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

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, 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, 100, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144

Offset: 1

Author

Peter Bala, Sep 25 2007

Keywords

• easy
• nonn
• tabl

Comments

Reading the triangle by rows produces the sequence 1,1,4,1,4,9,1,4,9,16,..., analogous to A002260.

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

Examples

			The triangle T(n, k) starts:
1;
1, 4;
1, 4, 9;
1, 4, 9, 16;
1, 4, 9, 16, 25;
		

Links

• Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened
• M. de Frenicle, Methode pour trouver la solutions des problemes par les exclusions, 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
• Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Crossrefs

Cf. A215630, A215631, A000290, A002260.

Cf. A048152, A060036.

Programs

Haskell

a133819 n k = a133819_tabl !! (n-1) !! (k-1)
a133819_row n = a133819_tabl !! (n-1)
a133819_tabl = map (`take` (tail a000290_list)) [1..]
-- Reinhard Zumkeller, Nov 11 2012

Mathematica

With[{sqs=Range[12]^2},Flatten[Table[Take[sqs,n],{n,12}]]] (* Harvey P. Dale, Sep 09 2012 *)

Formula

T(n, k) = k^2, n >= k >= 1. - Wolfdieter Lang, Dec 02 2014

O.g.f.: (1+qx)/((1-x)(1-qx)^3) = 1 + x(1 + 4q) + x^2(1 + 4q + 9q^2) + ... .

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

Extensions

Offset changed by Reinhard Zumkeller, Nov 11 2012