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%I A172076 #54 Sep 08 2022 08:45:50
%S A172076 0,1,17,62,150,295,511,812,1212,1725,2365,3146,4082,5187,6475,7960,
%T A172076 9656,11577,13737,16150,18830,21791,25047,28612,32500,36725,41301,
%U A172076 46242,51562,57275,63395,69936,76912,84337,92225,100590,109446,118807,128687
%N A172076 a(n) = n*(n+1)*(14*n-11)/6.
%C A172076 Generated by the formula n*(n+1)*(2*d*n-(2*d-3))/6 for d=7.
%C A172076 From _Bruno Berselli_, Dec 14 2010: (Start)
%C A172076 In fact, the sequence is related to A001106 by a(n) = n*A001106(n) - Sum_{k=0..n-1} A001106(k) and this is the case d=7 in the identity n*(n*(d*n-d+2)/2) - Sum_{k=0..n-1} k*(d*k-d+2)/2 = n*(n+1)*(2*d*n-2*d+3)/6.
%C A172076 Also 16-gonal (or hexadecagonal) pyramidal numbers.
%C A172076 Inverse binomial transform of this sequence: 0, 1, 15, 14, 0, 0 (0 continued). (End)
%D A172076 E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93. [From _Bruno Berselli_, Feb 13 2014]
%H A172076 Vincenzo Librandi, <a href="/load/view.php?a=aHR0cDovL29laXMub3JnL0ExNzIwNzYvYjE3MjA3Ni50eHQ">Table of n, a(n) for n = 0..1000</a>
%H A172076 B. Berselli, A description of the recursive method in Comments lines: website <a href="/load/view.php?a=aHR0cDovL3d3dy5sYW5vc3RyYS1tYXRlbWF0aWNhLm9yZy8yMDA4LzEyL3NlcXVlbnplLW51bWVyaWNoZS1lLXByb2NlZGltZW50aS5odG1s">Matem@ticamente</a> (in Italian), 2008.
%H A172076 <a href="/load/view.php?a=aHR0cDovL29laXMub3JnL2luZGV4L1BzI3B5cmFtaWRhbF9udW1iZXJz">Index to sequences related to pyramidal numbers</a>.
%H A172076 <a href="/load/view.php?a=aHR0cDovL29laXMub3JnL2luZGV4L1BzI3B5cmFtaWRhbF9udW1iZXJz">Index to sequences related to pyramidal numbers</a>
%H A172076 <a href="/load/view.php?a=aHR0cDovL29laXMub3JnL2luZGV4L1JlYyNvcmRlcl8wNA">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A172076 G.f.: x*(1+13*x)/(1-x)^4. - _Bruno Berselli_, Dec 15 2010
%F A172076 a(n) = Sum_{i=0..n} A051868(i). - _Bruno Berselli_, Dec 15 2010
%F A172076 a(n) = Sum_{i=0..n-1} (n-i)*(14*i+1), with a(0)=0. - _Bruno Berselli_, Feb 10 2014
%F A172076 E.g.f.: x*(6 + 45*x + 14*x^2)*exp(x)/6. - _G. C. Greubel_, Aug 30 2019
%p A172076 A172076:=n->n*(n+1)*(14*n-11)/6; seq(A172076(n), n=0..50); # _Wesley Ivan Hurt_, Feb 26 2014
%t A172076 LinearRecurrence[{4,-6,4,-1}, {0, 1, 17, 62}, 50] (* _Vincenzo Librandi_, Mar 01 2012 *)
%o A172076 (PARI) vector(40, n, n*(n-1)*(14*n-25)/6) \\ _G. C. Greubel_, Aug 30 2019
%o A172076 (Magma) [n*(n+1)*(14*n-11)/6: n in [0..40]] // _G. C. Greubel_, Aug 30 2019
%o A172076 (Sage) [n*(n+1)*(14*n-11)/6 for n in (0..40)] # _G. C. Greubel_, Aug 30 2019
%o A172076 (GAP) List([0..40], n-> n*(n+1)*(14*n-11)/6); # _G. C. Greubel_, Aug 30 2019
%Y A172076 Cf. similar sequences listed in A237616.
%K A172076 nonn,easy
%O A172076 0,3
%A A172076 _Vincenzo Librandi_, Jan 25 2010

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