A079500 -id:A079500 - cp's OEIS frontend

A188288 In lunar arithmetic in base 2, the number of divisors of the number 11...1101 (n digits, the binary expansion of 2^n-3).

0, 1, 0, 2, 2, 2, 4, 6, 10, 16, 31, 55, 100, 185, 345, 644, 1209, 2274, 4298, 8145, 15469, 29454, 56213, 107489, 205925, 395190, 759621, 1462282, 2818799, 5440705, 10513994, 20340794, 39393580, 76368240, 148185145, 287791544, 559386196, 1088144064, 2118283567, 4126561528, 8044217224

Offset: 0

Adam S. Jobson and N. J. A. Sloane, Mar 26 2011

a(1)=1 by convention. The g.f. is only a conjecture.

			a(6) = 4 since 111101 has the divisors 1, 101, 1101, 111101.
a(8) = 10 since 11111101 has the divisors 1, 101, 1001, 1101, 10101, 11001, 11101, 111001, 111101, 11111101.

N. J. A. Sloane, Table of n, a(n) for n = 0..255
D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A079500, A188524.

G.f.: x + x^3/(1-x) + Sum(x^l*(1-x)^2/(1-2*x+x^(l-1)-x^l+x^(l+2)), l=3..oo). - N. J. A. Sloane, Apr 19 2011

a(1) in b-file corrected by Andrew Howroyd, Feb 22 2018

A329128 Number of nonequivalent sets whose translations and reflections cover {1..n}.

Original entry on oeis.org

1, 2, 3, 6, 8, 17, 24, 52, 77, 171, 265, 593, 952, 2131, 3519, 7846, 13238, 29351, 50374, 111031, 193155, 423403, 744616, 1624302, 2881784, 6260030, 11186219, 24213106, 43522800, 93922741, 169653109, 365172178

Offset: 1

Andrew Howroyd, Nov 07 2019

Equivalence is up to translation and reflection. Only translations and reflections that are subsets of {1..n} are included.

			For n = 4 there are 6 sets (up to equivalence) that with their reflections and translations cover {1..4}:
  {{1}, {2}, {3}, {4}};
  {{1, 2}, {2, 3}, {3, 4}};
  {{1, 3}, {2, 4}};
  {{1, 2, 4}, {1, 3, 4}};
  {{1, 2, 3}, {2, 3, 4}};
  {{1, 2, 3, 4}}.
.
For n = 5 there are 8 sets (up to equivalence) that with their reflections and translations cover {1..5}:
  {{1}, {2}, {3}, {4}, {5}};
  {{1, 2}, {2, 3}, {3, 4}, {4, 5}};
  {{1, 3}, {2, 4}, {3, 5}};
  {{1, 2, 4}, {1, 3, 4}, {2, 3, 5}, {2, 4, 5}};
  {{1, 2, 3}, {2, 3, 4}, {3, 4, 5}};
  {{1, 2, 3, 5}, {1, 3, 4, 5}};
  {{1, 2, 3, 4}, {2, 3, 4, 5}};
  {{1, 2, 3, 4, 5}}.

Cf. A079500 (if only translations allowed).

Cf. A096202, A096203, A329235.

A369115 Expansion of (1 - x)^(-2) * Sum_{j>=0} (x^j / (1 - Sum_{k=1..j} x^k)).

Original entry on oeis.org

1, 3, 7, 14, 26, 46, 80, 138, 239, 417, 735, 1309, 2355, 4275, 7823, 14416, 26728, 49820, 93300, 175454, 331170, 627154, 1191204, 2268604, 4330915, 8286101, 15884857, 30507175, 58686513, 113066033, 218137531, 421391695, 814999229, 1578000229, 3058458885, 5933549906

Offset: 0

Peter Luschny, Jan 21 2024

nonn

Considering more generally the family of generating functions (1 - x)^n * Sum_{j>=0} (x^j / (1 - Sum_{k=1..j} x^k)) one finds several sequences related to compositions as indicated in the cross-references.

Cf. This sequence (n=-2), A186537 left shifted (n=-1), A079500 (n=0), A368279 (n=1), A369116 (n=2).

gf := (1 - x)^(-2) * add(x^j / (1 - add(x^k, k = 1..j)), j = 0..42):
ser := series(gf, x, 40): seq(coeff(ser, x, k), k = 0..38);

Partial sums of A186537 starting at n = 1.

A369116 Expansion of (1 - x)^2 * Sum_{j>=0} (x^j / (1 - Sum_{k=1..j} x^k)).

Original entry on oeis.org

1, -1, 1, 0, 1, 1, 3, 4, 9, 15, 29, 53, 100, 186, 352, 663, 1257, 2387, 4547, 8678, 16602, 31818, 61092, 117486, 226277, 436403, 842731, 1629297, 3153466, 6109704, 11848634, 22998892, 44680016, 86869392, 169024094, 329110519, 641254825, 1250261783, 2439155631

Offset: 0

Peter Luschny, Jan 21 2024

sign

Considering more generally the family of generating functions (1 - x)^n * Sum_{j>=0} (x^j / (1 - Sum_{k=1..j} x^k)) one finds several sequences related to compositions as indicated in the cross-references.

Cf. A369115 (n=-2), A186537 left shifted (n=-1), A079500 (n=0), A368279 (n=1), this sequence (n=2).

gf := (1 - x)^2 * add(x^j / (1 - add(x^k, k = 1..j)), j = 0..42):
ser := series(gf, x, 40): seq(coeff(ser, x, k), k = 0..38);

a(n) = A368279(n) - A368279(n-1) where A368279(-1) = 0.

A067150 Number of integers i=1,2,...,n such that (n,i) has Property F3, i.e., i and n are consecutive terms of a sequence b(k) satisfying b(1)=1, b(n) = (b(n-1) OR 2b(n-1)) + b(n-2), where the OR is taken bitwise.

Original entry on oeis.org

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

Offset: 1

John W. Layman, Jan 05 2002

nonn

Surprisingly, for k > 0, we find that a(2^k) = F(k-1), where {F(n)} is the sequence of Fibonacci numbers (A000045). Also, except for n = 2^3 = 8, these values are exactly those where new records in a(n) are made.
The definition can be restated as follows: a(n) is the number of integers i, 0 < i < n such that i and n are consecutive terms of some sequence b(k) satisfying b(1)=1 and b(k) = 3#b(k-1) + b(k-2), where # denotes OR-numbral multiplication (see A048888 for the definition).
If the OR-numbral multiplier 3 in the definition is replaced by 7, the resulting sequence has as record values the tribonacci numbers in A000073.

A. Frosini and S. Rinaldi, On the Sequence A079500 and Its Combinatorial Interpretations, J. Integer Seq., Vol. 9 (2006), Article 06.3.1.

Cf. A000045, A000073, A067148.

A188524 In base-2 lunar arithmetic, out of all odd numbers of length n, it appears that 111..1 (with n ones) has the most lunar divisors; the sequence gives the number of lunar divisors of the runner-up.

Original entry on oeis.org

2, 2, 4, 4, 6, 10, 16, 31, 55, 100, 185, 345, 644, 1209, 2274, 4298, 8145, 15469, 29454, 56213, 107489, 205925, 395190, 759621, 1462282, 2818799, 5440705, 10513994, 20340794, 39393580, 76368240, 148185145, 287791544, 559386196, 1088144064, 2118283567, 4126561528, 8044217224

Offset: 3

N. J. A. Sloane, Apr 16 2011

			For n = 3, 4, 5 the runner-ups are 101, 1101 or 1011, 11011; thereafter they appear to be the numbers 111...101 or their reversals (see A188288).

D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A079500, A188288.

A224960 Number of compositions [p(1), p(2), ..., p(k)] of n such that p(j) >= p(1) - 1.

Original entry on oeis.org

1, 1, 2, 4, 7, 14, 26, 52, 101, 200, 396, 787, 1564, 3117, 6214, 12398, 24749, 49427, 98740, 197303, 394323, 788201, 1575695, 3150265, 6298732, 12594595, 25184598, 50361842, 100711888, 201404839, 402779246, 805509560, 1610940381, 3221753990

Offset: 0

Joerg Arndt, Apr 21 2013

nonn

			The a(5) = 14 such compositions of 5 are
01:  [ 1 1 1 1 1 ]
02:  [ 1 1 1 2 ]
03:  [ 1 1 2 1 ]
04:  [ 1 1 3 ]
05:  [ 1 2 1 1 ]
06:  [ 1 2 2 ]
07:  [ 1 3 1 ]
08:  [ 1 4 ]
09:  [ 2 1 1 1 ]
10:  [ 2 1 2 ]
11:  [ 2 2 1 ]
12:  [ 2 3 ]
13:  [ 3 2 ]
14:  [ 5 ]
(the two forbidden compositions are [ 3 1 1 ] and [ 4 1 ]).

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

Cf. A171682 (compositions such that p(j) >= p(1)).
Cf. A079501 (compositions such that p(j) > p(1)).
Cf. A048888 (compositions such that p(j) <= p(1) + 1).
Cf. A007059 (compositions such that p(j) < p(1)).
Cf. A079500 (compositions such that p(j) <= p(1)).

b:= proc(n, i) option remember; `if`(n=0, 1, add(b(n-j,
      `if`(i=0, max(1, j-1), i)), j=`if`(i=0, 1, i)..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, May 02 2013

b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[b[n - j, If[i == 0, Max[1, j - 1], i]], {j, If[i == 0, 1, i], n}]];
a[n_] := b[n, 0];
a /@ Range[0, 50] (* Jean-François Alcover, Dec 20 2020, after Alois P. Heinz *)

a(n) ~ 3 * 2^(n-3). - Vaclav Kotesovec, May 01 2014

A238345 Triangle T(n,k) read by rows: T(n,k) is the number of compositions of n where the k-th part is the first occurrence of a largest part, n>=1, 1<=k<=n.

Original entry on oeis.org

1, 2, 0, 3, 1, 0, 5, 2, 1, 0, 8, 5, 2, 1, 0, 14, 9, 6, 2, 1, 0, 24, 18, 12, 7, 2, 1, 0, 43, 33, 25, 16, 8, 2, 1, 0, 77, 62, 49, 35, 21, 9, 2, 1, 0, 140, 115, 95, 73, 49, 27, 10, 2, 1, 0, 256, 215, 181, 148, 108, 68, 34, 11, 2, 1, 0, 472, 401, 346, 291, 230, 158, 93, 42, 12, 2, 1, 0, 874, 753, 657, 569, 470, 353, 228, 125, 51, 13, 2, 1, 0

Offset: 1

Joerg Arndt and Alois P. Heinz, Feb 25 2014

Column k=1: T(n,1) = A079500(n) = A007059(n+1).

Row sums are A011782.

			Triangle starts:
01:     1;
02:     2,    0;
03:     3,    1,    0;
04:     5,    2,    1,    0;
05:     8,    5,    2,    1,    0;
06:    14,    9,    6,    2,    1,    0;
07:    24,   18,   12,    7,    2,    1,    0;
08:    43,   33,   25,   16,    8,    2,    1,   0;
09:    77,   62,   49,   35,   21,    9,    2,   1,   0;
10:   140,  115,   95,   73,   49,   27,   10,   2,   1,   0;
11:   256,  215,  181,  148,  108,   68,   34,  11,   2,   1,  0;
12:   472,  401,  346,  291,  230,  158,   93,  42,  12,   2,  1,  0;
13:   874,  753,  657,  569,  470,  353,  228, 125,  51,  13,  2,  1, 0;
14:  1628, 1416, 1250, 1102,  943,  753,  533, 324, 165,  61, 14,  2, 1, 0;
15:  3045, 2673, 2380, 2126, 1866, 1558, 1188, 791, 453, 214, 72, 15, 2, 1, 0;
...

Joerg Arndt and Alois P. Heinz, Rows n = 1..141, flattened

g:= proc(n, m) option remember; `if`(n=0, 1,
       add(g(n-j, min(n-j, m)), j=1..min(n, m)))
    end:
h:= proc(n, t, m) option remember; `if`(n=0, 0,
      `if`(t=1, add(g(n-j, j), j=m+1..n),
       add(h(n-j, t-1, max(m, j)), j=1..n)))
    end:
T:= (n, k)-> h(n, k, 0):
seq(seq(T(n, k), k=1..n), n=1..15);

g[n_, m_] := g[n, m] = If[n == 0, 1, Sum[g[n-j, Min[n-j, m]], {j, 1, Min[n, m]}]]; h[n_, t_, m_] := h[n, t, m] = If[n == 0, 0, If[t == 1, Sum[g[n-j, j], {j, m+1, n}], Sum[h[n-j, t-1, Max[m, j]], {j, 1, n}]]]; T[n_, k_] := h[n, k, 0]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 15}] // Flatten (* Jean-François Alcover, Jan 12 2015, translated from Maple *)

A238346 Triangle T(n,k) read by rows: T(n,k) is the number of compositions of n where the k-th part is the last occurrence of a largest part, n>=1, 1<=k<=n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 5, 4, 1, 1, 8, 9, 8, 5, 1, 1, 14, 15, 15, 12, 6, 1, 1, 24, 27, 27, 24, 17, 7, 1, 1, 43, 47, 50, 46, 37, 23, 8, 1, 1, 77, 85, 90, 89, 75, 55, 30, 9, 1, 1, 140, 153, 165, 167, 152, 118, 79, 38, 10, 1, 1, 256, 279, 301, 313, 299, 250, 180, 110, 47, 11, 1, 1, 472, 511, 552, 582, 578, 516, 398, 267, 149, 57, 12, 1, 1

Offset: 1

Joerg Arndt and Alois P. Heinz, Feb 25 2014

Column k=1: T(n,1) = A079500(n-1) = A007059(n).

Row sums are A011782.

			Triangle starts:
01:  1,
02:  1, 1,
03:  2, 1, 1,
04:  3, 3, 1, 1,
05:  5, 5, 4, 1, 1,
06:  8, 9, 8, 5, 1, 1,
07:  14, 15, 15, 12, 6, 1, 1,
08:  24, 27, 27, 24, 17, 7, 1, 1,
09:  43, 47, 50, 46, 37, 23, 8, 1, 1,
10:  77, 85, 90, 89, 75, 55, 30, 9, 1, 1,
11:  140, 153, 165, 167, 152, 118, 79, 38, 10, 1, 1,
12:  256, 279, 301, 313, 299, 250, 180, 110, 47, 11, 1, 1,
13:  472, 511, 552, 582, 578, 516, 398, 267, 149, 57, 12, 1, 1,
...

Joerg Arndt, Table of n, a(n) for n = 1..465 (rows 1..30, flattened)

A245437 Expansion of x^5/(x^6-x^4-x^2-x+1).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 3, 6, 10, 17, 29, 50, 86, 147, 252, 432, 741, 1270, 2177, 3732, 6398, 10968, 18802, 32232, 55255, 94723, 162382, 278369, 477204, 818064, 1402395, 2404105, 4121322, 7065122, 12111635, 20762798, 35593360, 61017175, 104600848, 179315699

Offset: 0

Vincenzo Librandi, Jul 22 2014

G.f. taken from p. 12 of the Brlek et al. reference.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Srecko Brlek, Andrea Frosini, Simone Rinaldi, Laurent Vuillon, Tilings by translation: enumeration by a rational language approach, The Electronic Journal of Combinatorics, vol. 13 (2006).
Index entries for linear recurrences with constant coefficients, signature (1,1,0,1,0,-1).

Cf. A079500, A175331, A244521, A245185.

[n le 6 select Floor(n/6) else Self(n-1)+Self(n-2)+Self(n-4)-Self(n-6): n in [1..50]];

CoefficientList[Series[x^5/(x^6 - x^4 - x^2 - x + 1), {x, 0, 50}], x]
LinearRecurrence[{1, 1, 0, 1, 0, -1}, {0, 0, 0, 0, 0, 1}, 50] (* Bruno Berselli, Jul 22 2014 *)

G.f.: x^5/(x^6 - x^4 - x^2 - x + 1).

a(n) = a(n-1) + a(n-2) + a(n-4) - a(n-6) for n>5.

cp's OEIS Frontend

A188288 In lunar arithmetic in base 2, the number of divisors of the number 11...1101 (n digits, the binary expansion of 2^n-3).

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A329128 Number of nonequivalent sets whose translations and reflections cover {1..n}.

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A369115 Expansion of (1 - x)^(-2) * Sum_{j>=0} (x^j / (1 - Sum_{k=1..j} x^k)).

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A369116 Expansion of (1 - x)^2 * Sum_{j>=0} (x^j / (1 - Sum_{k=1..j} x^k)).

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A067150 Number of integers i=1,2,...,n such that (n,i) has Property F3*, i.e., i and n are consecutive terms of a sequence b(k) satisfying b(1)=1, b(n) = (b(n-1) OR 2*b(n-1)) + b(n-2), where the OR is taken bitwise.

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A188524 In base-2 lunar arithmetic, out of all odd numbers of length n, it appears that 111..1 (with n ones) has the most lunar divisors; the sequence gives the number of lunar divisors of the runner-up.

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A224960 Number of compositions [p(1), p(2), ..., p(k)] of n such that p(j) >= p(1) - 1.

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A238345 Triangle T(n,k) read by rows: T(n,k) is the number of compositions of n where the k-th part is the first occurrence of a largest part, n>=1, 1<=k<=n.

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Maple

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A238346 Triangle T(n,k) read by rows: T(n,k) is the number of compositions of n where the k-th part is the last occurrence of a largest part, n>=1, 1<=k<=n.

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A245437 Expansion of x^5/(x^6-x^4-x^2-x+1).

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A067150 Number of integers i=1,2,...,n such that (n,i) has Property F3, i.e., i and n are consecutive terms of a sequence b(k) satisfying b(1)=1, b(n) = (b(n-1) OR 2b(n-1)) + b(n-2), where the OR is taken bitwise.