A230407 -id:A230407 - cp's OEIS frontend

A230409 Partial sums of A230407.

0, -1, 0, 3, -2, -3, 0, 1, 4, -1, -2, 5, 4, -1, -2, -7, -2, 3, -2, -13, -12, -9, -20, -19, -22, -19, -18, -15, -20, -21, -14, -15, -20, -21, -26, -21, -16, -21, -32, -31, -28, -49, -48, -51, -54, -45, -44, -45, -50, -51, -56, -51, -46, -51, -62, -61, -58, -79

Offset: 0

Antti Karttunen, Nov 10 2013

sign

The term a(n) indicates approximately the "balance" of the factorial beanstalk (cf. A219666) at n steps up from the root, which in turn correlates with the behavior of such sequences as A219662 and A219663.
This sequence relates to the factorial base representation (A007623) in the same way as A218789 relates to the binary system.
Question: When will a negative term occur next time, after a(251) = -41 ?

Antti Karttunen, Table of n, a(n) for n = 0..21622

Cf. A230407 & A230408, A219662 & A219663.

a(0) = 0, a(n) = a(n-1) + A230407(n).

A230408 Partial sums of absolute values of A230407.

Original entry on oeis.org

0, 1, 2, 5, 10, 11, 14, 15, 18, 23, 24, 31, 32, 37, 38, 43, 48, 53, 58, 69, 70, 73, 84, 85, 88, 91, 92, 95, 100, 101, 108, 109, 114, 115, 120, 125, 130, 135, 146, 147, 150, 171, 172, 175, 178, 187, 188, 189, 194, 195, 200, 205, 210, 215, 226, 227, 230, 251, 252

Offset: 0

Antti Karttunen, Nov 10 2013

nonn

The term a(n) tells how many nodes there are in total in all side-trees ("tendrils") encountered (see A230430) after we have climbed n steps up along the infinite trunk of the factorial beanstalk, A219666.

This sequence relates to the factorial base representation (A007623) in the same way as A218785 relates to the binary system.

Antti Karttunen, Table of n, a(n) for n = 0..3149

Cf. A230407, A230409.

a(0) = 0, a(n) = a(n-1) + |A230407(n)| [Where | | indicates the absolute value].

A219666 The infinite trunk of factorial expansion beanstalk. The only infinite sequence such that a(n-1) = a(n) - sum of digits in factorial expansion of a(n).

Original entry on oeis.org

0, 1, 2, 5, 7, 10, 12, 17, 23, 25, 28, 30, 35, 40, 46, 48, 52, 57, 63, 70, 74, 79, 85, 92, 97, 102, 109, 119, 121, 124, 126, 131, 136, 142, 144, 148, 153, 159, 166, 170, 175, 181, 188, 193, 198, 204, 213, 221, 228, 238, 240, 244, 249, 255, 262, 266, 271, 277

Offset: 0

Antti Karttunen, Nov 25 2012

a(n) tells in what number we end in n steps, when we start climbing up the infinite trunk of the "factorial beanstalk" from its root (zero).
There are many finite sequences such as 0,1,2,4; 0,1,2,5,6; etc. obeying the same condition (see A219659) and as the length increases, so (necessarily) does the similarity to this infinite sequence.
See A007623 for the factorial number system representation.

Antti Karttunen, Table of n, a(n) for n = 0..21622

Cf. A007623, A034968, A219651, A230411, A226061. For all n, A219652(a(n)) = n and A219653(n) <= a(n) <= A219655(n).
Characteristic function: Χ_A219666(n) = A230418(n+1)-A230418(n).
The first differences: A230406.
Other derived sequences: A230425-A230427, A230430, A230407-A230409, A219662 & A219663, A231723 & A231724, A230420, A230410, A231717, A231719.
Subsets: A230428 & A230429.
Analogous sequence for binary system: A179016, for Fibonacci number system: A219648.

nn = 10^3; m = 1; While[m! < Floor[6 nn/5], m++]; m; t = TakeWhile[Reverse@ NestWhileList[# - Total@ IntegerDigits[#, MixedRadix[Reverse@ Range[2, m]]] &, Floor[6 nn/5], # > 0 &], # <= nn &] (* Michael De Vlieger, Jun 27 2016, Version 10.2 *)

;; Memoizing definec-macro from Antti Karttunen's IntSeq-library
(definec (A219666 n) (cond ((<= n 2) n) ((= (A226061 (A230411 n)) n) (- (A000142 (A230411 n)) 1)) (else (- (A219666 (+ n 1)) (A034968 (A219666 (+ n 1)))))))
;; Another variant, utilizing A230416 (which gives a more convenient way to compute large number of terms of this sequence):
(define (A219666 n) (A230416 (A230432 n)))
;; This function is for checking whether n belongs to this sequence:
(define (inA219666? n) (or (zero? n) (= 1 (- (A230418 (+ 1 n)) (A230418 n)))))

a(0) = 0, a(1) = 1, and for n>1, if A226061(A230411(n)) = n then a(n) = A230411(n)!-1, otherwise a(n) = a(n+1) - A034968(a(n+1)).

a(n) = A230416(A230432(n)).

A230427 a(n)=0 if n is in the infinite trunk of factorial beanstalk (in A219666), otherwise the number of nodes (including leaves and the node n itself) in that finite branch of the beanstalk.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 3, 0, 1, 1, 0, 5, 0, 1, 1, 3, 3, 0, 1, 1, 1, 1, 1, 0, 3, 0, 1, 1, 0, 5, 0, 1, 1, 3, 7, 0, 1, 1, 3, 3, 0, 1, 1, 1, 1, 1, 0, 5, 0, 1, 1, 3, 0, 5, 1, 1, 5, 0, 3, 1, 1, 3, 5, 0, 1, 1, 1, 1, 1, 3, 0, 5, 1, 1, 0, 11, 3, 1, 1, 0, 7, 3, 1, 1, 3, 0, 5

Offset: 0

Antti Karttunen, Nov 10 2013

nonn

This sequence relates to the factorial base representation (A007623) in the same way as A213727 relates to the binary system.

			From 11 sprouts the following finite side-tree of "factorial beanstalk":
    18  19
     \  /
  14  15
   \  /
    11
Its leaves are the numbers 14, 18 and 19 (which all occur in A219658), whose factorial base representations (see A007623) are '210', '300' and '301' respectively. The corresponding parent nodes are obtained by subtracting the sum of factorial base digits, thus we get 18-3 = 15 and also 19-4 = 15, thus 15 ('211' in factorial base) is the parent of 18 and 19. For 14 and 15 we get 14-3 = 15-4 = 11, thus 11 is the parent of both 14 and 15, and the common ancestor of all numbers 11, 14, 15, 18 and 19.
For numbers not occurring in A219666 this sequence gives 1 + the number of all descendant nodes in such subtrees. Thus a(11)=5, a(14)=1 (counting just the leaf 14 itself), a(15)=3 and a(18) = a(19) = 1.

Antti Karttunen, Table of n, a(n) for n = 0..10080

A219658 gives the position of ones (leaves). Cf. also A230407, A230423-A230424, A230425-A230426, A230430.

If A230412(n)=0, a(n)=1; otherwise, if n is in A219666, a(n)=0; otherwise a(n) = 1 + a(A230423(n)) + a(A230424(n)).

Alternatively, if A230426(n)=0, a(n)=0; otherwise (2*A230426(n))-1.

A230430 After zero, gives the numbers where finite side-trees (or "tendrils") of factorial beanstalk branch off from its infinite trunk (A219666).

Original entry on oeis.org

0, 3, 4, 6, 11, 13, 16, 22, 24, 29, 31, 34, 41, 47, 49, 53, 56, 62, 71, 75, 78, 84, 93, 96, 103, 108, 118, 120, 125, 127, 130, 137, 143, 145, 149, 152, 158, 167, 171, 174, 180, 189, 192, 199, 205, 212, 220, 229, 239, 241, 245, 248, 254, 263, 267, 270, 276, 285

Offset: 0

Antti Karttunen, Nov 10 2013

nonn

This sequence relates to the factorial base representation (A007623) in the same way as A213730 relates to the binary system.

Antti Karttunen, Table of n, a(n) for n = 0..3149

Cf. A230407.

(define (A230430 n) (+ (A230423 (A219666 n)) (- 1 (modulo (A219666 (1+ n)) 2))))
;; Alternative definition:
(define (A230430 n) (+ (A219666 (+ 1 n)) (expt -1 (modulo (A219666 (+ 1 n)) 2))))

a(n) = A219666(n+1) + (-1)^A219666(n+1).

a(n) = A230423(A219666(n)) + (1 - (A219666(n+1) mod 2)).

cp's OEIS Frontend

A230409 Partial sums of A230407.

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A230408 Partial sums of absolute values of A230407.

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A219666 The infinite trunk of factorial expansion beanstalk. The only infinite sequence such that a(n-1) = a(n) - sum of digits in factorial expansion of a(n).

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A230427 a(n)=0 if n is in the infinite trunk of factorial beanstalk (in A219666), otherwise the number of nodes (including leaves and the node n itself) in that finite branch of the beanstalk.

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A230430 After zero, gives the numbers where finite side-trees (or "tendrils") of factorial beanstalk branch off from its infinite trunk (A219666).

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