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A133819 Triangle whose rows are sequences of increasing squares: 1; 1,4; 1,4,9; ... .

%I A133819 #40 Feb 15 2022 13:03:09
%S A133819 1,1,4,1,4,9,1,4,9,16,1,4,9,16,25,1,4,9,16,25,36,1,4,9,16,25,36,49,1,
%T A133819 4,9,16,25,36,49,64,1,4,9,16,25,36,49,64,81,1,4,9,16,25,36,49,64,81,
%U A133819 100,1,4,9,16,25,36,49,64,81,100,121,1,4,9,16,25,36,49,64,81,100,121,144
%N A133819 Triangle whose rows are sequences of increasing squares: 1; 1,4; 1,4,9; ... .
%C A133819 Reading the triangle by rows produces the sequence 1,1,4,1,4,9,1,4,9,16,..., analogous to A002260.
%C A133819 Sequence B is called a reluctant sequence of sequence A, if B is triangle array read by rows: row number k coincides with first k elements of the sequence A. A133819 is reluctant sequence of A000290. - _Boris Putievskiy_, Jan 11 2013
%H A133819 Reinhard Zumkeller, <a href="/A133819/b133819.txt">Rows n = 1..120 of triangle, flattened</a>
%H A133819 M. de Frenicle, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k5493994j/">Methode pour trouver la solutions des problemes par les exclusions</a>, in: Divers ouvrages de mathematiques et de physique par messieurs de l'academie royale des sciences, (1693) pp 1-44, page 11. - _Paul Curtz_, Aug 18 2008
%H A133819 Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations Integer Sequences And Pairing Functions</a>, arXiv:1212.2732 [math.CO], 2012.
%F A133819 T(n, k) = k^2, n >= k >= 1. - _Wolfdieter Lang_, Dec 02 2014
%F A133819 O.g.f.: (1+qx)/((1-x)(1-qx)^3) = 1 + x(1 + 4q) + x^2(1 + 4q + 9q^2) + ... .
%F A133819 a(n) = A000290(m+1), where m = n-t(t+1)/2, t = floor((-1+sqrt(8*n-7))/2). - _Boris Putievskiy_, Jan 11 2013
%e A133819 The triangle T(n, k) starts:
%e A133819 1;
%e A133819 1, 4;
%e A133819 1, 4, 9;
%e A133819 1, 4, 9, 16;
%e A133819 1, 4, 9, 16, 25;
%t A133819 With[{sqs=Range[12]^2},Flatten[Table[Take[sqs,n],{n,12}]]] (* _Harvey P. Dale_, Sep 09 2012 *)
%o A133819 (Haskell)
%o A133819 a133819 n k = a133819_tabl !! (n-1) !! (k-1)
%o A133819 a133819_row n = a133819_tabl !! (n-1)
%o A133819 a133819_tabl = map (`take` (tail a000290_list)) [1..]
%o A133819 -- _Reinhard Zumkeller_, Nov 11 2012
%Y A133819 Cf. A215630, A215631, A000290, A002260.
%Y A133819 Cf. A048152, A060036.
%K A133819 easy,nonn,tabl
%O A133819 1,3
%A A133819 _Peter Bala_, Sep 25 2007
%E A133819 Offset changed by _Reinhard Zumkeller_, Nov 11 2012