A074279 - OEIS

%I #68 Nov 05 2024 12:17:36

%S 1,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,

%T 5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,

%U 6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6

%N n appears n^2 times.

%C Since the last occurrence of n comes one before the first occurrence of n+1 and the former is at Sum_{i=0..n} i^2 = A000330(n), we have a(A000330(n)) = a(n*(n+1)*(2n+1)/6) = n and a(1+A000330(n)) = a(1+(n*(n+1)*(2n+1)/6)) = n+1. So the current sequence is, loosely speaking, the inverse function of the square pyramidal sequence A000330. A000330 has many alternative formulas, thus yielding many alternative formulas for the current sequence. - _Jonathan Vos Post_, Mar 18 2006

%C Partial sums of A253903. - _Jeremy Gardiner_, Jan 14 2018

%H Antti Karttunen, <a href="/A074279/b074279.txt">Table of n, a(n) for n = 1..9455</a> (Table of squares from 1 X 1 to 30 X 30, flattened)

%H Y.-F. S. Petermann, J.-L. Remy and I. Vardi, <a href="http://dx.doi.org/10.1006/aama.2001.0750">Discrete derivatives of sequences</a>, Adv. in Appl. Math. 27 (2001), 562-84.

%F For 1 <= n <= 650, a(n) = floor((3n)^(1/3)+1/2). - _Mikael Aaltonen_, Jan 05 2015

%F a(n) = 1 + floor( t(n) + 1 / ( 12 * t(n) ) - 1/2 ), where t(n) = (sqrt(3888*(n-1)^2-1) / (8*3^(3/2)) + 3 * (n-1)/2 ) ^(1/3). - _Mikael Aaltonen_, Mar 01 2015

%F a(n) = floor(t + 1/(12*t) + 1/2), where t = (3*n - 1)^(1/3). - _Ridouane Oudra_, Oct 30 2023

%F a(n) = m+1 if n > m(m+1)(2m+1)/6 and a(n) = m otherwise where m = floor((3n)^(1/3)). - _Chai Wah Wu_, Nov 04 2024

%e This can be viewed also as an irregular table consisting of successively larger square matrices:

%e   1;

%e   2, 2;

%e   2, 2;

%e   3, 3, 3;

%e   3, 3, 3;

%e   3, 3, 3;

%e   4, 4, 4, 4;

%e   4, 4, 4, 4;

%e   4, 4, 4, 4;

%e   4, 4, 4, 4;

%e   etc.

%e When this is used with any similarly organized sequence, a(n) is the index of the matrix in whose range n is. A121997(n) (= A237451(n)+1) and A238013(n) (= A237452(n)+1) would then yield the index of the column and row within that matrix.

%t Table[n, {n, 0, 6}, {n^2}] // Flatten (* _Arkadiusz Wesolowski_, Jan 13 2013 *)

%o (Scheme, with Antti Karttunen's IntSeq-library. This uses starting offset=1)

%o (define A074279 (LEAST-GTE-I 1 1 A000330));; _Antti Karttunen_, Feb 08 2014

%o (PARI) A074279_vec(N=9)=concat(vector(N,i,vector(i^2,j,i))) \\ Note: This creates a vector; use A074279_vec()[n] to get the n-th term. - _M. F. Hasler_, Feb 17 2014

%o (Python)

%o from sympy import integer_nthroot

%o def A074279(n): return (m:=integer_nthroot(3*n,3)[0])+(6*n>m*(m+1)*((m<<1)+1)) # _Chai Wah Wu_, Nov 04 2024

%Y Cf. A000217, A000330, A002024, A006331, A050446, A050447, A000537, A006003, A005900, A064866, A237451, A237452.

%K nonn,tabf

%O 1,2

%A _Jon Perry_, Sep 21 2002

%E Offset corrected from 0 to 1 by _Antti Karttunen_, Feb 08 2014