A284556 -id:A284556 - cp's OEIS frontend

A284566 Odd bisection of A284556.

0, 1, 2, 1, 2, 3, 2, 2, 3, 3, 4, 4, 3, 4, 4, 2, 3, 5, 5, 5, 6, 6, 6, 5, 4, 6, 7, 5, 5, 6, 4, 3, 4, 5, 7, 7, 7, 9, 9, 6, 7, 10, 10, 9, 9, 9, 8, 6, 5, 8, 10, 8, 9, 11, 9, 7, 7, 8, 9, 8, 6, 7, 6, 3, 4, 7, 8, 8, 10, 11, 11, 9, 9, 13, 15, 12, 12, 14, 11, 8, 9, 12, 15, 14, 14, 17, 16, 11, 11, 15, 15, 13, 12, 12, 10, 7, 6

Offset: 0

Antti Karttunen, Apr 05 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8192

Cf. A001222, A007306, A284556, A284564, A284565.

a[n_] := Which[n < 2, n, EvenQ@ n, a[n/2], True, a[(n - 1)/2] + a[(n + 1)/2]]; Table[(a[#] - JacobiSymbol[#, 3])/2 &[2 n + 1], {n, 0, 96}] (* Michael De Vlieger, Apr 05 2017 *)

(define (A284566 n) (A284556 (+ n n 1)))
(define (A284566 n) (A001222 (A284564 n)))

a(n) = A284556((2*n)+1).
a(n) = A001222(A284564(n)).
Other identities. For all n >= 1:
A007306(n) = a(n-1) + A284565(n-1).

A000360 Distribution of nonempty triangles inside a fractal rep-4-tile.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 0, 2, 2, 2, 1, 3, 1, 2, 1, 2, 2, 4, 1, 4, 3, 3, 1, 4, 2, 4, 2, 3, 2, 3, 0, 3, 3, 4, 2, 6, 3, 5, 2, 5, 4, 7, 2, 6, 4, 4, 1, 5, 3, 6, 3, 6, 4, 6, 1, 5, 4, 5, 2, 5, 2, 3, 1, 3, 3, 6, 2, 7, 5, 6, 2, 8, 5, 9, 4, 8, 5, 7, 1, 7, 6, 9, 4, 11, 6, 9, 3, 8, 6, 10, 3, 8, 5, 5, 1, 6, 4, 8, 4, 9, 6, 9, 2

Offset: 0

M. Jeremie Lafitte (Levitas)

a(n) = Running count of congruent nonempty triangles along lines perpendicular to the base of the Gosper-Lafitte triangle.
Also, a(n) = Sum of the coefficients of the terms with an even exponent in the Stern polynomial B(n+1,t), or in other words, the sum of the even-indexed terms (the leftmost is at index 0) of the irregular triangle A125184, starting from its second row. - Antti Karttunen, Apr 20 2017
Back in May 1995, it was proved that a(n) = modulo 3 mapping, (+1,-1,+0)/2, of the Stern-Brocot sequence A002487, dropping its 1st term. - M. Jeremie Lafitte (Levitas), Apr 23 2017

M. J. Lafitte, Sur l'Effet Noah en Géométrie, rapport à l'INPI, mars 1995.

T. D. Noe, Table of n, a(n) for n = 0..10000
S. Klavzar, U. Milutinovic and C. Petr, Stern polynomials, Adv. Appl. Math. 39 (2007) 86-95.
M. J. Lafitte, Ensembles Auto-Similaires d'Intérieur Non-Vide, Preprint Hiver 1997, Chaire de Géometrie, Département de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, Switzerland. [Cached copy, with permission]
M. J. Lafitte, Fractal triangle underlying A000360, A000361, A000876
M. J. Lafitte, Notes on A000360, A000361, A000876 [Based on a latex file sent by M. Jeremie Lafitte (Levitas) to NJAS in 1995 - see file of emails below]
M. J. Lafitte, Latex source for the pdf file [Sent by MJL to NJAS in 1995 - see file of emails below]
M. J. Lafitte and N. J. A. Sloane, Emails, 1995-2007 (The three sequences mentioned in this correspondence are now A000360, A000361, A000876)

Cf. A002487, A000361, A000876.
Cf. A001222, A057078, A125184, A284553, A284556, A284565 (bisection).
Cf. also mutual recurrence pair A287729, A287730.

import Data.List (transpose)
a000360 n = a000360_list !! n
a000360_list = 1 : concat (transpose
   [zipWith (+) a000360_list $ drop 2 a057078_list,
    zipWith (+) a000360_list $ tail a000360_list])
-- Reinhard Zumkeller, Mar 22 2013
(Scheme, with memoization-macro definec):
(define (A000360 n) (A000360with_prep_0 (+ 1 n)))
(definec (A000360with_prep_0 n) (cond ((<= n 1) n) ((even? n) (A284556 (/ n 2))) (else (+ (A000360with_prep_0 (/ (- n 1) 2)) (A000360with_prep_0 (/ (+ n 1) 2))))))
(definec (A284556 n) (cond ((<= n 1) 0) ((even? n) (A000360with_prep_0 (/ n 2))) (else (+ (A284556 (/ (- n 1) 2)) (A284556 (/ (+ n 1) 2))))))
;; Antti Karttunen, Apr 07 2017

a[0] = 1; a[n_?EvenQ] := a[n] = a[n/2] + a[n/2-1]; a[n_?OddQ] := a[n] = a[(n-1)/2] - Mod[(n-1)/2-1, 3] + 1; Table[a[n], {n, 0, 103}] (* Jean-François Alcover, Jan 20 2015, after Ralf Stephan *)

a(n) = if(n==0, 1, if(n%2, a((n - 1)/2) - ((n - 1)/2 - 1)%3 + 1, a(n/2) + a(n/2 - 1))); \\ Indranil Ghosh, Apr 20 2017

a(3n) = (A002487(3n+1) + 1)/2, a(3n+1) = (A002487(3n+2) - 1)/2, a(3n+2) = A002487(3n+3)/2. - M. Jeremie Lafitte (Levitas), Apr 23 2017
a(0) = 1, a(2n) = a(n) + a(n-1), a(2n+1) = a(n) + 1 - (n-1 mod 3). - Ralf Stephan, Oct 05 2003; Note: according to Ralf Stephan, this formula was found empirically. It follows from that found for the Stern-Brocot sequence A002487 and the first formula. - Antti Karttunen, Apr 21 2017, M. Jeremie Lafitte (Levitas), Apr 23 2017
From Antti Karttunen, Apr 07 2017: (Start)
Ultimately equivalent to the above formulae, we have:
a(n) = A001222(A284553(1+n)).
a(n) = A002487(1+n) - A284556(1+n).
a(n) = b(1+n), with b from a mutual recurrence pair: b(0) = 0, b(1) = 1, b(2n) = c(n), b(2n+1) = b(n) + b(n+1), c(0) = c(1) = 0, c(2n) = b(n), c(2n+1) = c(n) + c(n+1). [c(n) = A284556(n), b(n)+c(n) = A002487(n).]
(End)

More terms from David W. Wilson, Aug 30 2000

Original relation to the Stern-Brocot sequence A002487 reformulated by M. Jeremie Lafitte (Levitas), Apr 23 2017

A287729 The c-fusc function c(n) = a(n): a(1)=1, a(2n) = s(n), a(2n+1) = s(n)+s(n+1), where s(n) = A287730(n).

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 1, 0, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 3, 2, 3, 1, 2, 1, 1, 0, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 1, 6, 5, 9, 4, 11, 7, 10, 3, 11, 8, 13, 5, 12, 7, 9, 2, 9, 7, 12, 5, 13, 8, 11, 3, 10, 7, 11, 4, 9, 5, 6, 1, 5, 4, 7, 3

Offset: 1

I. V. Serov, May 30 2017

Define a sequence chf(n) of Christoffel words over an alphabet {-,+}:
  chf(1) = '-',
  chf(2*n+0) = negate(chf(n)),
  chf(2*n+1) = negate(concatenate(chf(n),chf(n+1))).
Then the length of the chf(n) word is fusc(n) = A002487(n), the number of '-'-signs in the chf(n) word is c-fusc(n) = a(n) (the current sequence) and the number of '+'-signs in the chf(n) word is s-fusc(n) = A287730(n). See examples below.

			A000027(n) chf(n) A070939(n) A002487(n)    a(n)    A287730(n)
                                fusc       c-fusc     s-fusc
01         '-'       1          1          1          0
02         '+'       2          1          0          1
03         '+-'      2          2          1          1
04         '-'       3          1          1          0
05         '--+'     3          3          2          1
06         '-+'      3          2          1          1
07         '-++'     3          3          1          2
08         '+'       4          1          0          1
09         '+++-'    4          4          1          3
10         '++-'     4          3          1          2
11         '++-+-'   4          5          2          3
12         '+-'      4          2          1          1
13         '+-+--'   4          5          3          2
14         '+--'     4          3          2          1
15         '+---'    4          4          3          1
16         '-'       5          1          1          0
17         '----+'   5          5          4          1

I. V. Serov (terms 1..1025) & Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences related to Stern's sequences

Cf. A002487, A007814, A037227, A070939, A287730, A255738, A053644.

Cf. mutual recurrence pair A000360, A284556 and also A213369.

from sympy.core.cache import cacheit
@cacheit
def c(n): return 1 if n==1 else s(n//2) if n%2==0 else s((n - 1)//2) + s((n + 1)//2)
@cacheit
def s(n): return 0 if n==1 else c(n//2) if n%2==0 else c((n - 1)//2) + c((n + 1)//2)
print([c(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 08 2017

(definec (A287729 n) (cond ((= 1 n) n) ((even? n) (A287730 (/ n 2))) (else (+ (A287730 (/ (- n 1) 2)) (A287730 (/ (+ n 1) 2))))))
;; An implementation of memoization-macro definec can be found for example in: http://oeis.org/wiki/Memoization - Antti Karttunen, Jun 01 2017

The mutual diatomic recurrence pair c(n) (this sequence) and s(n) (A287730) are defined by c(1)=1, s(1)=0, c(2n) = s(n), c(2n+1) = s(n)+s(n+1), s(2n) = c(n), s(2n+1) = c(n)+c(n+1).
a(n) + A287730(n) = A002487(n). [c-fusc(n) + s-fusc(n) = fusc(n).]
gcd(a(n), A287730(n)) = gcd(a(n), A002487(n)) = 1.
Let k(n) = A037227(n) = 1 + 2*A007814(n) = 1 + 2*floor(A002487(n-1)/A002487(n)) for n > 1.
Let d(n) = 2*A255738(n)*(-1)^A070939(n) = 2*(n==2^(A070939(n)-1)+1)*(-1)^A070939(n) = 2*(n==A053644(n)+1)*(-1)^A070939(n) = 2*(A002487(n-1)==1)*(-1)^A070939(n) for n > 1;
then a(n) = k(n-1)*a(n-1) - a(n-2) + d(n) for n > 2 with a(1) = 1, a(2) = 0.
From Yosu Yurramendi, Apr 09 2019: (Start)
For m >= 0, m even, 0 <= k < 2^m, a(2^m+k) = A002487(2^m-k).
For m >= 0, m odd,  0 <= k < 2^m, a(2^m+k) = A002487(k).
(End)

A287730 The s-fusc function s(n) = a(n): a(1) = 0, a(2n) = A287729(n), a(2n+1) = A287729(n) + A287729(n+1).

Original entry on oeis.org

0, 1, 1, 0, 1, 1, 2, 1, 3, 2, 3, 1, 2, 1, 1, 0, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 1, 4, 3, 5, 2, 5, 3, 4, 1, 3, 2, 3, 1, 2, 1, 1, 0, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 1, 6, 5, 9, 4

Offset: 1

I. V. Serov, May 30 2017

Define a sequence chf(n) of Christoffel words over an alphabet {-,+}:
  chf(1) = '-',
  chf(2*n+0) = negate(chf(n)),
  chf(2*n+1) = negate(concatenate(chf(n),chf(n+1))).
Then the length of the chf(n) word is fusc(n) = A002487(n), the number of '-'-signs in the chf(n) word is c-fusc(n) = A287729(n) and the number of '+'-signs in the chf(n) word is the current sequence a(n) = s-fusc(n). See examples below.

			n         chf(n) A070939(n) A002487(n) A287729(n)    a(n)
                                fusc       c-fusc     s-fusc
1          '-'       1          1          1          0
2          '+'       2          1          0          1
3          '+-'      2          2          1          1
4          '-'       3          1          1          0
5          '--+'     3          3          2          1
6          '-+'      3          2          1          1
7          '-++'     3          3          1          2
8          '+'       4          1          0          1
9          '+++-'    4          4          1          3
10         '++-'     4          3          1          2
11         '++-+-'   4          5          2          3
12         '+-'      4          2          1          1
13         '+-+--'   4          5          3          2
14         '+--'     4          3          2          1
15         '+---'    4          4          3          1
16         '-'       5          1          1          0
17         '----+'   5          5          4          1

I. V. Serov (terms 1..1025) & Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences related to Stern's sequences

Cf. A002487, A007814, A070939, A037227, A287729, A053644.

Cf. mutual recurrence pair A000360, A284556 and also A213369.

from sympy.core.cache import cacheit
@cacheit
def c(n): return 1 if n==1 else s(n//2) if n%2==0 else s((n - 1)//2) + s((n + 1)//2)
@cacheit
def s(n): return 0 if n==1 else c(n//2) if n%2==0 else c((n - 1)//2) + c((n + 1)//2)
print([s(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 08 2017

(definec (A287730 n) (cond ((= 1 n) 0) ((even? n) (A287729 (/ n 2))) (else (+ (A287729 (/ (- n 1) 2)) (A287729 (/ (+ n 1) 2))))))
;; An implementation of memoization-macro definec can be found for example in: http://oeis.org/wiki/Memoization
;; Second version after the alternative formula given by the author:
(definec (A287730 n) (if (<= n 2) (- n 1) (- (* (A037227 (- n 1)) (A287730 (- n 1))) (A287730 (- n 2)) (* (if (= 1 (A002487 (- n 1))) 1 0) 2 (expt -1 (A070939 n)))))) ;; Antti Karttunen, Jun 01 2017

The mutual diatomic recurrence pair c(n) (A287729) and s(n) (this sequence) are defined by c(1)=1, s(1)=0, c(2n) = s(n), c(2n+1) = s(n)+s(n+1), s(2n) = c(n), s(2n+1) = c(n)+c(n+1).
a(n) + A287729(n) = A002487(n). [s-fusc(n) + c-fusc(n) = fusc(n).]
gcd(a(n), A287729(n)) = gcd(a(n), A002487(n)) = 1.
Let k(n) = A037227(n) = 1 + 2*A007814(n) = 1 + 2*floor(A002487(n-1)/A002487(n)) for n > 1.
Let d(n) = 2*A255738(n)*(-1)^A070939(n) = 2*(n==2^(A070939(n)-1)+1)*(-1)^A070939(n) = 2*(n==A053644(n)+1)*(-1)^A070939(n) = 2*(A002487(n-1)==1)*(-1)^A070939(n) for n > 1;
then a(n) = k(n-1)*a(n-1) - a(n-2) - d(n) for n > 2 with a(1) = 0, a(2) = 1.

A284565 Bisection of A000360.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 2, 2, 4, 4, 3, 4, 4, 3, 3, 3, 4, 6, 5, 5, 7, 6, 4, 5, 6, 6, 6, 5, 5, 5, 3, 3, 6, 7, 6, 8, 9, 8, 7, 7, 9, 11, 9, 8, 10, 8, 5, 6, 8, 9, 9, 9, 10, 10, 7, 6, 9, 9, 7, 7, 7, 5, 4, 4, 6, 9, 8, 9, 12, 11, 8, 10, 13, 14, 13, 12, 13, 12, 8, 8, 13, 15, 13, 15, 17, 15, 12, 11, 14, 16, 13, 11, 13, 10, 6, 7, 10

Offset: 0

Antti Karttunen, Apr 05 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8192

Cf. A000360, A001222, A007306, A284556, A284563, A284566.

a[n_] := a[n] = Which[n == 0, 1, EvenQ@ n, a[n/2] + a[n/2 - 1], True, a[(n - 1)/2] - Mod[(n - 1)/2 - 1, 3] + 1]; Table[a[2 n], {n, 0, 97}] (* Michael De Vlieger, Apr 05 2017, after Jean-François Alcover at A000360 *)

(define (A284565 n) (A001222 (A284563 n)))
(define (A284565 n) (A000360with_prep_0 (+ n n 1))) ;; See code in A284556.

a(n) = A000360(2n).
a(n) = A001222(A284563(n)).
Other identities. For all n >= 1:
A007306(n) = a(n-1) + A284566(n-1).

A284553 Prime factorization representation of Stern polynomials B(n,x) with only the even powers of x present: a(n) = A247503(A260443(n)).

Original entry on oeis.org

1, 2, 1, 2, 5, 2, 5, 10, 1, 10, 25, 10, 5, 50, 5, 10, 11, 10, 25, 250, 5, 250, 125, 50, 11, 250, 25, 250, 55, 50, 55, 110, 1, 110, 275, 250, 55, 6250, 125, 1250, 121, 1250, 625, 31250, 55, 6250, 1375, 550, 11, 2750, 275, 6250, 605, 6250, 1375, 13750, 11, 2750, 3025, 2750, 55, 6050, 55, 110, 17, 110, 275, 30250, 55, 68750, 15125, 13750, 121

Offset: 0

Antti Karttunen, Mar 29 2017

nonn

a(n) = Prime factorization representation of Stern polynomials B(n,x) where the coefficients of odd powers of x are replaced by zeros. In other words, only the constant term and other terms with even powers of x are present. See the examples.

Proof that A001222(a(1+n)) matches Ralf Stephan's formula for A000360(n): Consider functions A001222(a(n)) and A001222(A284554(n)) (= A284556(n)). They can be reduced to the following mutual recurrence pair: b(0) = 0, b(1) = 1, b(2n) = c(n), b(2n+1) = b(n) + b(n+1) and c(0) = c(1) = 0, c(2n) = b(n), c(2n+1) = c(n) + c(n+1). From the definitions it follows that the difference b(n) - c(n) for even n is b(2n) - c(2n) = -(b(n) - c(n)), and for odd n, b(2n+1) - c(2n+1) = (b(n)+b(n+1))-(c(n)+c(n+1)) = (b(n)-c(n)) + (b(n+1)-c(n+1)). Then by induction, if we assume that for 3n, 3n+1, 3n+2, ..., 6n, the value of difference b(n)-c(n) is always [0, +1, -1; repeated], it follows that from 6n to 12n the differences are [0, +1, -1; 0, +1, -1; repeated], which proves that b(n) - c(n) = A102283(n). As an implication, recurrence b can be defined without referring to c as: b(0) = 0, b(1) = 1, b(2n) = b(n) - A102283(n), b(2n+1) = b(n)+b(n+1), and this is equal to Ralf Stephan's Oct 05 2003 formula for A000360, but shifted once right, with prepended zero.

			n A260443(n)                      Stern            With odd powers
             prime factorization  polynomial       of x cleared  -> a(n)
------------------------------------------------------------------------
0       1    (empty)              B_0(x) = 0                    0  |  1
1       2    p_1                  B_1(x) = 1                    1  |  2
2       3    p_2                  B_2(x) = x                    0  |  1
3       6    p_2 * p_1            B_3(x) = x + 1                1  |  2
4       5    p_3                  B_4(x) = x^2                x^2  |  5
5      18    p_2^2 * p_1          B_5(x) = 2x + 1               1  |  2
6      15    p_3 * p_2            B_6(x) = x^2 + x            x^2  |  5
7      30    p_3 * p_2 * p_1      B_7(x) = x^2 + x + 1    x^2 + 1  | 10
8       7    p_4                  B_8(x) = x^3                  0  |  1
9      90    p_3 * p_2^2 * p_1    B_9(x) = x^2 + 2x + 1   x^2 + 1  | 10
10     75    p_3^2 * p_2          B_10(x) = 2x^2 + x         2x^2  | 25

Antti Karttunen, Table of n, a(n) for n = 0..8192
S. Klavzar, U. Milutinovic and C. Petr, Stern polynomials, Adv. Appl. Math. 39 (2007) 86-95.

Cf. A000360, A001222, A003961, A064989, A102283, A247503, A260443, A284554, A284556, A284563 (odd bisection).

a[n_] := a[n] = Which[n < 2, n + 1, EvenQ@ n, Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1] &@ a[n/2], True, a[#] a[# + 1] &[(n - 1)/2]]; Table[Times @@ (FactorInteger[#] /. {p_, e_} /; e > 0 :> (p^Mod[PrimePi@ p, 2])^e) &@ a@ n, {n, 0, 72}] (* Michael De Vlieger, Apr 05 2017 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From Michel Marcus
A260443(n) = if(n<2, n+1, if(n%2, A260443(n\2)*A260443(n\2+1), A003961(A260443(n\2)))); \\ Cf. Charles R Greathouse IV's code for "ps" in A186891 and A277013.
A247503(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 2] *= (primepi(f[i, 1]) % 2); ); factorback(f); } \\ After Michel Marcus
A284553(n) = A247503(A260443(n));

(define (A284553 n) (A247503 (A260443 n)))

a(0) = 1, a(1) = 2, a(2n) = A003961(A284554(n)), a(2n+1) = a(n)*a(n+1).
Other identities. For all n >= 0:
a(n) = A247503(A260443(n)).
a(n) = A260443(n) / A284554(n).
a(n) = A064989(A284554(2n)).
A001222(a(1+n)) = A000360(n). [Proof in Comments section.]

A284554 Prime factorization representation of Stern polynomials B(n,x) with only the odd powers of x present: a(n) = A248101(A260443(n)).

Original entry on oeis.org

1, 1, 3, 3, 1, 9, 3, 3, 7, 9, 3, 27, 7, 9, 21, 21, 1, 63, 21, 27, 49, 81, 21, 189, 7, 63, 147, 189, 7, 441, 21, 21, 13, 63, 21, 1323, 49, 567, 1029, 1323, 7, 3969, 1029, 1701, 343, 3969, 147, 1323, 13, 441, 1029, 9261, 49, 27783, 1029, 1323, 91, 3087, 147, 9261, 91, 441, 273, 273, 1, 819, 273, 1323, 637, 27783, 1029, 64827, 91, 27783, 50421, 583443, 343

Offset: 0

Antti Karttunen, Mar 29 2017

nonn

a(n) = Prime factorization representation of Stern polynomials B(n,x) where the coefficients of even powers of x (including the constant term) are replaced by zeros. In other words, only the terms with odd powers of x are present. See the examples.

			n A260443(n)                      Stern            With even powers
             prime factorization  polynomial       of x cleared  -> a(n)
------------------------------------------------------------------------
0       1    (empty)              B_0(x) = 0                    0  |  1
1       2    p_1                  B_1(x) = 1                    0  |  1
2       3    p_2                  B_2(x) = x                    x  |  3
3       6    p_2 * p_1            B_3(x) = x + 1                x  |  3
4       5    p_3                  B_4(x) = x^2                  0  |  1
5      18    p_2^2 * p_1          B_5(x) = 2x + 1              2x  |  9
6      15    p_3 * p_2            B_6(x) = x^2 + x              x  |  3
7      30    p_3 * p_2 * p_1      B_7(x) = x^2 + x + 1          x  |  3
8       7    p_4                  B_8(x) = x^3                x^3  |  7
9      90    p_3 * p_2^2 * p_1    B_9(x) = x^2 + 2x + 1        2x  |  9
10     75    p_3^2 * p_2          B_10(x) = 2x^2 + x            x  |  3

Antti Karttunen, Table of n, a(n) for n = 0..8192

Cf. A001222, A003961, A064989, A248101, A260443, A284553, A284556, A284564 (odd bisection).

a[n_] := a[n] = Which[n < 2, n + 1, EvenQ@ n, Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1] &@ a[n/2], True, a[#] a[# + 1] &[(n - 1)/2]]; Table[Times @@ (FactorInteger[#] /. {p_, e_} /; e > 0 :> (p^Mod[PrimePi@ p + 1, 2])^e) &@ a@ n, {n, 0, 76}] (* Michael De Vlieger, Apr 05 2017 *)

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From Michel Marcus
A260443(n) = if(n<2, n+1, if(n%2, A260443(n\2)*A260443(n\2+1), A003961(A260443(n\2)))); \\ Cf. Charles R Greathouse IV's code for "ps" in A186891 and A277013.
A248101(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 2] *= (primepi(f[i, 1])+1) % 2; ); factorback(f); } \\ After Michel Marcus
A284554(n) = A248101(A260443(n));

(define (A284554 n) (A248101 (A260443 n)))

a(0) = a(1) = 1, a(2n) = A003961(A284553(n)), a(2n+1) = a(n)*a(n+1).
Other identities. For all n >= 0:
a(n) = A248101(A260443(n)).
a(n) = A260443(n) / A284553(n).
a(n) = A064989(A284553(2n)).
A001222(a(n)) = A284556(n).

cp's OEIS Frontend

A284566 Odd bisection of A284556.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Scheme

Formula

A000360 Distribution of nonempty triangles inside a fractal rep-4-tile.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

Extensions

A287729 The c-fusc function c(n) = a(n): a(1)=1, a(2n) = s(n), a(2n+1) = s(n)+s(n+1), where s(n) = A287730(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

Scheme

Formula

A287730 The s-fusc function s(n) = a(n): a(1) = 0, a(2n) = A287729(n), a(2n+1) = A287729(n) + A287729(n+1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

Scheme

Formula

A284565 Bisection of A000360.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Scheme

Formula

A284553 Prime factorization representation of Stern polynomials B(n,x) with only the even powers of x present: a(n) = A247503(A260443(n)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Scheme

Formula

A284554 Prime factorization representation of Stern polynomials B(n,x) with only the odd powers of x present: a(n) = A248101(A260443(n)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs