A003520 -id:A003520 - cp's OEIS frontend

A226516 Number of (18,7)-reverse multiples with n digits.

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 3, 3, 4, 4, 6, 5, 8, 6, 10, 8, 13, 11, 17, 15, 23, 20, 31, 26, 41, 34, 54, 45, 71, 60, 94, 80, 125, 106, 166, 140, 220, 185, 291, 245, 385, 325, 510, 431, 676, 571, 896, 756, 1187, 1001, 1572, 1326, 2082, 1757, 2758, 2328, 3654, 3084, 4841, 4085, 6413

Offset: 0

N. J. A. Sloane, Jun 16 2013

Comment from Emeric Deutsch, Aug 21 2016 (Start):
Given an increasing sequence of positive integers S = {a0, a1, a2, ... }, let
          F(x) = x^{a0} + x^{a1} + x^{a2} + ... .
Then the g. f. for the number of palindromic compositions of n with parts in S is (see Hoggatt and Bicknell, Fibonacci Quarterly, 13(4), 1975):
      (1 + F(x))/(1 - F(x^2))
Playing with this, I have found easily that
1. number of palindromic compositions of n into {3,4,5,...} = A226916(n+4);
2. number of palindromic compositions of n into {1,4,7,10,13,...} = A226916(n+6);
3. number of palindromic compositions of n into {1,4} = A226517(n+10);
4. number of palindromic compositions of n into {1,5} = A226516(n+11).
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
V. E. Hogatt, M. Bicknell, Palindromic Compositions, Fib. Quart. 13(4) (1975) 350-356
N. J. A. Sloane, 2178 And All That, Fib. Quart., 52 (2014), 99-120.
N. J. A. Sloane, 2178 And All That [Local copy]
Index entries for linear recurrences with constant coefficients, signature (0,1,0,0,0,0,0,0,0,1).

Cf. A214927, A226517, A226916.

f:=proc(n) option remember;
if
n <= 5 then 0
elif n=6 then 1
elif n <= 10 then 0
elif n <= 12 then 1
else f(n-2)+f(n-10)
fi;
end;
[seq(f(n),n=0..100)]

CoefficientList[Series[x^6 (1 - x^2 + x^5 + x^6) / (1 - x^2 - x^10), {x, 0, 80}], x] (* Vincenzo Librandi, Jun 18 2013 *)
LinearRecurrence[{0,1,0,0,0,0,0,0,0,1},{0,0,0,0,0,0,1,0,0,0,0,1,1},80] (* Harvey P. Dale, Jun 17 2015 *)

G.f.: x^6*(1+x)*(1-x+x^5)/(1-x^2-x^10).
a(n) = a(n-2) + a(n-10) for n>12, with initial values a(0)-a(12) equal to 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1. [Bruno Berselli, Jun 17 2013]
a(2n+1) = A003520(n-5). a(2n) = A098523(n-6). - R. J. Mathar, Dec 13 2022

A243732 Irregular triangular array of denominators of the positive rational numbers ordered as in Comments.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jun 09 2014

Suppose that m >= 3, and define sets h(n) of positive rational numbers as follows: h(n) = {n} for n = 1..m, and thereafter, h(n) = Union({x+1: x in h(n-1)}, {x/(x+1) : x in h(n-m)}), with the numbers in h(n) arranged in decreasing order. Every positive rational lies in exactly one of the sets h(n). For the present array, put m = 5 and (row n) = h(n); the number of numbers in h(n) is A003520(n-1). (For m = 3, see A243712.)

			First 11 rows of the array:
  1/1
  2/1
  3/1
  4/1
  5/1
  6/1 ... 1/2
  7/1 ... 3/2 ... 2/3
  8/1 ... 5/2 ... 5/3 ... 3/4
  9/1 ... 7/2 ... 8/3 ... 7/4 ... 4/5
  10/1 .. 9/2 ... 11/3 .. 11/4 .. 9/5 ... 5/6
  11/1 .. 11/2 .. 14/3 .. 15/4 .. 14/5 .. 11/6 .. 6/7 .. 1/3
The denominators, by rows:  1,1,1,1,1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,6,1,2,3,4,5,6,7,3,...

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A243731, A243712, A243733, A003269.

z = 23; g[1] = {1}; g[2] = {2}; g[3] = {3}; g[4] = {4}; g[5] = {5};
g[n_] := Reverse[Union[1 + g[n - 1], g[n - 5]/(1 + g[n - 5])]]
Table[g[n], {n, 1, 9}]
v = Flatten[Table[g[n], {n, 1, z}]];
v1 = Denominator[v]; (* A243732 *)
v2 = Numerator[v];   (* A243733 *)

A247117 Number of tilings of a 10 X n rectangle using 2n pentominoes of shape I.

Original entry on oeis.org

1, 1, 1, 1, 1, 8, 17, 28, 41, 56, 144, 317, 609, 1060, 1716, 3324, 6713, 13188, 24624, 43620, 80464, 153645, 296025, 562097, 1037921, 1920661, 3600832, 6820873, 12920804, 24211457, 45173688, 84493668, 158848825, 299451277, 562923960, 1055117520, 1976475968

Offset: 0

Alois P. Heinz, Nov 19 2014

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Pentomino

Cf. A174249, A233427, A003520 (5 X n), A247218 (15 X n).

Column k=5 of A250662.

gf:= -(x^10+x^8-x^6-2*x^5-x^4-x^3+1) *(x-1)^4 *(x^4+x^3+x^2+x+1)^4 / (x^35 +x^33 -2*x^31 -7*x^30 -2*x^29 -6*x^28 +x^27 +9*x^26 +22*x^25 +8*x^24 +15*x^23 -4*x^22 -15*x^21 -39*x^20 -12*x^19 -20*x^18 +6*x^17 +10*x^16 +45*x^15 +8*x^14 +19*x^13 -4*x^12 -4*x^11 -33*x^10 -6*x^9 -10*x^8 +x^7 -3*x^6 +12*x^5 +x^3 +x-1):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..50);

G.f.: see Maple program.

A369838 Number of compositions of 5*n-3 into parts 1 and 5.

Original entry on oeis.org

1, 4, 15, 60, 245, 1001, 4085, 16665, 67985, 277350, 1131476, 4615966, 18831276, 76823991, 313410816, 1278589392, 5216127688, 21279691689, 86812537085, 354160046356, 1444829775128, 5894321227301, 24046447082350, 98099780277675, 400207434286276, 1632684497403029

Offset: 1

Seiichi Manyama, Feb 03 2024

nonn

Paolo Xausa, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-10,10,-5,1).

Cf. A079675, A369836, A369837, A369839.

Cf. A003520.

LinearRecurrence[{6, -10, 10, -5, 1}, {1, 4, 15, 60, 245}, 50] (* Paolo Xausa, Mar 15 2024 *)

a(n) = sum(k=0, n, binomial(n+1+4*k, n-1-k));

a(n) = A003520(5*n-3).
a(n) = Sum_{k=0..n} binomial(n+1+4*k,n-1-k).
a(n) = 6*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: x*(1-x)^2/((1-x)^5 - x).

A369839 Number of compositions of 5*n-4 into parts 1 and 5.

Original entry on oeis.org

1, 3, 11, 45, 185, 756, 3084, 12580, 51320, 209365, 854126, 3484490, 14215310, 57992715, 236586825, 965178576, 3937538296, 16063564001, 65532845396, 267347509271, 1090669728772, 4449491452173, 18152125855049, 74053333195325, 302107654008601, 1232477063116753

Offset: 1

Seiichi Manyama, Feb 03 2024

Paolo Xausa, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-10,10,-5,1).

Cf. A079675, A369836, A369837, A369838.

Cf. A003520.

LinearRecurrence[{6, -10, 10, -5, 1}, {1, 3, 11, 45, 185}, 50] (* Paolo Xausa, Mar 15 2024 *)

a(n) = sum(k=0, n, binomial(n+4*k, n-1-k));

a(n) = A003520(5*n-4).
a(n) = Sum_{k=0..n} binomial(n+4*k,n-1-k).
a(n) = 6*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: x*(1-x)^3/((1-x)^5 - x).

A243733 Irregular triangular array of numerators of the positive rational numbers ordered as in Comments.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 1, 7, 3, 2, 8, 5, 5, 3, 9, 7, 8, 7, 4, 10, 9, 11, 11, 9, 5, 11, 11, 14, 15, 14, 11, 6, 1, 12, 13, 17, 19, 19, 17, 13, 4, 7, 3, 2, 13, 15, 20, 23, 24, 23, 20, 7, 15, 8, 7, 8, 5, 5, 3, 14, 17, 23, 27, 29, 29, 27, 10, 23, 13, 12, 17, 12, 13

Offset: 1

Clark Kimberling, Jun 09 2014

Suppose that m >= 3, and define sets h(n) of positive rational numbers as follows: h(n) = {n} for n = 1..m, and thereafter, h(n) = Union({x+1: x in h(n-1)}, {x/(x+1) : x in h(n-m)}), with the numbers in h(n) arranged in decreasing order. Every positive rational lies in exactly one of the sets h(n). For the present array, put m = 5 and (row n) = h(n); the number of numbers in h(n) is A003520(n-1). (For m = 3, see A243712.)

			First 11 rows of the array:
  1/1
  2/1
  3/1
  4/1
  5/1
  6/1 ... 1/2
  7/1 ... 3/2 ... 2/3
  8/1 ... 5/2 ... 5/3 ... 3/4
  9/1 ... 7/2 ... 8/3 ... 7/4 ... 4/5
  10/1 .. 9/2 ... 11/3 .. 11/4 .. 9/5 ... 5/6
  11/1 .. 11/2 .. 14/3 .. 15/4 .. 14/5 .. 11/6 .. 6/7 .. 1/3
The numerators, by rows:  1,2,3,4,5,6,1,7,3,2,8,5,5,3,9,7,8,7,4,10,9,...

Clark Kimberling, Table of n, a(n) for n = 1..999

Cf. A243732, A243713, A243731, A003520.

z = 23; g[1] = {1}; g[2] = {2}; g[3] = {3}; g[4] = {4}; g[5] = {5};
g[n_] := Reverse[Union[1 + g[n - 1], g[n - 5]/(1 + g[n - 5])]]
Table[g[n], {n, 1, 9}]
v = Flatten[Table[g[n], {n, 1, z}]];
v1 = Denominator[v]; (* A243732 *)
v2 = Numerator[v];   (* A243733 *)

A246690 Number A(n,k) of compositions of n into parts of the k-th list of distinct parts in the order given by A246688; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 2, 0, 1, 0, 1, 1, 0, 3, 1, 1, 0, 1, 0, 1, 1, 5, 0, 1, 0, 1, 1, 0, 2, 0, 8, 1, 1, 0, 1, 0, 1, 0, 3, 0, 13, 0, 1, 0, 1, 0, 1, 1, 1, 4, 1, 21, 1, 1, 0, 1, 1, 0, 1, 2, 0, 6, 0, 34, 0, 1, 0, 1, 1, 2, 0, 1, 3, 0, 9, 0, 55, 1, 1, 0

Offset: 0

Alois P. Heinz, Sep 01 2014

The first lists of distinct parts in the order given by A246688 are: 0:[], 1:[1], 2:[2], 3:[1,2], 4:[3], 5:[1,3], 6:[4], 7:[1,4], 8:[2,3], 9:[5], 10:[1,2,3], 11:[1,5], 12:[2,4], 13:[6], 14:[1,2,4], 15:[1,6], 16:[2,5], 17:[3,4], 18:[7], 19:[1,2,5], 20:[1,3,4], ... .

			Square array A(n,k) begins:
  1, 1, 1,  1, 1,  1, 1,  1, 1, 1,   1, 1, 1, 1,   1, ...
  0, 1, 0,  1, 0,  1, 0,  1, 0, 0,   1, 1, 0, 0,   1, ...
  0, 1, 1,  2, 0,  1, 0,  1, 1, 0,   2, 1, 1, 0,   2, ...
  0, 1, 0,  3, 1,  2, 0,  1, 1, 0,   4, 1, 0, 0,   3, ...
  0, 1, 1,  5, 0,  3, 1,  2, 1, 0,   7, 1, 2, 0,   6, ...
  0, 1, 0,  8, 0,  4, 0,  3, 2, 1,  13, 2, 0, 0,  10, ...
  0, 1, 1, 13, 1,  6, 0,  4, 2, 0,  24, 3, 3, 1,  18, ...
  0, 1, 0, 21, 0,  9, 0,  5, 3, 0,  44, 4, 0, 0,  31, ...
  0, 1, 1, 34, 0, 13, 1,  7, 4, 0,  81, 5, 5, 0,  55, ...
  0, 1, 0, 55, 1, 19, 0, 10, 5, 0, 149, 6, 0, 0,  96, ...
  0, 1, 1, 89, 0, 28, 0, 14, 7, 1, 274, 8, 8, 0, 169, ...

Alois P. Heinz, Antidiagonals n = 0..140

Columns k=0-21, 23, 25-28 give: A000007, A000012, A059841, A000045(n+1), A079978, A000930, A121262, A003269(n+1), A182097, A079998, A000073(n+2), A003520, A079977, A079979, A060945, A005708, A001687(n+1), A017817, A082784, A079971, A006498, A005709, A052920, A120400, A060961, A005710, A013979.
Main diagonal gives A246691.
Cf. A246688, A246720 (the same for partitions).

b:= proc(n, i) b(n, i):= `if`(n=0, [[]], `if`(i>n, [],
      [map(x->[i, x[]], b(n-i, i+1))[], b(n, i+1)[]]))
    end:
f:= proc() local i, l; i, l:=0, [];
      proc(n) while n>=nops(l)
        do l:=[l[], b(i, 1)[]]; i:=i+1 od; l[n+1]
      end
    end():
g:= proc(n, l) option remember; `if`(n=0, 1,
      add(`if`(i>n, 0, g(n-i, l)), i=l))
    end:
A:= (n, k)-> g(n, f(k)):
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, i_] := b[n, i] = If[n == 0, {{}}, If[i>n, {}, Join[Prepend[#, i]& /@ b[n - i, i + 1], b[n, i + 1]]]];
f = Module[{i = 0, l = {}}, Function[n, While[n >= Length[l], l = Join[l, b[i, 1]]; i++]; l[[n + 1]]]];
g[n_, l_] := g[n, l] = If[n==0, 1, Sum[If[i>n, 0, g[n - i, l]], {i, l}]];
A[n_, k_] := g[n, f[k]];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

A247218 Number of tilings of a 15 X n rectangle using 3n pentominoes of shape I.

Original entry on oeis.org

1, 1, 1, 1, 1, 34, 95, 190, 325, 506, 3324, 10353, 25607, 55346, 108756, 389216, 1208901, 3281686, 8006108, 17950204, 51430928, 150609259, 419540401, 1090827453, 2651884943, 7077981621, 19691707908, 54499735145, 145671654672, 371632691473, 976543067070

Offset: 0

Alois P. Heinz, Nov 26 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Pentomino

Cf. A174249, A233427, A003520, A247117.

Column k=5 of A251072.

A247907 Expansion of (1 + x) / ((1 - x^4) * (1 - x - x^5)) in powers of x.

Original entry on oeis.org

1, 2, 2, 2, 3, 5, 7, 9, 12, 16, 21, 28, 38, 51, 67, 88, 117, 156, 207, 274, 363, 481, 637, 844, 1119, 1483, 1964, 2601, 3446, 4566, 6049, 8013, 10615, 14062, 18628, 24677, 32691, 43307, 57369, 75997, 100675, 133367, 176674, 234043, 310041, 410717, 544084

Offset: 0

Michael Somos, Sep 26 2014

			G.f. = 1 + 2*x + 2*x^2 + 2*x^3 + 3*x^4 + 5*x^5 + 7*x^6 + 9*x^7 + 12*x^8 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1,1,-1,1,-1).

Cf. A003520, A133872, A247918.

m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1 +x)/((1-x^4)*(1-x-x^5))));  // G. C. Greubel, Aug 04 2018

CoefficientList[Series[(1 + x)/((1 - x^4) (1 - x - x^5)), {x, 0, 100}], x] (* Vincenzo Librandi, Sep 27 2014 *)

{a(n) = if( n<0, n=-8-n; polcoeff( -1 / ((1 - x) * (1 - x + x^2) * (1 + x^2) * (1 + x - x^3)) + x * O(x^n), n), polcoeff( 1 / ((1 - x) * (1 - x + x^2) * (1 + x^2) * (1 - x^2 - x^3)) + x * O(x^n), n))};

G.f.: 1 / ((1 - x) * (1 - x + x^2) * (1 + x^2) * (1 - x^2 - x^3)).
a(n) = -A247918(-8-n) for all n in Z.
Convolution of A003520 and A133872.
0 = a(n) + a(n+4) - a(n+5) + mod(floor((n-1) / 2), 2) for all n in Z.
0 = a(n) - a(n+1) + a(n+2) - a(n+3) + a(n+4) - 2*a(n+5) + 2*a(n+6) - 2*a(n+7) + a(n+8) for all n in Z.

A339086 Number of compositions (ordered partitions) of n into distinct parts congruent to 1 mod 5.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 1, 2, 0, 0, 0, 1, 2, 0, 0, 0, 1, 4, 6, 0, 0, 1, 4, 6, 0, 0, 1, 6, 12, 0, 0, 1, 6, 18, 24, 0, 1, 8, 24, 24, 0, 1, 8, 30, 48, 0, 1, 10, 42, 72, 0, 1, 10, 48, 120, 120, 1, 12, 60, 144, 120, 1, 12, 72, 216, 240, 1, 14, 84, 264, 360, 1, 14, 96, 360, 600, 1, 16, 114

Offset: 0

Ilya Gutkovskiy, Nov 23 2020

nonn

			a(18) = 6 because we have [11, 6, 1], [11, 1, 6], [6, 11, 1], [6, 1, 11], [1, 11, 6] and [1, 6, 11].

Index entries for sequences related to compositions

Cf. A003520, A016861, A032020, A032021, A109697, A280454, A337547, A337548, A339059, A339060, A339087, A339088, A339089.

nmax = 78; CoefficientList[Series[Sum[k! x^(k (5 k - 3)/2)/Product[1 - x^(5 j), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

G.f.: Sum_{k>=0} k! * x^(k*(5*k - 3)/2) / Product_{j=1..k} (1 - x^(5*j)).

cp's OEIS Frontend

A226516 Number of (18,7)-reverse multiples with n digits.

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A243732 Irregular triangular array of denominators of the positive rational numbers ordered as in Comments.

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A247117 Number of tilings of a 10 X n rectangle using 2n pentominoes of shape I.

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A369838 Number of compositions of 5*n-3 into parts 1 and 5.

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PARI

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A369839 Number of compositions of 5*n-4 into parts 1 and 5.

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PARI

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A243733 Irregular triangular array of numerators of the positive rational numbers ordered as in Comments.

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A246690 Number A(n,k) of compositions of n into parts of the k-th list of distinct parts in the order given by A246688; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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Maple

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A247218 Number of tilings of a 15 X n rectangle using 3n pentominoes of shape I.

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A247907 Expansion of (1 + x) / ((1 - x^4) * (1 - x - x^5)) in powers of x.

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