cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

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

Views

Author

Reinhard Zumkeller, Jan 22 2004

Keywords

Examples

			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.

Links

Crossrefs

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

Programs

  • Maple
    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

Extensions

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

A088670 Number of partitions of n into distinct decimal repdigit numbers.

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

Views

Author

Reinhard Zumkeller, Oct 03 2003

Keywords

Comments

a(n) <= A000009(n).
Not the same as A091581: a(n) < A091581(n) for n > 101.
A109967(n) = a(n+1) - a(n). - Reinhard Zumkeller, Jul 06 2005

Links

Crossrefs

Cf. A010785, A088669, A109968, A131364.

Programs

  • Haskell
    a088670 = p $ tail a010785_list where
   p _      0 = 1
   p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m
-- Reinhard Zumkeller, Dec 10 2011

Extensions

a(0)=1 added and offset adjusted by Reinhard Zumkeller, Dec 10 2011

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

Views

Author

Ilya Gutkovskiy, Feb 19 2017

Keywords

Examples

			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].

Links

Crossrefs

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

Programs

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

Formula

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

Views

Author

Ilya Gutkovskiy, Feb 19 2017

Keywords

Comments

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.

Examples

			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].

Links

Crossrefs

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

Programs

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

Formula

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

A091585 Number of partitions of the n-th decimal palindrome into distinct decimal palindromes.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 6, 8, 11, 34, 47, 58, 82, 105, 140, 186, 234, 238, 307, 401, 491, 603, 723, 891, 1058, 1302, 1543, 1858, 2228, 2626, 3113, 3666, 4274, 5026, 5823, 6815, 7879, 9251, 10727, 12347, 14219, 16340, 18740, 21465, 24473, 27960, 31819
Offset: 1

Views

Author

Reinhard Zumkeller, Jan 22 2004

Keywords

Comments

a(n) = A091581(A002113(n)).

Links

Crossrefs

Cf. A002113, A091581, A091583.

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

Views

Author

Ilya Gutkovskiy, Nov 11 2020

Keywords

Links

Crossrefs

Cf. A002113, A091580, A091581, A282584.
Showing 1-6 of 6 results.

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