A131205 -id:A131205 - cp's OEIS frontend

A278163 a(n) = the least k such that A131205(1+k) >= n; each n occurs A000123(n) times.

0, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8

Offset: 1

Antti Karttunen, Nov 15 2016

nonn

Greatest monotonic left inverse of A131205(1+n).

Antti Karttunen, Table of n, a(n) for n = 1..9669

Cf. A000123, A131205, A277903, A278164.

(define (A278163 n) (let loop ((k 0)) (if (>= (A131205 (+ 1 k)) n) k (loop (+ 1 k)))))
(definec (A131205 n) (if (= 1 n) 1 (+ (A000123 (- n 1)) (A131205 (- n 1)))))
;; For other code, see A277903.

a(A131205(1+n)) = n.

A152560 a(n) = A131205(A005578(n)).

Original entry on oeis.org

1, 1, 3, 7, 37, 225, 2707, 47991, 1566965, 85002865, 8346008131, 1441358958439, 452666387604933, 257384373709193473, 268828878795481175283, 516988085848458847554135, 1844678455777198460320221077

Offset: 0

Paul D. Hanna, Dec 22 2008

nonn

Related to binary partitions (A000123); A131205 is defined by a(n) = a(n-1) + a(floor(n/2)) + a(ceiling(n/2)) and A005578(n) = (2^(n+1)+3+(-1)^n)/6.

Cf. A131205, A005578, A000123.

{a(n)=local(N=(2^(n+1)+3+(-1)^n)/6,X=x+O(x^N));polcoeff(1/((1-X)^2*prod(m=0,n-1,1-X^(2^m))),N-1)}

A033485 a(n) = a(n-1) + a(floor(n/2)), a(1) = 1.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 13, 18, 23, 30, 37, 47, 57, 70, 83, 101, 119, 142, 165, 195, 225, 262, 299, 346, 393, 450, 507, 577, 647, 730, 813, 914, 1015, 1134, 1253, 1395, 1537, 1702, 1867, 2062, 2257, 2482, 2707, 2969, 3231, 3530, 3829, 4175, 4521, 4914, 5307, 5757

Offset: 1

N. J. A. Sloane. This was in the 1973 "Handbook", but was then dropped from the database. Resubmitted by Philippe Deléham. Entry revised by N. J. A. Sloane, Jun 10 2012

Sequence gives the number of partitions of 2n into "strongly decreasing" parts (see the function s*(n) in the paper by Bessenrodt, Olsson, and Sellers); see the example in A040039.
a(A036554(n)) is even, a(A003159(n)) is odd. - Benoit Cloitre, Oct 23 2002
Partial sums of the sequence a(1)=1, a(1), a(1), a(2), a(2), a(3), a(3), a(4), a(4), a(5), a(5), a(6), ...; example: a(1) = 1, a(2) = 1+1 = 2, a(3) = 1+1+1 = 3, a(4) = 1+1+1+2 = 5, a(5) = 1+1+1+2+2 = 7, ... - Philippe Deléham, Jan 02 2004
The number of odd numbers before the n-th even number in this sequence is A003156(n). - Philippe Deléham, Mar 27 2004
There are no terms divisible by 4 and there are infinitely many terms divisible by {2,3,5,6,7,9,10,11,13,14,15} (see van Doorn link). - Ivan N. Ianakiev, Aug 06 2022 and Wouter van Doorn, Sep 17 2024
a(n) = A001401(n), for 1..14. A001401(15) = 84. - Wolfdieter Lang, Jan 09 2023

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..1000
J. Arkin, Problem H-102: Gone but not forgotten, Fibonacci Quarterly, Vol. 9 (1971), page 135.
Christine Bessenrodt, Jorn B. Olsson, and James A. Sellers, Unique path partitions: Characterization and Congruences, arXiv:1107.1156 [math.CO], 2011-2012.
Philippe Deléham, Letter to N. J. A. Sloane, Apr 20 1998
William Gasarch, What is known about that sequence, Computational Complexity blog, Jul 20 2022.
Wouter van Doorn, On the congruence properties and growth rate of a recursively defined sequence, arXiv:2406.09532 [math.NT], 2024.

Cf. A040039 (first differences), A178855 (partial sums).
Also half of A000123 (with first term omitted).
Cf. A022907.
Cf. A036554, A003159, A003156.
Cf. A007413, A178855, A131205.

import Data.List (transpose)
a033485 n = a033485_list !! (n-1)
a033485_list = 1 : zipWith (+)
   a033485_list (concat $ transpose [a033485_list, a033485_list])
-- Reinhard Zumkeller, Nov 15 2012

[n le 1 select 1 else Self(n-1) + Self(Floor(n/2)) : n in [1..60]]; // Vincenzo Librandi, Nov 20 2015

a:= proc(n) option remember;
      `if`(n<2, n, a(n-1)+a(iquo(n, 2)))
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, Dec 16 2019

b[1]=1; b[n_] := b[n]=Sum[b[k], {k, 1, n/2}]; Table[b[n], {n, 3, 105, 2}] (* Robert G. Wilson v, Apr 22 2001 *)

PARI
```
a(n)=if(n<2,1,a(floor(n/2))+a(n-1))
```

from itertools import islice
from collections import deque
def A033485_gen(): # generator of terms
    aqueue, f, b, a = deque([2]), True, 1, 2
    yield from (1, 2)
    while True:
        a += b
        yield a
        aqueue.append(a)
        if f: b = aqueue.popleft()
        f = not f
A033485_list = list(islice(A033485_gen(),40)) # Chai Wah Wu, Jun 07 2022

a(n) = A000123(n)/2, for n >= 1.
Conjecture: lim_{n->oo} a(2n)/a(n)*log(n)/n = c = 1.64.... and a(n)/A(n) is bounded where A(n)=1 if n is a power of 2, otherwise A(n) = sqrt(n)*Product_{kBenoit Cloitre, Jan 26 2003
G.f.: A(x) satisfies x + (1+x)*A(x^2) = (1-x)*A(x). a(n) modulo 2 = A035263(n). - Philippe Deléham, Feb 25 2004
G.f.: (1/2)*(((1-x)*Product_{n>=0} (1-x^(2^n)))^(-1)-1). a(n) modulo 4 = A007413(n). - Philippe Deléham, Feb 28 2004
Sum_{k=1..n} a(k) = (a(2n+1)-1)/2 = A178855(n). - Philippe Deléham, Mar 18 2004
a(2n-1) = A131205(n). - Jean-Paul Allouche, Aug 11 2021
There exists a function f(n) such that n^f(n) < a(n) < n^(f(n) + epsilon) for all epsilon > 0 and all large enough n. - Wouter van Doorn, Sep 17 2024

A278164 Irregular triangle read by rows: row n (n >= 0) is obtained by listing numbers 0 .. A000123(n)-1.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 3, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12

Offset: 1

Antti Karttunen, Nov 15 2016

			A000123 -> range -> terms on row n
   1       [0,0]:    0;
   2       [0,1]:    0, 1;
   4       [0,3]:    0, 1, 2, 3;
   6       [0,5]:    0, 1, 2, 3, 4, 5;
  10       [0,9]:    0, 1, 2, 3, 4, 5, 6, 7, 8, 9;
etc.

Antti Karttunen, Table of n, a(n) for n = 1..9669

Obtained by taking the every second row of similar table A277904.

Cf. A000123, A131205, A278163.

(define (A278164 n) (if (= 1 n) 0 (- n (A131205 (A278163 n)) 1)))

a(1) = 0; for n > 1, a(n) = n - A131205(A278163(n)) - 1.

cp's OEIS Frontend

A278163 a(n) = the least k such that A131205(1+k) >= n; each n occurs A000123(n) times.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Scheme

Formula

A152560 a(n) = A131205(A005578(n)).

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

A033485 a(n) = a(n-1) + a(floor(n/2)), a(1) = 1.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Python

Formula

A278164 Irregular triangle read by rows: row n (n >= 0) is obtained by listing numbers 0 .. A000123(n)-1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Scheme

Formula