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%I #56 Feb 26 2023 19:34:26
%S 4,16,28,40,52,64,76,88,100,112,124,136,148,160,172,184,196,208,220,
%T 232,244,256,268,280,292,304,316,328,340,352,364,376,388,400,412,424,
%U 436,448,460,472,484,496,508,520,532,544,556,568,580,592,604,616,628
%N a(n) = 12*n + 4.
%C Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0( 46 ).
%C Number of 6 X n 0-1 matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01;0), (11;0) and (01;1). An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1<i2, j1<j2 and these elements are in the same relative order as those in the triple (x,y,z). In general, the number of m X n 0-1 matrices in question is given by 2^m+2m(n-1). Cf. m=2: A008574; m=3: A016933; m=4: A022144; m=5: A017293. - _Sergey Kitaev_, Nov 13 2004
%C Except for 4, exponents e such that x^e-x^2+1 is reducible.
%C If Y and Z are 2-blocks of a (3n+1)-set X then a(n-1) is the number of 3-subsets of X intersecting both Y and Z. - _Milan Janjic_, Oct 28 2007
%C Terms are perfect squares iff n is a generalized octagonal number (A001082), then n = k*(3*k-2) and a(n) = (2*(3k-1))^2. - _Bernard Schott_, Feb 26 2023
%H Vincenzo Librandi, <a href="/A017569/b017569.txt">Table of n, a(n) for n = 0..5000</a>
%H Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>.
%H Sergey Kitaev, <a href="http://www.emis.de/journals/INTEGERS/papers/e21/e21.Abstract.html">On multi-avoidance of right angled numbered polyomino patterns</a>, Integers: Electronic Journal of Combinatorial Number Theory, Vol. 4 (2004), Article A21, 20pp.
%H William A. Stein, <a href="http://wstein.org/Tables/dimskg0n.gp">Dimensions of the spaces S_k(Gamma_0(N))</a>.
%H William A. Stein, <a href="http://wstein.org/Tables/">The modular forms database</a>.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F A089911(a(n)) = 3. - _Reinhard Zumkeller_, Jul 05 2013
%F Sum_{n>=0} (-1)^n/a(n) = sqrt(3)*Pi/36 + log(2)/12. - _Amiram Eldar_, Dec 12 2021
%F From _Stefano Spezia_, Feb 25 2023: (Start)
%F O.g.f.: 4*(1 + 2*x)/(1 - x)^2.
%F E.g.f.: 4*exp(x)*(1 + 3*x). (End)
%t 12*Range[0,200]+4 (* _Vladimir Joseph Stephan Orlovsky_, Feb 19 2011 *)
%o (Magma) [12*n+4: n in [0..50]]; // _Vincenzo Librandi_, May 04 2011
%o (Haskell)
%o a017569 = (+ 4) . (* 12) -- _Reinhard Zumkeller_, Jul 05 2013
%Y Cf. A008594, A017533, A017545, A089911.
%Y Cf. A016933, A016777, A017293, A022144.
%Y Cf. A001082.
%K nonn,easy
%O 0,1
%A _N. J. A. Sloane_
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