cp's OEIS Frontend

A074279 n appears n^2 times.

%I A074279 #72 Jun 30 2025 04:24:13
%S A074279 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 A074279 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 A074279 6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6
%N A074279 n appears n^2 times.
%C A074279 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 A074279 Partial sums of A253903. - _Jeremy Gardiner_, Jan 14 2018
%H A074279 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 A074279 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 A074279 For 1 <= n <= 650, a(n) = floor((3n)^(1/3)+1/2). - _Mikael Aaltonen_, Jan 05 2015
%F A074279 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 A074279 a(n) = floor(t + 1/(12*t) + 1/2), where t = (3*n - 1)^(1/3). - _Ridouane Oudra_, Oct 30 2023
%F A074279 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
%F A074279 Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/4 (A003881). - _Amiram Eldar_, Jun 30 2025
%e A074279 This can be viewed also as an irregular table consisting of successively larger square matrices:
%e A074279   1;
%e A074279   2, 2;
%e A074279   2, 2;
%e A074279   3, 3, 3;
%e A074279   3, 3, 3;
%e A074279   3, 3, 3;
%e A074279   4, 4, 4, 4;
%e A074279   4, 4, 4, 4;
%e A074279   4, 4, 4, 4;
%e A074279   4, 4, 4, 4;
%e A074279   etc.
%e A074279 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 A074279 Table[n, {n, 0, 6}, {n^2}] // Flatten (* _Arkadiusz Wesolowski_, Jan 13 2013 *)
%o A074279 (Scheme, with Antti Karttunen's IntSeq-library. This uses starting offset=1)
%o A074279 (define A074279 (LEAST-GTE-I 1 1 A000330));; _Antti Karttunen_, Feb 08 2014
%o A074279 (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 A074279 (Python)
%o A074279 from sympy import integer_nthroot
%o A074279 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 A074279 Cf. A000217, A000330, A002024, A003881, A006331, A050446, A050447, A000537, A006003, A005900, A064866, A237451, A237452.
%K A074279 nonn,tabf
%O A074279 1,2
%A A074279 _Jon Perry_, Sep 21 2002
%E A074279 Offset corrected from 0 to 1 by _Antti Karttunen_, Feb 08 2014