A319468 -id:A319468 - cp's OEIS frontend

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.

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).

A341155 Number of partitions of n into 2 distinct nonzero decimal palindromes.

Original entry on oeis.org

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

Offset: 3

Ilya Gutkovskiy, Feb 06 2021

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

Cf. A002113, A136522, A260254, A319468, A341156, A341157, A341158, A341159.

A341191 Number of ways to write n as an ordered sum of 2 nonzero decimal palindromes.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 9, 8, 7, 6, 5, 4, 3, 2, 2, 0, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 6, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 7, 2, 2

Offset: 2

Ilya Gutkovskiy, Feb 06 2021

Alois P. Heinz, Table of n, a(n) for n = 2..20000

Cf. A002113, A136522, A260254, A319468, A341155, A341192, A341193.

nmax = 90; CoefficientList[Series[Sum[Boole[PalindromeQ[k]] x^k, {k, 1, nmax}]^2, {x, 0, nmax}], x] // Drop[#, 2] &

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.

A337853 a(n) is the number of partitions of n as the sum of two Niven numbers.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 5, 5, 5, 4, 4, 3, 3, 3, 3, 3, 4, 3, 4, 4, 4, 4, 5, 4, 5, 4, 4, 4, 3, 2, 4, 3, 3, 4, 3, 3, 5, 3, 4, 5, 4, 4, 7, 4, 5, 6, 5, 3, 7, 4, 4, 6, 4, 2, 7, 3, 4, 5, 4, 3, 7, 3, 4, 5, 4, 3, 8, 3, 4, 6, 3, 3, 6, 2, 5, 6, 5, 3, 8, 4, 4, 6

Offset: 0

Marius A. Burtea, Sep 26 2020

a(n) >= 1 for n >= 2 ?.

For n <= 200000, a(n) = 1 only for n = 2, 3, 299, (2 = 1 + 1, 3 = 1 + 2, 299 = 1 + 288) and a(n) = 2 only for n in {4, 5, 35, 59, 79, 95, 97, 149, 169, 179, 389}.

			0 and 1 cannot be decomposed as the sum of two Niven numbers, so a(0) = a(1) = 0.
4 = 1 + 3 = 2 + 2 and 1, 2, 3 are in A005349, so a(4) = 2.
15 = 3 + 12 = 5 + 10 = 6 + 9 = 7 + 8 and 3, 5, 6, 7, 8, 9, 10, 12 are in A005349, so a(15) = 4.

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Cf. A005349, A116357, A172151, A172398, A180149, A280829, A319468, A302479, A319395, A337854.

niven:=func; [#RestrictedPartitions(n,2,{k: k in [1..n-1] | niven(k)}): n in [0..100]];

m = 100; nivens = Select[Range[m], Divisible[#, Plus @@ IntegerDigits[#]] &]; a[n_] := Length[IntegerPartitions[n, {2}, nivens]]; Array[a, m, 0] (* Amiram Eldar, Sep 27 2020 *)

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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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A341155 Number of partitions of n into 2 distinct nonzero decimal palindromes.

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A341191 Number of ways to write n as an ordered sum of 2 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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A337853 a(n) is the number of partitions of n as the sum of two Niven numbers.

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