A047967 -id:A047967 - cp's OEIS frontend

A261982 Number of compositions of n with some part repeated.

0, 0, 1, 1, 5, 11, 21, 51, 109, 229, 455, 959, 1947, 3963, 7999, 16033, 32333, 64919, 130221, 260967, 522733, 1045825, 2093855, 4189547, 8382315, 16768455, 33543127, 67093261, 134193413, 268404995, 536829045, 1073686083, 2147408773, 4294869253, 8589803783

Offset: 0

Alois P. Heinz, Sep 07 2015

nonn

Also compositions matching the pattern (1,1). - Gus Wiseman, Jun 23 2020

			a(2) = 1: 11.
a(3) = 1: 111.
a(4) = 5: 22, 211, 121, 112, 1111.

Alois P. Heinz, Table of n, a(n) for n = 0..3322
Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Row sums of A261981 and of A262191.
Cf. A262047.
The version for patterns is A019472.
The (1,1)-avoiding version is A032020.
The case of partitions is A047967.
(1,1,1)-matching compositions are counted by A335455.
Patterns matched by compositions are counted by A335456.
(1,1)-matching compositions are ranked by A335488.
Cf. A011782, A056986, A106356, A181796, A238279, A269134, A333755.

b:= proc(n, k) option remember; `if`(k<0 or n<0, 0,
      `if`(k=0, `if`(n=0, 1, 0), b(n-k, k) +k*b(n-k, k-1)))
    end:
a:= n-> ceil(2^(n-1))-add(b(n, k), k=0..floor((sqrt(8*n+1)-1)/2)):
seq(a(n), n=0..40);

b[n_, k_] := b[n, k] = If[k<0 || n<0, 0, If[k==0, If[n==0, 1, 0], b[n-k, k] + k*b[n-k, k-1]]]; a[n_] := Ceiling[2^(n-1)]-Sum[b[n, k], {k, 0, Floor[ (Sqrt[8n+1]-1)/2]}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 08 2017, translated from Maple *)
Table[Length[Join@@Permutations/@Select[IntegerPartitions[n],Length[#]>Length[Split[#]]&]],{n,0,10}] (* Gus Wiseman, Jun 24 2020 *)

a(n) = A011782(n) - A032020(n).

G.f.: (1 - x) / (1 - 2*x) - Sum_{k>=0} k! * x^(k*(k + 1)/2) / Product_{j=1..k} (1 - x^j). - Ilya Gutkovskiy, Jan 30 2020

A336866 Number of integer partitions of n without all distinct multiplicities.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 4, 5, 9, 15, 21, 28, 46, 56, 80, 114, 149, 192, 269, 337, 455, 584, 751, 943, 1234, 1527, 1944, 2422, 3042, 3739, 4699, 5722, 7100, 8668, 10634, 12880, 15790, 19012, 23093, 27776, 33528, 40102, 48264, 57469, 68793, 81727, 97372, 115227

Offset: 0

Gus Wiseman, Aug 09 2020

nonn

			The a(0) = 0 through a(9) = 15 partitions (empty columns shown as dots):
  .  .  .  (21)  (31)  (32)  (42)    (43)    (53)     (54)
                       (41)  (51)    (52)    (62)     (63)
                             (321)   (61)    (71)     (72)
                             (2211)  (421)   (431)    (81)
                                     (3211)  (521)    (432)
                                             (3221)   (531)
                                             (3311)   (621)
                                             (4211)   (3321)
                                             (32111)  (4221)
                                                      (4311)
                                                      (5211)
                                                      (32211)
                                                      (42111)
                                                      (222111)
                                                      (321111)

A098859 counts the complement.
A130092 gives the Heinz numbers of these partitions.
A001222 counts prime factors with multiplicity.
A013929 lists nonsquarefree numbers.
A047966 counts uniform partitions.
A047967 counts non-strict partitions.
A071625 counts distinct prime multiplicities.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A327498 gives the maximum divisor with distinct prime multiplicities.
Cf. A000009, A000041, A008284, A116608, A118914, A124010, A242882, A255231, A325280, A325242, A329739, A336422, A336571.

Table[Length[Select[IntegerPartitions[n],!UnsameQ@@Length/@Split[#]&]],{n,0,30}]

a(n) = A000041(n) - A098859(n).

A086543 Number of partitions of n with at least one odd part.

Original entry on oeis.org

0, 1, 1, 3, 3, 7, 8, 15, 17, 30, 35, 56, 66, 101, 120, 176, 209, 297, 355, 490, 585, 792, 946, 1255, 1498, 1958, 2335, 3010, 3583, 4565, 5428, 6842, 8118, 10143, 12013, 14883, 17592, 21637, 25525, 31185, 36711, 44583, 52382, 63261, 74173, 89134, 104303, 124754, 145698, 173525, 202268

Offset: 0

Vladeta Jovovic, Sep 10 2003

nonn

From Gus Wiseman, Oct 12 2023: (Start)
Also the number of integer partitions of n whose greatest part is not n/2, ranked by A366319. The a(1) = 1 through a(7) = 15 partitions are:
  (1)  (2)  (3)    (4)     (5)      (6)       (7)
            (21)   (31)    (32)     (42)      (43)
            (111)  (1111)  (41)     (51)      (52)
                           (221)    (222)     (61)
                           (311)    (411)     (322)
                           (2111)   (2211)    (331)
                           (11111)  (21111)   (421)
                                    (111111)  (511)
                                              (2221)
                                              (3211)
                                              (4111)
                                              (22111)
                                              (31111)
                                              (211111)
                                              (1111111)
Compare to the a(1) = 1 through a(7) = 15 partitions with at least one odd part, ranked by A366322:
  (1)  (11)  (3)    (31)    (5)      (33)      (7)
             (21)   (211)   (32)     (51)      (43)
             (111)  (1111)  (41)     (321)     (52)
                            (221)    (411)     (61)
                            (311)    (2211)    (322)
                            (2111)   (3111)    (331)
                            (11111)  (21111)   (421)
                                     (111111)  (511)
                                               (2221)
                                               (3211)
                                               (4111)
                                               (22111)
                                               (31111)
                                               (211111)
                                               (1111111)
(End)

			a(4)=3 because we have [3,1],[2,1,1] and [1,1,1] ([4] and [2,2] do not qualify).

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A038348, A047967.
The complement is counted by A035363, ranks A344415.
These partitions have ranks A366322.
A025065 counts partitions with sum <= twice length, ranks A344296.
A110618 counts partitions with sum >= twice maximum, ranks A344291.
Cf. A238628, A257991, A300061, A320924, A340387, A344414, A366319.

g:=sum(x^(2*k-1)/product(1-x^j,j=1..2*k-1)/product(1-x^(2*j),j=k..70),k=1..70): gser:=series(g,x=0,50): seq(coeff(gser,x,n),n=0..45); # Emeric Deutsch, Mar 30 2006

nn=50;CoefficientList[Series[Sum[x^(2k-1)/Product[1-x^j,{j,1,2k-1}] /Product[(1-x^(2j)),{j,k,nn}],{k,1,nn}],{x,0,nn}],x] (* Geoffrey Critzer, Sep 28 2013 *)
Table[Length[Select[IntegerPartitions[n],Max[#]!=n/2&]],{n,0,30}] (* Gus Wiseman, Oct 12 2023 *)

x='x+O('x^66); concat([0], Vec(1/eta(x)-1/eta(x^2)) ) \\ Joerg Arndt, May 04 2013

A000041(n) if n is odd; otherwise, A000041(n) - A000041(n/2).
G.f.: Sum_{k>=1} x^(2k-1)/((Product_{j=1..2k-1} (1-x^j))*(Product_{j>=k} (1-x^(2j)))). - Emeric Deutsch, Mar 30 2006
G.f.: 1/E(x) - 1/E(x^2) where E(x) = prod(n>=1, 1-x^n ); see Pari code. - Joerg Arndt, May 04 2013

A364913 Number of integer partitions of n having a part that can be written as a nonnegative linear combination of the other (possibly equal) parts.

Original entry on oeis.org

0, 0, 1, 2, 4, 5, 10, 12, 20, 27, 39, 51, 74, 95, 130, 169, 225, 288, 378, 479, 617, 778, 990, 1239, 1560, 1938, 2419, 2986, 3696, 4538, 5575, 6810, 8319, 10102, 12274, 14834, 17932, 21587, 25963, 31120, 37275, 44513, 53097, 63181, 75092, 89030, 105460, 124647

Offset: 0

Gus Wiseman, Aug 20 2023

nonn

Includes all non-strict partitions (A047967).

			The a(0) = 0 through a(7) = 12 partitions:
  .  .  (11)  (21)   (22)    (41)     (33)      (61)
              (111)  (31)    (221)    (42)      (322)
                     (211)   (311)    (51)      (331)
                     (1111)  (2111)   (222)     (421)
                             (11111)  (321)     (511)
                                      (411)     (2221)
                                      (2211)    (3211)
                                      (3111)    (4111)
                                      (21111)   (22111)
                                      (111111)  (31111)
                                                (211111)
                                                (1111111)
The partition (5,4,3) has no part that can be written as a nonnegative linear combination of the others, so is not counted under a(12).
The partition (6,4,3,2) has 6 = 4+2, or 6 = 3+3, or 6 = 2+2+2, or 4 = 2+2, so is counted under a(15).

The strict case is A364839.
For sums instead of combinations we have A364272, binary A364670.
The complement in strict partitions is A364350.
For subsets instead of partitions we have A364914, complement A326083.
Allowing equal parts gives A365068, complement A364915.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A116861 and A364916 count linear combinations of strict partitions.
A365006 = no strict partitions w/ pos linear combination.
Cf. A085489, A151897, A236912, A237113, A237667, A275972, A364346.

combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[IntegerPartitions[n],!UnsameQ@@#||Or@@Table[combs[#[[k]],Delete[#,k]]!={},{k,Length[#]}]&]],{n,0,15}]

a(n) + A364915(n) = A000041(n).

A365924 Number of incomplete integer partitions of n, meaning not every number from 0 to n is the sum of some submultiset.

Original entry on oeis.org

0, 0, 1, 1, 3, 3, 6, 7, 12, 14, 22, 25, 38, 46, 64, 76, 106, 124, 167, 199, 261, 309, 402, 471, 604, 714, 898, 1053, 1323, 1542, 1911, 2237, 2745, 3201, 3913, 4536, 5506, 6402, 7706, 8918, 10719, 12364, 14760, 17045, 20234, 23296, 27600, 31678, 37365, 42910, 50371, 57695, 67628, 77300, 90242, 103131, 119997

Offset: 0

Gus Wiseman, Sep 26 2023

nonn

The complement (complete partitions) is A126796.

			The a(0) = 0 through a(8) = 12 partitions:
  .  .  (2)  (3)  (4)    (5)    (6)      (7)      (8)
                  (2,2)  (3,2)  (3,3)    (4,3)    (4,4)
                  (3,1)  (4,1)  (4,2)    (5,2)    (5,3)
                                (5,1)    (6,1)    (6,2)
                                (2,2,2)  (3,2,2)  (7,1)
                                (4,1,1)  (3,3,1)  (3,3,2)
                                         (5,1,1)  (4,2,2)
                                                  (4,3,1)
                                                  (5,2,1)
                                                  (6,1,1)
                                                  (2,2,2,2)
                                                  (5,1,1,1)

Joerg Arndt, Table of n, a(n) for n = 0..10000

For parts instead of sums we have A047967/A365919, ranks A080259/A055932.
The complement is A126796, ranks A325781, strict A188431.
These partitions have ranks A365830.
The strict case is A365831.
Row sums of A365923 without the first column, strict A365545.
A000041 counts integer partitions, strict A000009.
A046663 counts partitions w/o a submultiset summing to k, strict A365663.
A276024 counts positive subset-sums of partitions, strict A284640.
A325799 counts non-subset-sums of prime indices.
A364350 counts combination-free strict partitions.
A365543 counts partitions with a submultiset summing to k, strict A365661.
Cf. A002865, A006827, A018818, A264401, A299701, A304792, A364272, A365658, A365918, A365921.

nmz[y_]:=Complement[Range[Total[y]],Total/@Subsets[y]];
Table[Length[Select[IntegerPartitions[n],Length[nmz[#]]>0&]],{n,0,15}]

a(n) = A000041(n) - A126796(n).

A365831 Number of incomplete strict integer partitions of n, meaning not every number from 0 to n is the sum of some submultiset.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 3, 4, 6, 8, 9, 11, 13, 16, 21, 25, 31, 36, 43, 50, 59, 69, 82, 96, 113, 131, 155, 179, 208, 239, 276, 315, 362, 414, 472, 539, 614, 698, 795, 902, 1023, 1158, 1311, 1479, 1672, 1881, 2118, 2377, 2671, 2991, 3354, 3748, 4194, 4679, 5223, 5815

Offset: 0

Gus Wiseman, Sep 28 2023

nonn

			The strict partition (14,5,4,2,1) has no subset summing to 13 so is counted under a(26).
The a(2) = 1 through a(10) = 9 strict partitions:
  (2)  (3)  (4)    (5)    (6)    (7)    (8)      (9)      (10)
            (3,1)  (3,2)  (4,2)  (4,3)  (5,3)    (5,4)    (6,4)
                   (4,1)  (5,1)  (5,2)  (6,2)    (6,3)    (7,3)
                                 (6,1)  (7,1)    (7,2)    (8,2)
                                        (4,3,1)  (8,1)    (9,1)
                                        (5,2,1)  (4,3,2)  (5,3,2)
                                                 (5,3,1)  (5,4,1)
                                                 (6,2,1)  (6,3,1)
                                                          (7,2,1)

For parts instead of sums we have ranks A080259, A055932.
The strict complement is A188431, non-strict A126796 (ranks A325781).
Row sums of A365545 without the first column, non-strict A365923.
The non-strict version is A365924, ranks A365830.
A000041 counts integer partitions, strict A000009.
A046663 counts partitions w/o a submultiset summing to k, strict A365663.
A276024 counts positive subset-sums of partitions, strict A284640.
A325799 counts non-subset-sums of prime indices.
A365543 counts partitions with a submultiset summing to k, strict A365661.
Cf. A006827, A047967, A299701, A304792, A364350, A365658, A365918, A365921.

nmz[y_]:=Complement[Range[Total[y]], Total/@Subsets[y]];
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Length[nmz[#]]>0&]],{n,0,15}]

A090858 Number of partitions of n such that there is exactly one part which occurs twice, while all other parts occur only once.

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 2, 4, 6, 7, 8, 13, 15, 21, 25, 30, 39, 50, 58, 74, 89, 105, 129, 156, 185, 221, 264, 309, 366, 433, 505, 593, 696, 805, 941, 1090, 1258, 1458, 1684, 1933, 2225, 2555, 2922, 3346, 3823, 4349, 4961, 5644, 6402, 7267, 8234, 9309, 10525, 11886, 13393

Offset: 0

Vladeta Jovovic, Feb 12 2004

Number of solutions (p(1),p(2),...,p(n)), p(i)>=0,i=1..n, to p(1)+2*p(2)+...+n*p(n)=n such that |{i: p(i)<>0}| = p(1)+p(2)+...+p(n)-1.

Also number of partitions of n such that if k is the largest part, then, with exactly one exception, all the integers 1,2,...,k occur as parts. Example: a(7)=4 because we have [4,2,1], [3,3,1], [3,2,2] and [3,1,1,1,1]. - Emeric Deutsch, Apr 18 2006

			a(7) = 4 because we have 4 such partitions of 7: [1,1,2,3], [1,1,5], [2,2,3], [1,3,3].
From _Gus Wiseman_, Apr 19 2019: (Start)
The a(2) = 1 through a(11) = 13 partitions described in the name are the following (empty columns not shown). The Heinz numbers of these partitions are given by A060687.
  (11)  (22)   (221)  (33)   (322)   (44)    (441)   (55)    (443)
        (211)  (311)  (411)  (331)   (332)   (522)   (433)   (533)
                             (511)   (422)   (711)   (442)   (551)
                             (3211)  (611)   (3321)  (622)   (722)
                                     (3221)  (4221)  (811)   (911)
                                     (4211)  (4311)  (5221)  (4322)
                                             (5211)  (5311)  (4331)
                                                     (6211)  (4421)
                                                             (5411)
                                                             (6221)
                                                             (6311)
                                                             (7211)
                                                             (43211)
The a(2) = 1 through a(10) = 8 partitions described in Emeric Deutsch's comment are the following (empty columns not shown). The Heinz numbers of these partitions are given by A325284.
  (2)  (22)  (32)   (222)   (322)    (332)     (432)      (3322)
       (31)  (311)  (3111)  (331)    (431)     (3222)     (3331)
                            (421)    (2222)    (4221)     (22222)
                            (31111)  (3311)    (4311)     (42211)
                                     (4211)    (33111)    (43111)
                                     (311111)  (42111)    (331111)
                                               (3111111)  (421111)
                                                          (31111111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A047967, A265251.
Column k=2 of A266477.
Cf. A008284, A046660, A060687, A116608, A117571, A127002, A325241, A325244.

g:=sum(x^(k*(k+1)/2)*((1-x^k)/x^(k-1)/(1-x)-k)/product(1-x^i,i=1..k),k=1..15): gser:=series(g,x=0,64): seq(coeff(gser,x,n),n=1..54); # Emeric Deutsch, Apr 18 2006
# second Maple program:
b:= proc(n, i, t) option remember; `if`(n>i*(i+3-2*t)/2, 0,
     `if`(n=0, t, b(n, i-1, t)+`if`(i>n, 0, b(n-i, i-1, t)+
     `if`(t=1 or 2*i>n, 0, b(n-2*i, i-1, 1)))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 28 2015

b[n_, i_, t_] := b[n, i, t] = If[n > i*(i + 3 - 2*t)/2, 0, If[n == 0, t, b[n, i - 1, t] + If[i > n, 0,  b[n - i, i - 1, t] + If[t == 1 || 2*i > n, 0, b[n - 2*i, i - 1, 1]]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 100} ] (* Jean-François Alcover, Jan 20 2016, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],Length[#]-Length[Union[#]]==1&]],{n,0,30}] (* Gus Wiseman, Apr 19 2019 *)

alist(n)=concat([0,0],Vec(sum(k=1,n\2,(x^(2*k)+x*O(x^n))/(1+x^k)*prod(j=1,n-2*k,1+x^j+x*O(x^n))))) \\ Franklin T. Adams-Watters, Nov 02 2015

G.f.: Sum_{k>0} x^(2*k)/(1+x^k) * Product_{k>0} (1+x^k). Convolution of 1-A048272(n) and A000009(n). a(n) = A036469(n) - A015723(n).
G.f.: sum(x^(k(k+1)/2)[(1-x^k)/x^(k-1)/(1-x)-k]/product(1-x^i,i=1..k), k=1..infinity). - Emeric Deutsch, Apr 18 2006
a(n) ~ c * exp(Pi*sqrt(n/3)) / n^(1/4), where c = 3^(1/4) * (1 - log(2)) / (2*Pi) = 0.064273294789... - Vaclav Kotesovec, May 24 2018

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 26 2004

a(0) added by Franklin T. Adams-Watters, Nov 02 2015

A140106 Number of noncongruent diagonals in a regular n-gon.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35, 36, 36, 37

Offset: 1

Andrew McFarland, Jun 03 2008

Number of double-stars (diameter 3 trees) with n nodes. For n >= 3, number of partitions of n-2 into two parts. - Washington Bomfim, Feb 12 2011
Number of roots of the n-th Bernoulli polynomial in the left half-plane. - Michel Lagneau, Nov 08 2012
From Gus Wiseman, Oct 17 2020: (Start)
Also the number of 3-part non-strict integer partitions of n - 1. The Heinz numbers of these partitions are given by A285508. The version for partitions of any length is A047967, with Heinz numbers A013929. The a(4) = 1 through a(15) = 6 partitions are (A = 10, B = 11, C = 12):
  111  211  221  222  322  332  333  433  443  444  544  554
            311  411  331  422  441  442  533  552  553  644
                      511  611  522  622  551  633  661  662
                                711  811  722  822  733  833
                                          911  A11  922  A22
                                                    B11  C11
(End)

			The square (n=4) has two congruent diagonals; so a(4)=1. The regular pentagon also has congruent diagonals; so a(5)=1. Among all the diagonals in a regular hexagon, there are two noncongruent ones; hence a(6)=2, etc.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Washington Bomfim, Double-star corresponding to the partition [3,7]
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).
Index entries for sequences related to trees

Cf. A000554, A007304, A007997, A013929, A022998.
Cf. A047967, A129194, A235451, A285508, A321773.
A001399(n-3) = A069905(n) = A211540(n+2) counts 3-part partitions.
Essentially the same as A004526.

A140106:= func< n | n eq 1 select 0 else Floor((n-2)/2) >;
[A140106(n): n in [1..80]]; // G. C. Greubel, Feb 10 2023

with(numtheory): for n from 1 to 80 do:it:=0:
y:=[fsolve(bernoulli(n,x) , x, complex)] : for m from 1 to nops(y) do : if Re(y[m])<0 then it:=it+1:else fi:od: printf(`%d, `,it):od:

a[1]=0; a[n_?OddQ] := (n-3)/2; a[n_] := n/2-1; Array[a, 100] (* Jean-François Alcover, Nov 17 2015 *)

a(n)=if(n>1,n\2-1,0) \\ Charles R Greathouse IV, Oct 16 2015

def A140106(n): return n-2>>1 if n>1 else 0 # Chai Wah Wu, Sep 18 2023

def A140106(n): return 0 if (n==1) else (n-2)//2
[A140106(n) for n in range(1,81)] # G. C. Greubel, Feb 10 2023

a(n) = floor((n-2)/2), for n > 1, otherwise 0. - Washington Bomfim, Feb 12 2011
G.f.: x^4/(1-x-x^2+x^3). - Colin Barker, Jan 31 2012
a(n) = floor(A129194(n-1)/A022998(n)), for n > 1. - Paul Curtz, Jul 23 2017
a(n) = A001399(n-3) - A001399(n-6). Compare to A007997(n) = A001399(n-3) + A001399(n-6). - Gus Wiseman, Oct 17 2020

More terms from Joseph Myers, Sep 05 2009

A027934 a(0)=0, a(1)=1, a(2)=2; for n > 2, a(n) = 3a(n-1) - a(n-2) - 2a(n-3).

Original entry on oeis.org

0, 1, 2, 5, 11, 24, 51, 107, 222, 457, 935, 1904, 3863, 7815, 15774, 31781, 63939, 128488, 257963, 517523, 1037630, 2079441, 4165647, 8342240, 16702191, 33433039, 66912446, 133899917, 267921227, 536038872, 1072395555, 2145305339

Offset: 0

Clark Kimberling

Number of compositions of n with at least one even part (offset 2). - Vladeta Jovovic, Dec 29 2004
First differences of A008466. a(n) = A008466(n+2) - A008466(n+1). - Alexander Adamchuk, Apr 06 2006
Starting with "1" = eigensequence of a triangle with the Fibonacci series as the left border and the rest 1's. - Gary W. Adamson, Jul 24 2010
An elephant sequence, see A175654. For the corner squares 24 A[5] vectors, with decimal values between 11 and 416, lead to this sequence (without the leading 0). For the central square these vectors lead to the companion sequence A099036 (without the first leading 1). - Johannes W. Meijer, Aug 15 2010
a(n) = Sum_{k=1..n} A108617(n,k) / 2. - Reinhard Zumkeller, Oct 07 2012
a(n) is the number of binary strings that contain the substring 11 or end in 1. a(3) = 5 because we have: 001, 011, 101, 110, 111. - Geoffrey Critzer, Jan 04 2014
a(n-1), n >= 1, is the number of nonexisting (due to the maturation delay) "[male-female] pairs of Fibonacci rabbits" at the beginning of the n-th month. - Daniel Forgues, May 06 2015
a(n-1) is the number of subsets of {1,2,..,n} that contain n that have at least one pair of consecutive integers. For example, for n=5, a(4) = 11 and the 11 subsets are {4,5}, {1,2,5}, {1,4,5}, {2,3,5}, {2,4,5}, {3,4,5}, {1,2,3,5}, {1,2,4,5}, {1,3,4,5}, {2,3,4,5}, {1,2,3,4,5}.  Note that A008466(n) is the number of all subsets of {1,2,..,n} that have at least one pair of consecutive integers. - Enrique Navarrete, Aug 15 2020

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Patrick Letendre, Polynomials with integer roots, arXiv:1911.00480 [math.NT], 2019. See p. 4.
Toufik Mansour and Mark Shattuck, Pattern avoidance in flattened derangements, Disc. Math. Lett. (2025) Vol. 15, 67-74. See p. 74.
Augustine O. Munagi, Integer Compositions and Higher-Order Conjugation, J. Int. Seq., Vol. 21 (2018), Article 18.8.5.
Denis Neiter and Amsha Proag, Links Between Sums Over Paths in Bernoulli's Triangles and the Fibonacci Numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.8.3.
OEIS Wiki, Fibonacci rabbits
Index entries for linear recurrences with constant coefficients, signature (3,-1,-2).

Row sums of triangle A131767. - Gary W. Adamson, Jul 13 2007
a(n) = A101220(1, 2, n+1).
T(n, n) + T(n, n+1) + ... + T(n, 2n), T given by A027926.
Diagonal sums of A055248.
Cf. A000045, A000079, A008466, A059570, A099036, A047967, A000032.

List([0..35], n-> 2^n - Fibonacci(n+1) ); # G. C. Greubel, Sep 27 2019

a027934 n = a027934_list !! n
a027934_list = 0 : 1 : 2 : zipWith3 (\x y z -> 3 * x - y - 2 * z)
               (drop 2 a027934_list) (tail a027934_list) a027934_list
-- Reinhard Zumkeller, Oct 07 2012

[2^n - Fibonacci(n+1): n in [0..35]]; // G. C. Greubel, Sep 27 2019

A027934:= proc(n) local K; K:= Matrix ([[2,0,0], [0,1,1], [0,1,0]])^n; K[1,1]-K[2,2] end: seq (A027934(n), n=0..31); # Alois P. Heinz, Jul 28 2008
a := n -> 2^n - combinat:-fibonacci(n+1): seq(a(n),n=0..31); # Peter Luschny, May 09 2015

nn=31; a:=1/(1-x-x^2); b:=1/(1-2x); CoefficientList[Series[a*x*(1+x*b), {x,0,nn}], x] (* Geoffrey Critzer, Jan 04 2014 *)
LinearRecurrence[{3,-1,-2}, {0,1,2}, 32] (* Jean-François Alcover, Jan 09 2016 *)
nxt[{a_,b_,c_}]:={b,c,3c-b-2a}; NestList[nxt,{0,1,2},40][[;;,1]] (* Harvey P. Dale, Feb 02 2025 *)

a(n)=2^n-fibonacci(n+1) \\ Charles R Greathouse IV, Jun 11 2015

[2^n - fibonacci(n+1) for n in (0..35)] # G. C. Greubel, Sep 27 2019

a(n) = Sum_{j=0..floor(n/2)} Sum_{k=0..n-2*j} binomial(n-j, n-2*j-k). - Paul Barry, Feb 07 2003
From Paul Barry, Jan 23 2004: (Start)
Row sums of A105809.
G.f.: x*(1-x)/((1-2*x)*(1-x-x^2)).
a(n) = 2^n - Fibonacci(n+1). (End) - corrected Apr 06 2006 and Oct 05 2012
a(n) = Sum_{j=0..n} Sum_{k=0..n} binomial(n-k, k+j). - Paul Barry, Aug 29 2004
a(n) = (Sum of (n+1)-th row of the triangle in A108617) / 2. - Reinhard Zumkeller, Jun 12 2005
a(n) = term (1,1) - term (2,2) in the 3 X 3 matrix [2,0,0; 0,1,1; 0,1,0]^n. - Alois P. Heinz, Jul 28 2008
a(n) = 2^n - A000045(n+1). - Geoffrey Critzer, Jan 04 2014
a(n) ~ 2^n. - Daniel Forgues, May 06 2015
From Bob Selcoe, Mar 29 2016: (Start)
a(n) = 2*a(n-1) + A000045(n-2).
a(n) = 4*a(n-2) + A000032(n-2). (End)
a(n) = 2^(n-1) - ( ((1+sqrt(5))/2)^n - ((1-sqrt(5))/2)^n)/sqrt(5). - Haider Ali Abdel-Abbas, Aug 17 2019

Simpler definition from Miklos Kristof, Nov 24 2003
Initial zero added by N. J. A. Sloane, Feb 13 2008
Definition fixed by Reinhard Zumkeller, Oct 07 2012

A353863 Number of integer partitions of n whose weak run-sums cover an initial interval of nonnegative integers.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 6, 7, 10, 11, 16, 20, 24, 30, 43, 47, 62, 79, 94, 113, 143, 170, 211, 256, 307, 372, 449, 531, 648, 779, 926, 1100, 1323, 1562, 1864, 2190, 2595, 3053, 3611, 4242, 4977, 5834, 6825, 7973, 9344, 10844, 12641, 14699, 17072, 19822

Offset: 0

Gus Wiseman, Jun 04 2022

nonn

A weak run-sum of a sequence is the sum of any consecutive constant subsequence. For example, the weak run-sums of (3,2,2,1) are {1,2,3,4}.

This is a kind of completeness property, cf. A126796.

			The a(1) = 1 through a(8) = 7 partitions:
  (1)  (11)  (21)   (211)   (311)    (321)     (3211)     (3221)
             (111)  (1111)  (2111)   (3111)    (4111)     (32111)
                            (11111)  (21111)   (22111)    (41111)
                                     (111111)  (31111)    (221111)
                                               (211111)   (311111)
                                               (1111111)  (2111111)
                                                          (11111111)

For parts instead of weak run-sums we have A000009.
For multiplicities instead of weak run-sums we have A317081.
If weak run-sums are distinct we have A353865, the completion of A353864.
A003242 counts anti-run compositions, ranked by A333489, complement A261983.
A005811 counts runs in binary expansion.
A165413 counts distinct run-lengths in binary expansion, sums A353929.
A300273 ranks collapsible partitions, counted by A275870, comps A353860.
A353832 represents taking run-sums of a partition, compositions A353847.
A353833 ranks partitions with all equal run-sums, counted by A304442.
A353835 counts distinct run-sums of prime indices.
A353837 counts partitions with distinct run-sums, ranked by A353838.
A353840-A353846 pertain to partition run-sum trajectory.
A353861 counts distinct weak run-sums of prime indices.
A353932 lists run-sums of standard compositions.
Rulers: A103295, A103300, A169942, A325768.
Complete: A002033, A325780, A126796, A276024, A325781, A188431, A353866.
Cf. A047967, A073093, A181819, A237685, A353844, A353867, A353930.

normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
msubs[s_]:=Join@@@Tuples[Table[Take[t,i],{t,Split[s]},{i,0,Length[t]}]];
wkrs[y_]:=Union[Total/@Select[msubs[y],SameQ@@#&]];
Table[Length[Select[IntegerPartitions[n],normQ[Rest[wkrs[#]]]&]],{n,0,15}]

\\ isok(p) tests the partition.
isok(p)={my(b=0, s=0, t=0); for(i=1, #p, if(p[i]<>t, t=p[i]; s=0); s += t; b = bitor(b, 1<<(s-1))); bitand(b,b+1)==0}
a(n) = {my(r=0); forpart(p=n, r+=isok(p)); r} \\ Andrew Howroyd, Jan 15 2024

a(31) onwards from Andrew Howroyd, Jan 15 2024

cp's OEIS Frontend

A261982 Number of compositions of n with some part repeated.

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A336866 Number of integer partitions of n without all distinct multiplicities.

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A086543 Number of partitions of n with at least one odd part.

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A364913 Number of integer partitions of n having a part that can be written as a nonnegative linear combination of the other (possibly equal) parts.

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A365924 Number of incomplete integer partitions of n, meaning not every number from 0 to n is the sum of some submultiset.

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A365831 Number of incomplete strict integer partitions of n, meaning not every number from 0 to n is the sum of some submultiset.

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A090858 Number of partitions of n such that there is exactly one part which occurs twice, while all other parts occur only once.

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A140106 Number of noncongruent diagonals in a regular n-gon.

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A027934 a(0)=0, a(1)=1, a(2)=2; for n > 2, a(n) = 3a(n-1) - a(n-2) - 2a(n-3).