A091580 -id:A091580 - 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).

A091581 Number of partitions of n into distinct decimal palindromes.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 11, 13, 14, 17, 19, 21, 23, 26, 27, 30, 32, 34, 36, 37, 39, 40, 42, 42, 44, 44, 45, 45, 47, 47, 47, 49, 48, 50, 50, 52, 52, 55, 55, 58, 60, 60, 64, 65, 68, 69, 73, 73, 77, 78, 82, 84, 84, 88, 88, 92, 92, 96, 96, 100, 100, 105, 107, 107, 113

Offset: 0

Reinhard Zumkeller, Jan 22 2004

Not the same as A088670: a(n) > A088670(n) for n > 101.

			n=13: there are A000009(13)=18 partitions of 13 into distinct integers, 4 of them contain non-palindromes: 13=12+1, 13=10+3, 13=10+2+1 and 13 itself, therefore a(13)=18-4=14;
for n=14 there are a(14)=17 partitions into palindromes: 11+3 = 11+2+1 = 9+5 = 9+4+1 = 9+3+2 = 8+6 = 8+5+1 = 8+4+2 = 8+3+2+1 = 7+6+1 = 7+5+2 = 7+4+3 = 7+4+2+1 = 6+5+3 = 6+5+2+1 = 6+4+3+1 = 5+4+3+2.

Alois P. Heinz, Table of n, a(n) for n = 0..20000
Eric Weisstein's World of Mathematics, Palindromic Number
Eric Weisstein's World of Mathematics, Partition

Cf. A091580, A046489.

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

A088669 Number of partitions of n into decimal repdigit numbers.

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, Oct 03 2003

Different from A091580: a(101) is one less than A091580(101) and thereafter the difference steadily increases. - Joshua Zucker, May 10 2007

a(n) <= A000041(n).

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

Cf. A010785, A088670. Different from A091580.

Cf. A131361.

a(0)=1 prepended by Alois P. Heinz, Jan 18 2016

A282584 Number of compositions (ordered partitions) of n into decimal palindromes (A002113).

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1022, 2042, 4081, 8156, 16300, 32576, 65104, 130112, 260032, 519681, 1038595, 2075660, 4148259, 8290402, 16568581, 33112734, 66176648, 132255728, 264316464, 528243231, 1055707644, 2109858797, 4216606912, 8426997041, 16841569684, 33658308890, 67266993433

Offset: 0

Ilya Gutkovskiy, Feb 19 2017

			a(4) = 8 because we have [4], [3, 1], [2, 2], [2, 1, 1], [1, 3], [1, 2, 1], [1, 1, 2] and [1, 1, 1, 1].

Eric Weisstein's World of Mathematics, Palindromic Number
Index entries for sequences related to palindromes
Index entries for sequences related to compositions

Cf. A002113, A091580, A091581, A260254, A261422.

nmax = 37; CoefficientList[Series[1/(1 - Sum[Boole[PalindromeQ[k]] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

G.f.: 1/(1 - Sum_{k>=2} x^A002113(k)).

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.

A091584 Number of partitions of the n-th decimal palindrome into decimal palindromes.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 55, 788, 5747, 28915, 114064, 378823, 1105429, 2916219, 7093435, 8275126, 17341313, 34721974, 66837990, 124308827, 224290376, 393929311, 675393430, 1133103539, 1864009161, 3157186031, 5011108578, 7836124010, 12086660443

Offset: 1

Reinhard Zumkeller, Jan 22 2004

a(n) = A091580(A002113(n)).

Eric Weisstein's World of Mathematics, Palindromic Number
Eric Weisstein's World of Mathematics, Partition

Cf. A091582, A046488, A046486, A046497, A046498, A046499.

More terms from Alois P. Heinz, Nov 13 2018

A338847 Number of compositions (ordered partitions) of n into distinct decimal palindromes.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 11, 13, 19, 27, 56, 63, 98, 122, 180, 324, 376, 538, 710, 962, 1130, 2202, 2341, 3410, 4226, 5276, 6692, 8324, 13940, 15476, 21692, 22394, 32952, 33555, 43418, 47738, 93002, 92282, 101666, 136346, 140810, 176186, 176336, 212448

Offset: 0

Ilya Gutkovskiy, Nov 11 2020

Eric Weisstein's World of Mathematics, Palindromic Number
Index entries for sequences related to palindromes
Index entries for sequences related to compositions

Cf. A002113, A091580, A091581, A282584.

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.

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A091581 Number of partitions of n into distinct decimal palindromes.

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A088669 Number of partitions of n into decimal repdigit numbers.

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A282584 Number of compositions (ordered partitions) of n into decimal palindromes (A002113).

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

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A091584 Number of partitions of the n-th decimal palindrome into decimal palindromes.

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A338847 Number of compositions (ordered partitions) of n into distinct decimal palindromes.

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