, 10, 9, 10, 8, 8, 7, 6, 4, 4, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 4, 5, 6, 6, 6, 6, 6, 6, 6, 6, 7, 4, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 6

Ilya Gutkovskiy, Feb 06 2021

Alois P. Heinz, Table of n, a(n) for n = 6..20000

Cf. A002113, A136522, A261131, A261132, A261422, A341155, A341157, A341158, A341159.

A341192 Number of ways to write n as an ordered sum of 3 nonzero decimal palindromes.

Original entry on oeis.org

1, 3, 6, 10, 15, 21, 28, 36, 45, 52, 60, 66, 70, 72, 72, 70, 66, 60, 55, 45, 39, 34, 30, 27, 25, 24, 24, 24, 27, 27, 25, 27, 27, 27, 27, 27, 27, 27, 27, 30, 27, 27, 30, 30, 30, 30, 30, 30, 30, 30, 33, 27, 30, 33, 33, 33, 33, 33, 33, 33, 33, 36, 27, 34, 36, 36, 36, 36

Offset: 3

Ilya Gutkovskiy, Feb 06 2021

Alois P. Heinz, Table of n, a(n) for n = 3..20000

Cf. A002113, A136522, A261131, A261132, A261422, A341156, A341191, A341193.

nmax = 70; CoefficientList[Series[Sum[Boole[PalindromeQ[k]] x^k, {k, 1, nmax}]^3, {x, 0, nmax}], x] // Drop[#, 3] &

A319477 Nonnegative integers which cannot be obtained by adding exactly two nonzero decimal palindromes.

Original entry on oeis.org

0, 1, 21, 32, 43, 54, 65, 76, 87, 98, 111, 131, 141, 151, 161, 171, 181, 191, 201, 1031, 1041, 1042, 1051, 1052, 1053, 1061, 1062, 1063, 1064, 1071, 1072, 1073, 1074, 1075, 1081, 1082, 1083, 1084, 1085, 1086, 1091, 1092, 1093, 1094, 1095, 1096, 1097, 1099

Offset: 1

Alois P. Heinz, Sep 19 2018

Every integer larger than two can be obtained by adding exactly three nonzero decimal palindromes.

The nonzero palindromes of this sequence are in A213879.

Alois P. Heinz, Table of n, a(n) for n = 1..65536
Javier Cilleruelo, Florian Luca and Lewis Baxter, Every positive integer is a sum of three palindromes, arXiv: 1602.06208 [math.NT], 2017, Math. Comp. 87 (2018), 3023-3055.
James Grime and Brady Haran, Every Number is the Sum of Three Palindromes, Numberphile video (2018)

Cf. A002113, A035137 (allowing zero), A213879, A261131, A319453, A319468, A319586.

p:= proc(n) option remember; local i, s; s:= ""||n;
      for i to iquo(length(s), 2) do if
        s[i]<>s[-i] then return false fi od; true
    end:
h:= proc(n) option remember; `if`(n<1, 0,
     `if`(p(n), n, h(n-1)))
    end:
b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(t*i (k-> b(n, h(n), k)-b(n, h(n), k-1))(2):
a:= proc(n) option remember; local j; for j from 1+
      `if`(n=1, -1, a(n-1)) while g(j)<>0 do od; j
    end:
seq(a(n), n=1..80);

A319468(a(n)) = 0.

A282585 Number of ways to write n as an ordered sum of 3 squarefree palindromes (A071251).

Original entry on oeis.org

0, 0, 0, 1, 3, 6, 7, 9, 12, 19, 21, 21, 18, 24, 27, 28, 18, 18, 19, 24, 15, 10, 6, 12, 12, 12, 9, 9, 12, 15, 18, 12, 9, 7, 15, 15, 15, 9, 12, 15, 18, 18, 12, 9, 9, 18, 15, 12, 0, 9, 9, 9, 0, 0, 0, 6, 6, 9, 12, 9, 12, 15, 18, 18, 12, 9, 13, 18, 18, 18, 9, 15, 18, 21, 18, 12, 9, 15, 21, 21, 21, 9, 18, 21, 24, 18

Offset: 0

Ilya Gutkovskiy, Feb 19 2017

Every number can be written as the sum of 3 palindromes (see A261132 and A261422).
Conjecture: a(n) > 0 for any sufficiently large n.
Additional conjecture: every number > 3 can be written as the sum of 4 squarefree palindromes.

			a(22) = 6 because we have [11, 6, 5], [11, 5, 6] [6, 11, 5], [6, 5, 11], [5, 11, 6] and [5, 6, 11].

Ilya Gutkovskiy, Extended graphical example
Ilya Gutkovskiy, Extended graphical example for additional conjecture
Eric Weisstein's World of Mathematics, Palindromic Number
Eric Weisstein's World of Mathematics, Squarefree
Index entries for sequences related to palindromes

Cf. A002113, A005117, A035137, A071251, A091580, A091581, A260254, A261131, A261132, A261422, A280210, A282584.

nmax = 85; CoefficientList[Series[Sum[Boole[SquareFreeQ[k] && PalindromeQ[k]] x^k, {k, 1, nmax}]^3, {x, 0, nmax}], x]

G.f.: (Sum_{k>=1} x^A071251(k))^3.

A282845 Number of ways to write n as an ordered sum of 6 prime power palindromes (A084092).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 6, 21, 56, 126, 246, 432, 702, 1077, 1576, 2232, 3072, 4112, 5352, 6801, 8422, 10197, 12102, 14117, 16146, 18177, 20112, 21882, 23382, 24661, 25566, 26136, 26316, 26181, 25560, 24677, 23436, 21981, 20226, 18486, 16536, 14642, 12702, 10962, 9166, 7662, 6222, 5042, 3912, 3096, 2306, 1746, 1236, 921, 600

Offset: 0

Ilya Gutkovskiy, Feb 22 2017

Is there k which satisfies a(n) > 0 for all n > k?

			a(7) = 6 because we have:
[2, 1, 1, 1, 1, 1]
[1, 2, 1, 1, 1, 1]
[1, 1, 2, 1, 1, 1]
[1, 1, 1, 2, 1, 1]
[1, 1, 1, 1, 2, 1]
[1, 1, 1, 1, 1, 2]

Ilya Gutkovskiy, Extended graphical example
Eric Weisstein's World of Mathematics, Prime Power
Eric Weisstein's World of Mathematics, Palindromic Number
Index entries for sequences related to palindromes

Cf. A000961, A002113, A002385, A084092, A260254, A261131, A261132, A261422, A282585.

nmax = 55; CoefficientList[Series[(x + Sum[Boole[PrimePowerQ[k] && PalindromeQ[k]] x^k, {k, 1, nmax}])^6, {x, 0, nmax}], x]

G.f.: (Sum_{k>=1} x^A084092(k))^6.

cp's OEIS Frontend

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

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A319453 Number T(n,k) of partitions of n into exactly k nonzero decimal palindromes; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A341156 Number of partitions of n into 3 distinct nonzero decimal palindromes.

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A341192 Number of ways to write n as an ordered sum of 3 nonzero decimal palindromes.

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A319477 Nonnegative integers which cannot be obtained by adding exactly two nonzero decimal palindromes.

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A282585 Number of ways to write n as an ordered sum of 3 squarefree palindromes (A071251).

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A282845 Number of ways to write n as an ordered sum of 6 prime power palindromes (A084092).

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