A033485 -id:A033485 - cp's OEIS frontend

A003156 A self-generating sequence (see Comments for definition).

1, 4, 5, 6, 9, 12, 15, 16, 17, 20, 21, 22, 25, 26, 27, 30, 33, 36, 37, 38, 41, 44, 47, 48, 49, 52, 55, 58, 59, 60, 63, 64, 65, 68, 69, 70, 73, 76, 79, 80, 81, 84, 85, 86, 89, 90, 91, 94, 97, 100, 101, 102, 105, 106, 107, 110, 111, 112, 115, 118, 121, 122, 123, 126, 129, 132

Offset: 1

N. J. A. Sloane

nonn

From N. J. A. Sloane, Dec 26 2020: (Start)
The best definitions of the triple [this sequence, A003157, A003158] are as the rows a(n), b(n), c(n) of the table:
n: 1, 2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, ...
a: 1, 4,  5,  6,  9, 12, 15, 16, 17, 20, 21, 22, ...
b: 3, 8, 11, 14, 19, 24, 29, 32, 35, 40, 43, 46, ...
c: 2, 7, 10, 13, 18, 23, 28, 31, 34, 39, 42, 45, ...
where a(1)=1, b(1)=3, c(1)=2, and thereafter
a(n) = mex{a(i), b(i), c(i), ib(n) = a(n) + 2*n,
c(n) = b(n) - 1.
Then a,b,c form a partition of the positive integers.
Note that there is another triple of sequences (A003144, A003145, A003146) also called a, b, c and also a partition of the positive integers, in a different paper by the same authors (Carlitz-Scovelle-Hoggatt) in the same volume of the same journal.
(End)
a(n) is the number of ones before the n-th zero in the Feigenbaum sequence A035263. - Philippe Deléham, Mar 27 2004
Number of odd numbers before the n-th even number in A007413, A007913, A001511, A029883, A033485, A035263, A036585, A065882, A065883, A088172, A092412. - Philippe Deléham, Apr 03 2004
Indices of a in the sequence closed under a -> abc, b -> a, c -> a, starting with a(1) = a; see A092606 where a = 0, b = 2, c = 1. - Philippe Deléham, Apr 12 2004

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Jean-Paul Allouche, On the morphism 1 -> 121, 2 -> 12221, Preprint, 2024 [Local copy, with permission]. See also Optimization, Discrete Mathematics and Applications to Data Sciences, Springer Optimization and Its Applications, Springer, Cham, 2025, and author's site, CNRS France, 2024. See p. 6.
L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., Fibonacci representations of higher order, Fib. Quart., 10 (1972), 43-69. [Here a, b, c are A003144, A003145, A003146.]
L. Carlitz, R. Scoville, and V. E. Hoggatt, Jr., Representations for a special sequence, Fibonacci Quarterly 10.5 (1972), 499-518, 550. [A003156, A003157, A003158 are on page 500.]
Robbert Fokkink and Gandhar Joshi, On Cloitre's hiccup sequences, arXiv:2507.16956 [math.CO], 2025. See pp. 8, 17

Cf. A001511, A003157, A003158, A003159, A007413, A007913, A029883, A033485, A035263, A036554, A036585, A056196, A065882, A065883, A072939, A079523, A072939, A080426, A088172, A092412, A092606.

A102378 a(n) = a(n-1) + a([n/2]) + 1, a(1) = 1.

Original entry on oeis.org

1, 3, 5, 9, 13, 19, 25, 35, 45, 59, 73, 93, 113, 139, 165, 201, 237, 283, 329, 389, 449, 523, 597, 691, 785, 899, 1013, 1153, 1293, 1459, 1625, 1827, 2029, 2267, 2505, 2789, 3073, 3403, 3733, 4123, 4513, 4963, 5413, 5937, 6461, 7059, 7657, 8349

Offset: 1

Mitch Harris, Jan 05 2005

nonn

From Gus Wiseman, Mar 23 2019: (Start)
The offset could safely be changed to zero by setting the boundary condition to a(0) = 0.
Also the number of integer partitions of 2n into powers of 2 with at least one part > 1. The Heinz numbers of these partitions are given by A324927. For example, the a(1) = 1 through a(5) = 13 integer partitions are:
  (2)  (4)    (42)     (8)        (82)
       (22)   (222)    (44)       (442)
       (211)  (411)    (422)      (811)
              (2211)   (2222)     (4222)
              (21111)  (4211)     (4411)
                       (22211)    (22222)
                       (41111)    (42211)
                       (221111)   (222211)
                       (2111111)  (421111)
                                  (2221111)
                                  (4111111)
                                  (22111111)
                                  (211111111)
(End)

Cf. A000123, A033485.

Cf. A018819, A318400, A324927, A324928.

Table[Length[Select[IntegerPartitions[n],And[Max@@#>1,And@@IntegerQ/@Log[2,#]]&]],{n,0,30,2}] (* Gus Wiseman, Mar 23 2019 *)

from itertools import islice
from collections import deque
def A102378_gen(): # generator of terms
    aqueue, f, b, a = deque([2]), True, 1, 2
    yield from (1, 3)
    while True:
        a += b
        yield 2*a - 1
        aqueue.append(a)
        if f: b = aqueue.popleft()
        f = not f
A102378_list = list(islice(A102378_gen(),40)) # Chai Wah Wu, Jun 08 2022

a(n) - a(n-1) = A018819(n+1)
G.f. A(x) satisfies (1-x)*A(x) = 2(1 + x)*B(x^2), where B(x) is the gf of A033485
a(n) = A000123(n) - 1. - Gus Wiseman, Mar 23 2019
G.f. A(x) satisfies: A(x) = (x + (1 - x^2) * A(x^2)) / (1 - x)^2. - Ilya Gutkovskiy, Aug 11 2021

A091785 Evil numbers (see A001969) in A003159.

Original entry on oeis.org

3, 5, 9, 12, 15, 17, 20, 23, 27, 29, 33, 36, 39, 43, 45, 48, 51, 53, 57, 60, 63, 65, 68, 71, 75, 77, 80, 83, 85, 89, 92, 95, 99, 101, 105, 108, 111, 113, 116, 119, 123, 125, 129, 132, 135, 139, 141, 144, 147, 149, 153, 156, 159, 163, 165, 169, 172, 175, 177, 180, 183

Offset: 1

Philippe Deléham, Mar 16 2004

Also n such that A033485(n) == 3 (mod 4); see A007413.
Also n such that A029883(n-1) = -1, A036577(n-1) = 0, A036585(n-1) = 1. - Philippe Deléham, Mar 25 2004
The number of odd numbers before the n-th even number in this sequence is a(n). - Philippe Deléham, Mar 27 2004
Numbers n such that {A010060(n-1), A010060(n)}={1,0} where A010060 is the Thue-Morse sequence. - Benoit Cloitre, Jun 16 2006

G. C. Greubel, Table of n, a(n) for n = 1..10000
Aviezri S. Fraenkel, The vile, dopey, evil and odious game players, Discrete Math. 312 (2012), pp. 42-46.
Index entries for 2-automatic sequences.

A036554 := Select[Range[750], OddQ[IntegerExponent[#, 2]] &]; Table[ A036554[[2*n]]/2, {n, 1, 50}] (* G. C. Greubel, Jan 14 2018 *)

is(n)=hammingweight(n-1)%2 && hammingweight(n)%2==0 \\ Charles R Greathouse IV, Mar 26 2013

a(n) = A003159(2*n) = A036554(2*n)/2.
a(n) is asymptotic to 3*n. - Benoit Cloitre, Mar 22 2004
A050292(a(n)) = 2n. - Philippe Deléham, Mar 26 2004

More terms from Benoit Cloitre, Mar 22 2004

A022907 The sequence m(n) in A022905.

Original entry on oeis.org

0, 2, 5, 8, 14, 20, 29, 38, 53, 68, 89, 110, 140, 170, 209, 248, 302, 356, 425, 494, 584, 674, 785, 896, 1037, 1178, 1349, 1520, 1730, 1940, 2189, 2438, 2741, 3044, 3401, 3758, 4184, 4610, 5105, 5600, 6185, 6770, 7445, 8120, 8906, 9692, 10589

Offset: 1

Clark Kimberling

nonn

T. D. Noe, Table of n, a(n) for n=1..1000
J. M. Dover, On two OEIS conjectures, arXiv:1606.08033 [math.CO], 2016.

a123[n_] := a123[n] = If[n == 0, 1, a123[Floor[n/2]] + a123[n-1]];
a[n_] := If[n == 1, 0, (3/2) a123[n-1] - 1]; Array[a, 50] (* Jean-François Alcover, Dec 04 2018 *)

from itertools import islice
from collections import deque
def A022907_gen(): # generator of terms
    aqueue, f, b, a = deque([2]), True, 1, 2
    yield from (0, 2, 5)
    while True:
        a += b
        yield 3*a-1
        aqueue.append(a)
        if f: b = aqueue.popleft()
        f = not f
A022907_list = list(islice(A022907_gen(),40)) # Chai Wah Wu, Jun 08 2022

a(n) = 3 * A033485(n-1) - 1 = (3/2) * A000123(n-1) - 1, n>1. Proved by Jeremy Dover. - Ralf Stephan, Dec 08 2004

A131205 a(n) = a(n-1) + a(floor(n/2)) + a(ceiling(n/2)).

Original entry on oeis.org

1, 3, 7, 13, 23, 37, 57, 83, 119, 165, 225, 299, 393, 507, 647, 813, 1015, 1253, 1537, 1867, 2257, 2707, 3231, 3829, 4521, 5307, 6207, 7221, 8375, 9669, 11129, 12755, 14583, 16613, 18881, 21387, 24177, 27251, 30655, 34389, 38513, 43027, 47991

Offset: 1

Reinhard Zumkeller, Oct 22 2007

nonn

From Gary W. Adamson, Dec 16 2009: (Start)
Let M = an infinite lower triangular matrix with (1, 3, 4, 4, 4, ...) in every column shifted down twice, with the rest zeros:
  1;
  3, 0;
  4, 1, 0;
  4, 3, 0, 0;
  4, 4, 1, 0, 0;
  4, 4, 3, 0, 0, 0;
  ...
A131205 = lim_{n->infinity} M^n, the left-shifted vector considered as a sequence. (End)
The subsequence of primes in this sequence begins with 5 in a row: 3, 7, 13, 23, 37, 83, 647, 1867, 2707, 88873, 388837, 655121, 754903, 928621, 1062443. - Jonathan Vos Post, Apr 25 2010

T. D. Noe, Table of n, a(n) for n = 1..1000
Cristina Ballantine, George Beck, and Mircea Merca, Partitions and elementary symmetric polynomials -- an experimental approach, arXiv:2408.13346 [math.CO], 2024. See p. 13.
Cristina Ballantine, George Beck, Mircea Merca, and Bruce Sagan, Elementary symmetric partitions, arXiv:2409.11268 [math.CO], 2024. See pp. 5, 7.

Cf. A000123, A008619. Bisection of A033485.

a131205 n = a131205_list !! (n-1)
a131205_list = scanl1 (+) a000123_list -- Reinhard Zumkeller, Oct 10 2013

A[1]:= 1:
for n from 2 to 100 do A[n]:= A[n-1] + A[floor(n/2)] + A[ceil(n/2)] od:
seq(A[n],n=1..100); # Robert Israel, Sep 06 2016

Nest[Append[#1, #1[[-1]] + #1[[Floor@ #3]] + #[[Ceiling@ #3]] ] & @@ {#1, #2, #2/2} & @@ {#, Length@ # + 1} &, {1}, 42] (* Michael De Vlieger, Jan 16 2020 *)

Partial sums of A000123. - Gary W. Adamson, Oct 26 2007
G.f.: r(x) * r(x^2) * r(x^4) * r(x^8) * ... where r(x) = (1 + 3x + 4x^2 + 4x^3 + 4x^4 + ...) is the g.f. of A113311. - Gary W. Adamson, Sep 01 2016
G.f.: (x/(1 - x))*Product_{k>=0} (1 + x^(2^k))/(1 - x^(2^k)). - Ilya Gutkovskiy, Jun 05 2017
a(n) = A033485(2n-1). - Jean-Paul Allouche, Aug 11 2021

A322156 Irregular triangle where row n includes all decreasing sequences S = {k_0 = n, k_1, k_2, ..., k_m} in reverse lexicographic order such that the sum of subsequent terms k_j for all i < j <= m does not exceed any k_i.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 3, 3, 1, 3, 1, 1, 3, 2, 3, 2, 1, 3, 3, 4, 4, 1, 4, 1, 1, 4, 2, 4, 2, 1, 4, 2, 1, 1, 4, 2, 2, 4, 3, 4, 3, 1, 4, 4, 5, 5, 1, 5, 1, 1, 5, 2, 5, 2, 1, 5, 2, 1, 1, 5, 2, 2, 5, 3, 5, 3, 1, 5, 3, 1, 1, 5, 3, 2, 5, 4, 5, 4, 1, 5, 5, 6, 6, 1, 6, 1, 1, 6, 2, 6, 2, 1, 6, 2, 1, 1, 6, 2, 2, 6

Offset: 1

Michael De Vlieger, Dec 11 2018

Algorithm:
Let S be a sequence starting with n. Let k be the index of a term in S, with n at position k = 0. Let S_r be the r-th sequence in row n.
Starting with S_1 = {n}, we either (A) append a 1 to the left of S_r, or (B) we drop the most recently-appended term S_(k) and increment the rightmost term (k - 1).
By default we execute (A) and test according to the following. Consider the reversed accumulation A_(r + 1) = Sum(reverse(S_(k + 1))) = Sum(k_m, k_(m - 1), ..., k_2, k_1). If S_r - A_(r + 1) contains nothing less than 0, then S_(k + 1) is retained, otherwise we execute (B).
We end after k_1 = n, since otherwise we would enter an endless loop that also increments k_0 ad infinitum.
The first sequence S in row n is {n} while the last is {n, n}.
All rows n contain {{n}, {n, 1}, {n, n}}.
Only one repeated term k may appear at the end of any S in row n.
The longest possible sequence S in row n has 2 + floor(log_2(n)) terms = 2 + A113473(n).
The sequence S describes unique integer partitions L that are recursively symmetrical. Example: We can convert S = {4, 2, 1} into the partition (7, 6, 5, 4, 3, 2, 1), a partition of N = 28. We set a 4X Durfee square with its upper-left corner at origin. Then we set 2^k = 2^1 = 2 2X squares, each with its upper-left corner in any coordinate bounded at left and top by either a previously-lain square or an axis. Finally, we set 2^2 = 4 1X squares as above once again. We obtain a Ferrer diagram as below, with the k marked, i.e., the 1st term 4X, the 2nd term 2X, the 3rd term 1X squares:
    0  0  0  0  1  1  2
    0  0  0  0  1  1
    0  0  0  0  2
    0  0  0  0
    1  1  2
    1  1
    2
The resulting partition L is recursively self-conjugate; its arms are identical to its legs. We can eliminate the Durfee square and the other appendage and have a symmetrical partition L_1 with Durfee square of k_1 units, etc.
Were we to admit either more than 1 repeated k or a term such that S_k - A_(k + 1) had differences less than 1, we would have overlapping squares in the Ferrer diagram. Such diagrams are generated by larger n and all resulting diagrams are unique given the described algorithm.
The sequences S in row n, converted into integer partitions L, sum to n^2 <= N <= 3 * n^2.

			Triangle begins:
1; 1,1;
2; 2,1; 2,1,1; 2,2;
3; 3,1; 3,1,1; 3,2; 3,2,1; 3,3;
4; 4,1; 4,1,1; 4,2; 4,2,1; 4,2,1,1; 4,2,2; 4,3; 4,3,1; 4,4;
...
Row n = 5 starts with S_1 = 5. We append 1 to get {5,1}. 1 does not exceed 5, thus S_2 = {5,1}. We append 1 to get {5,1,1}. A = {1,2}; {5,1}-{2,1} = {3,0}, thus S_3 = {5,1,1} and we drop the last term and increment the new last term to get {5,2}. S_4 = {5,2}, and the ensuing terms {5,2,1}, {5,2,1,1}, {5,2,2} enter into the row. Since there are repeated terms at the last sequence, we drop the last term and increment the new last to get {5,3}. The terms {5,3,1}, {5,3,1,1}, {5,3,2}, {5,3,2,1}, are admitted. {5,3,2,1,1} has A = {1,2,4,6}. {5,3,2,1}-{6,4,2,1} = {-1,1,0,0}: {5,3,2,1,1} cannot be admitted, so we drop the last term and increment to {5,3,2,2} but the sum of the last two terms exceeds the second and we drop the last term and increment to {5,3,3}. For similar reasons, this cannot be admitted, so we drop the last term and increment to {5,4}. This enters as well as {5,4,1}. Since any appendage or increment proves invalid, we end up incrementing to {5,5}. The two terms are the same, therefore we end the row n = 5.

Michael De Vlieger, Table of n, a(n) for n = 1..10055 (rows 1 <= n <= 21, flattened)
Michael De Vlieger, Illustration for A322156
Michael De Vlieger, Rows 1 <= n <= 30, sequences S in row n have terms k that are space delimited

Cf. A000123, A033485, A190899, A190900, A321223, A322457.

(* Generate sequence: *)
f[n_] := Block[{w = {n}, c}, c[x_] := Apply[Times, Most@ x - Reverse@ Accumulate@ Reverse@ Rest@ x]; Reap[Do[Which[And[Length@ w == 2, SameQ @@ w], Sow[w]; Break[], Length@ w == 1, Sow[w]; AppendTo[w, 1], c[w] > 0, Sow[w]; AppendTo[w, 1], True, Sow[w]; w = MapAt[1 + # &, Drop[w, -1], -1]], {i, Infinity}] ][[-1, 1]] ]; Array[f, 6] // Flatten
(* Convert S = row n to standard partition: *)
g[w_] := Block[{k}, k = Total@ w; Total@ Map[Apply[Function[{s, t}, s Array[Boole[t <= # <= s + t - 1] &, k] ], #] &, Apply[Join, Prepend[Table[Function[{v, c}, Map[{w[[k]], # + 1} &, Map[Total[v #] &, Tuples[{0, 1}, {Length@ v}]]]] @@ {Most@ #, ConstantArray[1, Length@ # - 1]} &@ Take[w, k], {k, 2, Length@ w}], {{w[[1]], 1}}]]] ]

Row n contains A000123(n) = 2*A033485(n) sequences S.

A062188 a(n+1) = a(n) + a(floor(n/2)), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 1, 2, 3, 4, 5, 7, 9, 12, 15, 19, 23, 28, 33, 40, 47, 56, 65, 77, 89, 104, 119, 138, 157, 180, 203, 231, 259, 292, 325, 365, 405, 452, 499, 555, 611, 676, 741, 818, 895, 984, 1073, 1177, 1281, 1400, 1519, 1657, 1795, 1952, 2109, 2289, 2469, 2672, 2875, 3106

Offset: 0

Henry Bottomley, Jun 13 2001

nonn

			a(6) = a(5)+a(2) = 4+1 = 5.
a(7) = a(6)+a(3) = 5+2 = 7.

Ivan Neretin, Table of n, a(n) for n = 0..10000

Cf. A033485, A062186, A062187.

[n le 2 select n-1 else  Self(n-1)+Self(Floor(n/2)): n in [1..60]]; // Vincenzo Librandi, Mar 03 2016

Join[{0}, Nest[Append[#, #[[-1]] + #[[Quotient[Length@#, 2]]]] &, {1, 1}, 53]] (* Ivan Neretin, Mar 03 2016 *)

from itertools import islice
from collections import deque
def A062188_gen(): # generator of terms
    aqueue, f, b, a = deque([1]), True, 0, 1
    yield from (0,1)
    while True:
        a += b
        yield a
        aqueue.append(a)
        if f: b = aqueue.popleft()
        f = not f
A062188_list = list(islice(A062188_gen(),40)) # Chai Wah Wu, Jun 08 2022

G.f. A(x) satisfies: A(x) = x * (1 + (1 + x)*A(x^2))/(1 - x). - Ilya Gutkovskiy, May 04 2019

A092412 Fixed point of the morphism 0->11, 1->12, 2->13, 3->10, starting from a(1) = 1.

Original entry on oeis.org

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

Offset: 1

Philippe Deléham, Mar 22 2004

Paolo Xausa, Table of n, a(n) for n = 1..16384 (terms 1..1024 from Andrew Howroyd)
Index entries for sequences that are fixed points of mappings.

Cf. A001511, A033485, A035263, A346070 (same with values 3,0,1,2).

Nest[ Function[ l, {Flatten[(l /. {0 -> {1, 1}, 1 -> {1, 2}, 2 -> {1, 3}, 3 -> {1, 0}})] }], {0}, 7] (* Robert G. Wilson v, Mar 04 2005 *)
SubstitutionSystem[{0 -> {1, 1}, 1 -> {1, 2}, 2 -> {1, 3}, 3 -> {1, 0}}, {1}, 7] // Last (* Jean-François Alcover, Sep 20 2019 *)
Mod[IntegerExponent[Range[100], 2] + 1, 4] (* Paolo Xausa, Feb 25 2025 *)

a(n)=(1 + valuation(n, 2)) %4; \\ Andrew Howroyd, Aug 06 2018

def A092412(n): return (n&-n).bit_length()&3 # Chai Wah Wu, Jul 13 2022

a(n) = A001511(n) mod 4.
a(2n+1) = 1; a(2n) = a(n) + 1 mod 4.
a(n) == A035263(n) (mod 2); a(n) == A033485(n) (mod 2).
Multiplicative with a(2^e) = (1 + e) mod 4, a(p^e) = 1 for odd prime p. - Andrew Howroyd, Aug 06 2018
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 22/15. - Amiram Eldar, Nov 29 2022
Dirichlet g.f.: zeta(s)*(3*2^s+2^(2*s+1)+2^(3*s))/(1+2^s+4^s+8^s). - Amiram Eldar, Jan 04 2023

A022905 a(n) = M(n) + m(n) for n >= 2, where M(n) = max{ a(i) + a(n-i): i = 1..n-1 }, m(n) = min{ a(i) + a(n-i): i = 1..n-1 }.

Original entry on oeis.org

1, 4, 10, 19, 34, 55, 85, 124, 178, 247, 337, 448, 589, 760, 970, 1219, 1522, 1879, 2305, 2800, 3385, 4060, 4846, 5743, 6781, 7960, 9310, 10831, 12562, 14503, 16693, 19132, 21874, 24919, 28321, 32080, 36265, 40876, 45982, 51583, 57769

Offset: 1

Clark Kimberling

nonn

T. D. Noe, Table of n, a(n) for n=1..1000
J. M. Dover, On two OEIS conjectures, arXiv:1606.08033 [math.CO], 2016.

Cf. A022907, A022908.

a:= proc(n) option remember; `if`(n=1, 1,
      a(n-1)+1+a(floor(n/2))+a(ceil(n/2)))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Sep 17 2013

a[n_] := a[n] = If[n==1, 1, a[n-1]+1+a[Floor[n/2]]+a[Ceiling[n/2]]]; Array[a,100] (* Jean-François Alcover, Aug 07 2017, after Alois P. Heinz *)

from itertools import islice
from collections import deque
def A022905_gen(): # generator of terms
    aqueue, f, b, a = deque([2]), True, 1, 2
    yield 1
    while True:
        a += b
        aqueue.append(a)
        if f:
            yield (3*a-1)//2
            b = aqueue.popleft()
        f = not f
A022905_list = list(islice(A022905_gen(),40)) # Chai Wah Wu, Jun 08 2022

a(n) = n + Sum_{k=2..n} A022907(k).
a(n+1) = 1+3*Sum_{k=1..n} A033485(k). - Philippe Deléham, Jun 17 2010
a(n) = a(n-1) + 1 + a(floor(n/2)) + a(ceiling(n/2)) for n>1, a(1) = 1. - Alois P. Heinz, Sep 17 2013
a(n+1) = (3*A033485(2n+1)-1)/2. - Chai Wah Wu, Jun 08 2022

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cp's OEIS Frontend

A091855 Odious numbers (see A000069) in A003159.

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A003156 A self-generating sequence (see Comments for definition).

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A102378 a(n) = a(n-1) + a([n/2]) + 1, a(1) = 1.

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A091785 Evil numbers (see A001969) in A003159.

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Mathematica

PARI

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A022907 The sequence m(n) in A022905.

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A131205 a(n) = a(n-1) + a(floor(n/2)) + a(ceiling(n/2)).

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Haskell

Maple

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A322156 Irregular triangle where row n includes all decreasing sequences S = {k_0 = n, k_1, k_2, ..., k_m} in reverse lexicographic order such that the sum of subsequent terms k_j for all i < j <= m does not exceed any k_i.

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