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%I #62 Sep 08 2022 08:44:32
%S 0,1,9,31,74,145,251,399,596,849,1165,1551,2014,2561,3199,3935,4776,
%T 5729,6801,7999,9330,10801,12419,14191,16124,18225,20501,22959,25606,
%U 28449,31495,34751,38224,41921,45849,50015,54426,59089,64011
%N a(n) = n*(7*n^2 - 1)/6.
%C 3-dimensional analog of centered polygonal numbers.
%C Sum of n triangular numbers starting from T(n), where T = A000217. E.g., a(4) = T(4) + T(5) + T(6) + T(7) = 10 + 15 + 21 + 28 = 74. - _Amarnath Murthy_, Jul 16 2004
%C Also as a(n) = (1/6)*(7*n^3-n), n>0: structured heptagonal diamond numbers (vertex structure 8). Cf. A100179 = alternate vertex; A000447 = structured diamonds; A100145 for more on structured numbers. - James A. Record (james.record(AT)gmail.com), Nov 07 2004
%C Partial sums of A069099, centered heptagonal numbers (A000566). - _Jonathan Vos Post_, Mar 16 2006
%C Binomial transform of (0, 1, 7, 7, 0, 0, 0, ...) and third partial sum of (0, 1, 6, 7, 7, 7, ...). - _Gary W. Adamson_, Oct 05 2015
%D E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 140.
%H Vincenzo Librandi, <a href="/A004126/b004126.txt">Table of n, a(n) for n = 0..5000</a>
%H T. P. Martin, <a href="http://dx.doi.org/10.1016/0370-1573(95)00083-6">Shells of atoms</a>, Phys. Reports, 273 (1996), 199-241, eq. (11).
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = C(2*n+1,3)-C(n+1,3), n>=0. - _Zerinvary Lajos_, Jan 21 2007
%F a(n) = A000447(n) - A000292(n). - _Zerinvary Lajos_, Jan 21 2007
%F G.f.: x*(1+5*x+x^2)/(1-x)^4. - _Colin Barker_, Mar 02 2012
%F E.g.f.: (x/6)*(7*x^2 + 21*x + 6)*exp(x). - _G. C. Greubel_, Oct 05 2015
%F a(n) = Sum_{i = n..2*n-1} A000217(i). - _Bruno Berselli_, Sep 06 2017
%F a(n) = n^3 + Sum_{k=0..n-1} k*(k+1)/2. Alternately, a(n) = A000578(n) + A000292(n-1) for n>0. - _Bruno Berselli_, May 23 2018
%p seq(binomial(2*n+1,3)-binomial(n+1,3), n=0..38); # _Zerinvary Lajos_, Jan 21 2007
%t Table[n (7 n^2 - 1)/6, {n, 0, 80}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 18 2011 *)
%o (Magma) [n*(7*n^2-1)/6: n in [0..50]]; // _Vincenzo Librandi_, May 15 2011
%o (Maxima) makelist(n*(7*n^2-1)/6,n,0,30); /* _Martin Ettl_, Jan 08 2013 */
%o (PARI) vector(100, n, n--; n*(7*n^2 - 1)/6) \\ _Altug Alkan_, Oct 06 2015
%Y 1/12*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.
%Y Cf. A000217, A000566, A016993, A069099.
%Y Cf. A000447, A000292.
%K nonn,easy
%O 0,3
%A Albert D. Rich (Albert_Rich(AT)msn.com)
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