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%I #58 Sep 08 2022 08:45:03
%S 0,1,10,39,100,205,366,595,904,1305,1810,2431,3180,4069,5110,6315,
%T 7696,9265,11034,13015,15220,17661,20350,23299,26520,30025,33826,
%U 37935,42364,47125,52230,57691,63520,69729,76330,83335,90756,98605,106894,115635,124840
%N a(n) = n*(2*n^2 - 2*n + 1).
%C Mean of the first four nonnegative powers of 2n+1, i.e., ((2n+1)^0 + (2n+1)^1 + (2n+1)^2 + (2n+1)^3)/4. E.g., a(2) = (1 + 3 + 9 + 27)/4 = 10.
%C Equatorial structured meta-diamond numbers, the n-th number from an equatorial structured n-gonal diamond number sequence. There are no 1- or 2-gonal diamonds, so 1 and (n+2) are used as the first and second terms since all the sequences begin as such. - James A. Record (james.record(AT)gmail.com), Nov 07 2004
%C Starting with offset 1 = row sums of triangle A143803. - _Gary W. Adamson_, Sep 01 2008
%C Form an array from the antidiagonals containing the terms in A002061 to give antidiagonals 1; 3,3; 7,4,7; 13,8,8,13; 21,14,9,14,21; and so on. The difference between the sum of the terms in n+1 X n+1 matrices and those in n X n matrices is a(n) for n>0. - _J. M. Bergot_, Jul 08 2013
%C Sum of the numbers from (n-1)^2 to n^2. - _Wesley Ivan Hurt_, Sep 08 2014
%H Harry J. Smith, <a href="/A059722/b059722.txt">Table of n, a(n) for n = 0..1000</a>
%H Bruno Berselli, <a href="/A059722/a059722_1.jpg">Table of sequences with the formula m*(m+1)^2 - (k+2)*m^2</a> (table includes this sequence for k = 2-m, m >= 0).
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = A053698(2*n-1)/4.
%F a(n) = Sum_{j=1..n} ((n+j-1)^2-j^2+1). - _Zerinvary Lajos_, Sep 13 2006
%F From _R. J. Mathar_, Sep 02 2008: (Start)
%F G.f.: x*(1 + x)*(1 + 5*x)/(1 - x)^4.
%F a(n) = A002414(n-1) + A002414(n).
%F a(n+1) - a(n) = A136392(n+1). (End)
%F a(n) = (A000290(n) + A000290(n+1)) * (A000217(n+1) - A000217(n)). - _J. M. Bergot_, Nov 02 2012
%F a(n) = n * A001844(n-1). - _Doug Bell_, Aug 18 2015
%F a(n) = A000217(n^2) - A000217(n^2-2*n). - _Bruno Berselli_, Jun 26 2018
%F E.g.f.: exp(x)*x*(1 + 4*x + 2*x^2). - _Stefano Spezia_, Jun 20 2021
%p A059722:=n->n*(2*n^2 - 2*n + 1): seq(A059722(n), n=0..50); # _Wesley Ivan Hurt_, Sep 08 2014
%t Table[n (2 n^2 - 2 n + 1), {n, 0, 50}] (* _Wesley Ivan Hurt_, Sep 08 2014 *)
%o (PARI) { for (n=0, 1000, write("b059722.txt", n, " ", n*(2*n^2 - 2*n + 1)); ) } \\ _Harry J. Smith_, Jun 28 2009
%o (Magma) [n*(2*n^2 - 2*n + 1) : n in [0..50]]; // _Wesley Ivan Hurt_, Sep 08 2014
%Y Cf. A000330, A005900, A081436, A100178, A100179, A059722: "equatorial" structured diamonds; A000447: "polar" structured meta-diamond; A006484 for other structured meta numbers; and A100145 for more on structured numbers. - James A. Record (james.record(AT)gmail.com), Nov 07 2004
%Y Cf. A000217, A000290, A001844, A002061, A002414, A053698, A059721, A059723, A136392, A143803.
%K nonn,easy
%O 0,3
%A _Henry Bottomley_, Feb 07 2001
%E Edited with new definition by _N. J. A. Sloane_, Aug 29 2008
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