A319453 -id:A319453 - 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

A091580 Number of partitions of n into decimal palindromes.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 41, 55, 74, 96, 126, 162, 208, 263, 333, 415, 518, 639, 788, 962, 1174, 1420, 1716, 2060, 2468, 2940, 3497, 4137, 4886, 5747, 6744, 7885, 9203, 10702, 12424, 14379, 16611, 19136, 22009, 25245, 28915, 33037, 37688, 42901, 48765

Offset: 0

Reinhard Zumkeller, Jan 22 2004

			n=12: there are A000041(12)=77 partitions of 12, 3 of them contain non-palindromes: 12=10+2, 12=10+1+1 and 12 itself, therefore a(12)=77-3=74.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Palindromic Number
Eric Weisstein's World of Mathematics, Partition

Cf. A002113, A091581.
Different from A088669 and from A000041.
Row sums of A319453.

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, 1,
      b(n, h(i-1))+b(n-i, h(min(n-i, i))))
    end:
a:= n-> b(n, h(n)):
seq(a(n), n=0..100);  # Alois P. Heinz, Sep 19 2018

a(0)=1 prepended by Alois P. Heinz, Sep 17 2018

A261131 Number of ways to write n as the sum of 3 positive palindromes.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 13, 15, 14, 15, 14, 14, 12, 12, 9, 9, 8, 7, 6, 6, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 5, 8, 7, 7, 7, 7, 7

Offset: 0

Giovanni Resta, Aug 10 2015

Conjecture: a(n) > 0 for n > 2, i.e., every number greater than 2 can be written as the sum of 3 positive palindromes.

The conjecture is true (see links). - Giorgos Kalogeropoulos, May 10 2025

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

Giovanni Resta, Table of n, a(n) for n = 0..10000
Javier Cilleruelo, Florian Luca, Lewis Baxter, Every positive integer is a sum of three palindromes, arXiv:1602.06208, 2017.

Cf. A002113, A260254, A261132.

Column k=3 of A319453.

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`(i<1 or
      t<1, 0, b(n, h(i-1), t)+b(n-i, h(min(n-i, i)), t-1)))
    end:
a:= n-> (k-> b(n, h(n), k)-b(n, h(n), k-1))(3):
seq(a(n), n=0..120);  # Alois P. Heinz, Sep 19 2018

pal=Select[Range@ 1000, (d = IntegerDigits@ #; d == Reverse@ d)&]; a[n_] := Length@ IntegerPartitions[n, {3}, pal]; a /@ Range[0, 1000]
Table[Count[IntegerPartitions[n,{3}],?(AllTrue[#,PalindromeQ]&)],{n,0,90}] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale, Mar 26 2021 *)

A319468 Number of partitions of n into exactly two nonzero decimal palindromes.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 4, 5, 4, 4, 3, 3, 2, 2, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Alois P. Heinz, Sep 19 2018

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Column k=2 of A319453.

Cf. A002113, A319477.

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):
seq(a(n), n=0..100);

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

a(n) = 0 <=> n in { A319477 }.

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.

A319469 Number of partitions of n into exactly four nonzero decimal palindromes.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 3, 5, 6, 9, 11, 15, 17, 22, 24, 29, 31, 35, 36, 40, 39, 41, 39, 40, 37, 37, 33, 33, 29, 28, 25, 25, 22, 22, 21, 21, 20, 20, 20, 20, 20, 20, 20, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 24, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 26

Offset: 0

Alois P. Heinz, Sep 19 2018

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Column k=4 of A319453.

Cf. A002113.

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))(4):
seq(a(n), n=0..100);

a(n) = [x^n y^4] 1/Product_{j>=2} (1-y*x^A002113(j)).

A319470 Number of partitions of n into exactly five nonzero decimal palindromes.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 10, 13, 18, 22, 29, 34, 42, 48, 57, 63, 72, 77, 85, 88, 95, 96, 100, 99, 101, 98, 98, 93, 92, 87, 85, 80, 79, 74, 73, 70, 69, 67, 67, 65, 66, 65, 66, 66, 67, 68, 69, 70, 72, 73, 75, 76, 78, 79, 81, 81, 83, 83, 84, 84, 85, 84

Offset: 0

Alois P. Heinz, Sep 19 2018

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Column k=5 of A319453.

Cf. A002113.

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))(5):
seq(a(n), n=0..100);

Table[Count[IntegerPartitions[n,{5}],?(AllTrue[#,PalindromeQ]&)],{n,0,70}] (* _Harvey P. Dale, Oct 20 2024 *)

a(n) = [x^n y^5] 1/Product_{j>=2} (1-y*x^A002113(j)).

A319471 Number of partitions of n into exactly six nonzero decimal palindromes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 14, 20, 25, 34, 41, 53, 62, 76, 88, 104, 116, 134, 145, 163, 174, 189, 197, 211, 215, 225, 226, 232, 229, 233, 227, 228, 221, 219, 212, 211, 203, 201, 195, 194, 189, 190, 186, 187, 186, 188, 188, 192, 192, 197, 199, 204

Offset: 0

Alois P. Heinz, Sep 19 2018

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Column k=6 of A319453.

Cf. A002113.

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))(6):
seq(a(n), n=0..100);

a(n) = [x^n y^6] 1/Product_{j>=2} (1-y*x^A002113(j)).

A319472 Number of partitions of n into exactly seven nonzero decimal palindromes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, 21, 27, 37, 46, 60, 73, 91, 108, 131, 151, 178, 201, 231, 256, 287, 311, 342, 365, 393, 412, 437, 450, 470, 479, 493, 496, 505, 503, 508, 503, 504, 496, 496, 487, 487, 479, 478, 472, 473, 469, 472, 471, 476, 477

Offset: 0

Alois P. Heinz, Sep 19 2018

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Column k=7 of A319453.

Cf. A002113.

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))(7):
seq(a(n), n=0..100);

a(n) = [x^n y^7] 1/Product_{j>=2} (1-y*x^A002113(j)).

A319473 Number of partitions of n into exactly eight nonzero decimal palindromes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, 22, 28, 39, 49, 65, 80, 102, 123, 152, 179, 215, 248, 292, 331, 380, 423, 477, 522, 578, 623, 679, 721, 773, 811, 859, 889, 929, 953, 985, 1000, 1025, 1032, 1050, 1051, 1063, 1060, 1068, 1062, 1068, 1062, 1068

Offset: 0

Alois P. Heinz, Sep 19 2018

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Column k=8 of A319453.

Cf. A002113.

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))(8):
seq(a(n), n=0..100);

a(n) = [x^n y^8] 1/Product_{j>=2} (1-y*x^A002113(j)).

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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A091580 Number of partitions of n into decimal palindromes.

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A261131 Number of ways to write n as the sum of 3 positive palindromes.

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A319468 Number of partitions of n into exactly two 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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A319469 Number of partitions of n into exactly four nonzero decimal palindromes.

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A319470 Number of partitions of n into exactly five nonzero decimal palindromes.

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A319471 Number of partitions of n into exactly six nonzero decimal palindromes.

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