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A032527 Concentric pentagonal numbers: floor( 5*n^2 / 4 ).

%I A032527 #67 Jan 16 2023 08:17:41
%S A032527 0,1,5,11,20,31,45,61,80,101,125,151,180,211,245,281,320,361,405,451,
%T A032527 500,551,605,661,720,781,845,911,980,1051,1125,1201,1280,1361,1445,
%U A032527 1531,1620,1711,1805,1901,2000,2101,2205,2311,2420,2531,2645,2761,2880,3001
%N A032527 Concentric pentagonal numbers: floor( 5*n^2 / 4 ).
%C A032527 Also A033429 and A062786 interleaved. - _Omar E. Pol_, Sep 28 2011
%C A032527 Partial sums of A047209. - _Reinhard Zumkeller_, Jan 07 2012
%C A032527 From _Wolfdieter Lang_, Aug 06 2013: (Start)
%C A032527 a(n) = -N(-floor(n/2),n) with the N(a,b) = ((2*a+b)^2 - b^2*5)/4, the norm for integers a + b*omega(5), a, b rational integers, in the quadratic number field Q(sqrt(5)), where omega(5) = (1 + sqrt(5))/2 (golden section).
%C A032527 a(n) = max({|N(a,n)|,a = -n..+n}) = |N(-floor(n/2),n)| = n^2 + n*floor(n/2) - floor(n/2)^2 = floor(5*n^2/4) (the last eq. checks for even and odd n). (End)
%H A032527 Vincenzo Librandi, <a href="/A032527/b032527.txt">Table of n, a(n) for n = 0..10000</a>
%H A032527 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F A032527 a(n) = 5*n^2/4+((-1)^n-1)/8. - _Omar E. Pol_, Sep 28 2011
%F A032527 G.f.: x*(1+3*x+x^2)/(1-2*x+2*x^3-x^4). - _Colin Barker_, Jan 06 2012
%F A032527 a(n) = a(-n); a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n>0, a(-1) = 1, a(0) = 0, a(1) = 1, a(2) = 5, n >= 3. (See the _Bruno Berselli_ recurrence and a general comment for primes 1 (mod 4) under A227541). - _Wolfdieter Lang_, Aug 08 2013
%F A032527 a(n) = Sum_{j=1..n} Sum{i=1..n} ceiling((i+j-n+1)/2). - _Wesley Ivan Hurt_, Mar 12 2015
%F A032527 Sum_{n>=1} 1/a(n) = Pi^2/30 + tan(Pi/(2*sqrt(5)))*Pi/sqrt(5). - _Amiram Eldar_, Jan 16 2023
%e A032527 From _Omar E. Pol_, Sep 28 2011 (Start):
%e A032527 Illustration of initial terms (In a precise representation the pentagons should appear strictly concentric):
%e A032527 .
%e A032527 .                                             o
%e A032527 .                                           o   o
%e A032527 .                            o            o   o   o
%e A032527 .                          o   o        o   o   o   o
%e A032527 .               o        o   o   o    o   o   o   o   o
%e A032527 .             o   o    o   o   o   o   o   o     o   o
%e A032527 .      o    o   o   o   o   o o   o     o   o o o   o
%e A032527 .    o   o   o     o     o       o       o         o
%e A032527 . o   o o     o o o       o o o o         o o o o o
%e A032527 .
%e A032527 . 1    5        11          20                31
%e A032527 .
%e A032527 (End)
%p A032527 A032527:=n->5*n^2/4+((-1)^n-1)/8: seq(A032527(n), n=0..100); # _Wesley Ivan Hurt_, Mar 12 2015
%t A032527 Table[Round[5n^2/4], {n, 0, 39}] (* _Alonso del Arte_, Sep 28 2011 *)
%o A032527 (PARI) a(n)=5*n^2>>2 \\ _Charles R Greathouse IV_, Sep 28 2011
%o A032527 (Magma) [5*n^2/4+((-1)^n-1)/8: n in [0..50]]; // _Vincenzo Librandi_, Sep 29 2011
%o A032527 (Haskell)
%o A032527 a032527 n = a032527_list !! n
%o A032527 a032527_list = scanl (+) 0 a047209_list
%o A032527 -- _Reinhard Zumkeller_, Jan 07 2012
%o A032527 (Python)
%o A032527 def A032527(n): return 5*n**2>>2  # _Chai Wah Wu_, Jul 30 2022
%Y A032527 Cf. A000290, A032528, A077043, A195041. Column 5 of A195040. [_Omar E. Pol_, Sep 28 2011]
%K A032527 nonn,easy
%O A032527 0,3
%A A032527 _N. J. A. Sloane_
%E A032527 New name from _Omar E. Pol_, Sep 28 2011