cp's OEIS Frontend

Self-inverse when considered as a permutation or function, i.e., a(a(n)) = n. - Howard A. Landman, Sep 25 2001

That each initial segment has an integer average is trivially equivalent to the sum of the first n elements always being divisible by n. - Franklin T. Adams-Watters, Jul 07 2014

Also, a lexicographically minimal sequence of distinct positive integers such that all values of a(n)-n are also distinct. - Ivan Neretin, Apr 18 2015

Comments from N. J. A. Sloane, Mar 29 2025 (Start):

Let d(n) = number of 1 <= i <= n such that a(i) < i. The d(i) sequence begins 0, 0, 1, 1, 1, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 6, ..., and appears to be A060144 without its initial term.

Let r(n) = 1 if a(n) < a(n+1), otherwise 0, and let f(n) = 1 if a(n) > a(n+1), otherwise 0. Then R = partial sums of r(n) and F = partial sums of f(n) count the rises and falls, respectively, in the present sequence. It appears that R and F are essentially A060143 and A060144 (again).

If a(n) is the k-th term in a monotonically strictly increasing rum of terms, set R(n) = k. It appears that the sequence R(n), n>=1, is essentially A270788.

For other sequences derived from the present one, see A382162, A382168, and A382169.

(End)

For n > 2, a(n) is also the number of partitions of n into parts <= n-2: a(n) = A026820(n+1, n-1). - Reinhard Zumkeller, Jan 21 2010

Also, the number of partitions of 2*n in which n-1 is the maximal part; see the Mathematica section. - Clark Kimberling, Mar 13 2012

This is column 2 of the matrix A in Sect. 2.3 of the Govindarajan preprint, cf. references and A096651. - M. F. Hasler, Apr 12 2012

Average of first n terms of A019444, which is defined to be a permutation of the positive integers, p_1, p_2, ..., such that the average of each initial segment is an integer, using the greedy algorithm to define p_n.

Number of pairs (i,j) of nonnegative integers such that n-1=floor(i+j*tau). - Clark Kimberling, Jun 18 2002

The terms that occur exactly once are 1,3,6,8,..., given by A026352(n)=n+1+floor(n*tau). - Clark Kimberling, Jun 18 2002

The number n appears A001468(n) times. - Reinhard Zumkeller, Feb 02 2012

It seems that the indices of the terms that occur exactly once are listed in A276885. - Ivan N. Ianakiev, Aug 30 2018

From Michel Dekking, Oct 13 2020: (Start)

Here is a proof of the conjecture by Ivan N. Ianakiev. Let b = (b(n)) be the sequence of occurrences of the "singleton terms" in (a(n)). We have to show that b = A276885.

In the following phi := (1+sqrt(5))/2 (so phi = tau).

By its definition, the sequence (a(n)) is a generalized Beatty sequence with terms a(n) = floor(phi*n)-n+1, since 1/phi = phi-1. So by Lemma 8 in the paper by Allouche and Dekking, its sequence of first differences Delta = 1011010110..., given by Delta(n) = a(n+1)-a(n), is equal to y, where y = A005614 is the binary complement of the Fibonacci word. By definition, y is the fixed point of the morphism nu: 0->1, 1->10.

The crucial observation is that a term occurs exactly once in (a(n)) if and only if the word 11 of length 2 occurs in Delta (with an exception for a(1)=1). So to obtain the sequence b of occurrences of these "singleton terms", we have to study the return words of 11 in y. (The return words of 11 in y are the words occurring in y that start with 11, and having no other occurrences of 11.)

The return words of 11 are the words A:=11010, and B:=110. Since

nu(A) = nu(11010) = 10101101, nu(B) = nu(110) = 10101,

the morphism nu induces a descendant morphism tau given by

tau(A) = BA, tau(B) = A.

So tau is nothing else but the Fibonacci morphism on the alphabet {B,A}.

Since the words A and B have lengths 5 and 3, the first differences b(n+1)-b(n) are given by the fixed point z = 5353353533... of the Fibonacci morphism on the alphabet {5,3}.

From Lemma 8 in the paper by Allouche and Dekking we then obtain that the sequence b is a generalized Beatty sequence

V(n) = (5-3)*floor(phi*n)+(2*3-5)*n+r = 2*floor(phi*n)+n+r, for some integer r.

Starting at the value 4, filling in n=1, we obtain that r=1, and so V(n) = 2*floor(phi*n)+n+1. To incorporate also the first "singleton term" a(1)=1, we take

b(n) = V(n-1) = 2*floor(phi*(n-1))+n-1+1 = 2*floor(phi*(n-1))+n.

Then, indeed, b(n) = A276885(n), for n=1,2,... (see my Comment in A276885).

(End)

It seems that the indices of the records are listed in A026351. - Ivan N. Ianakiev, Mar 25 2021

Binomial transform of [1, 10, 17, 8, 0, 0, 0, ...] = (1, 11, 38, 90, ...). - Gary W. Adamson, Mar 18 2009

This sequence is related to A000384 by a(n) = n*A000384(n) - Sum_{i=0..n-1} A000384(i) and this is the case d=4 in the identity n*(n*(d*n-d+2)/2) - Sum_{k=0..n-1} k*(d*k-d+2)/2 = n*(n+1)*(2*d*n - 2*d + 3)/6. - Bruno Berselli, Apr 21 2010

For n>0, (a(n)) is the principal diagonal of the convolution array A213750. - Clark Kimberling, Jun 20 2012

From Ant King, Oct 30 2012: (Start)

The partial sums of the figurate decagonal numbers A001107.

For n>1, the digital roots of this sequence A010888(A007585(n)) form the purely periodic 27-cycle {1,2,2,9,4,4,8,6,6,7,8,8,6,1,1,5,3,3,4,5,5,3,7,7,2,9,9}.

For n>1, the units’ digits of this sequence A010879(A007585(n)) form the purely periodic 20-cycle {1,1,8,0,5,1,6,8,5,5,6,6,3,5,0,6,1,3,0,0}.

(End)

From the definition, assign label value "1" to an origin triangle; at n-th generation add a triangle at each vertex. Each non-overlapping triangle will have the same label value as that of the predecessor triangle to which it is connected; for the overlapping ones, the label value will be the sum of the label values of predecessors. a(n) is the sum of all label values at the n-th generation. The triangle count is A005448. See illustration. For n >= 1, (a(n) - a(n-1))/3 is A027383. - Kival Ngaokrajang, Sep 05 2014

The sequence appears with a different offset in some other sources. - Michael Somos, Apr 02 2006

Also known as Göbel's (or Goebel's) Sequence. Asymptotically, a(n) ~ n*C^(2^n) where C=1.0478... (A115632). A more precise asymptotic formula is given in A116603. - M. F. Hasler, Dec 12 2007

Let s(n) = (n-1)*a(n). By considering the p-adic representation of s(n) for primes p=2,3,...,43, one finds that a(44) is the first nonintegral value in this sequence. Furthermore, for n>44, the valuation of s(n) w.r.t. 43 is -2^(n-44), implying that both s(n) and a(n) are nonintegral. - M. F. Hasler and Max Alekseyev, Mar 03 2009

a(44) is approximately 5.4093*10^178485291567. - Hans Havermann, Nov 14 2017.

The fractional part is simply 24/43 (see page 709 of Guy (1988)).

The more precise asymptotic formula is a(n+1) ~ C^(2^n) * (n + 2 - 1/n + 4/n^2 - 21/n^3 + 138/n^4 - 1091/n^5 + ...). - Michael Somos, Mar 17 2012

Because d(k) <= 2*sqrt(k), it suffices to check k from 1 to 4*n^2. - Nathaniel Johnston, May 04 2011

A051521(a(n)) = 1. - Reinhard Zumkeller, Dec 28 2011

Because d(k) <= 2*sqrt(k), it suffices to check k from 1 to 4*n^2. - Nathaniel Johnston, May 04 2011

A051521(a(n)) = 2. - Reinhard Zumkeller, Dec 28 2011

a(n) is also the number of permutations in the symmetric group S_n that are pure cycles, see example. - Avi Peretz (njk(AT)netvision.net.il), Mar 24 2001

Also the number of elementary circuits in a complete directed graph with n nodes [D. B. Johnson, 1975]. - N. J. A. Sloane, Mar 24 2014

If one takes 1,2,3,4, ..., n and starts creating parenthetic products of k-tuples and adding, one gets a(n+1). For 1,2,3,4 one gets (1)+(2)+(3)+(4) = 10; (1*2)+(2*3)+(3*4) = 20; (1*2*3)+(2*3*4) = 30; (1*2*3*4) = 24; and 10+20+30+24 = 84 = a(5). - J. M. Bergot, Apr 24 2014

Let P_n be the set of probability distributions over orderings of n objects that can be obtained by drawing n real numbers from independent probability distributions and sorting. Then a(n) is conjectured to be the dimension of P_n, as a semi-algebraic subset of R^(n!). - Jamie Tucker-Foltz, Jul 29 2024

Negatives of odd fundamental discriminants D := b^2-4*a*c<0 of definite integer binary quadratic forms F=a*x^2+b*x*y+c*y^2. See Buell reference pp. 224-230. See 4*A089269 = A191483 for the even case and A003657 for combined even and odd numbers. - Wolfdieter Lang, Nov 07 2003

The asymptotic density of this sequence is 2/Pi^2 = 0.202642... (A185197). - Amiram Eldar, Feb 10 2021