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A254749 1-gonal pyramidal numbers.

%I A254749 #22 Aug 04 2025 21:11:40
%S A254749 1,2,2,0,-5,-14,-28,-48,-75,-110,-154,-208,-273,-350,-440,-544,-663,
%T A254749 -798,-950,-1120,-1309,-1518,-1748,-2000,-2275,-2574,-2898,-3248,
%U A254749 -3625,-4030,-4464,-4928,-5423,-5950,-6510,-7104,-7733,-8398,-9100,-9840,-10619,-11438
%N A254749 1-gonal pyramidal numbers.
%C A254749 Not strictly pyramidal numbers, but the result of using the Wikipedia formula with r = 1.
%C A254749 Essentially the same as A129936 and A005586.
%H A254749 Colin Barker, <a href="/A254749/b254749.txt">Table of n, a(n) for n = 1..1000</a>
%H A254749 Wikipedia, <a href="http://en.wikipedia.org/wiki/Pyramidal_number">Pyramidal number</a>.
%H A254749 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A254749 a(n) = n*(4 + 3*n - n^2)/6.
%F A254749 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
%F A254749 G.f.: x*(1 - 2*x)/(x-1)^4.
%F A254749 a(n) = A005581(-n) = -A005586(n-4) = -A129936(n-2) for all n in Z. - _Michael Somos_, Jul 28 2015
%F A254749 E.g.f.: -exp(x)*x*(x^2 - 6)/6. - _Elmo R. Oliveira_, Aug 04 2025
%e A254749 G.f. = x + 2*x^2 + 2*x^3 - 5*x^5 - 14*x^6 - 28*x^7 - 48*x^8 - 75*x^9 + ...
%t A254749 Table[(n*(4+3*n-n^2))/6, {n,1,60}] (* or *) LinearRecurrence[{4,-6,4,-1}, {1, 2, 2, 0}, 60] (* _G. C. Greubel_, Aug 03 2018 *)
%o A254749 (PARI) ppg(r, n) = (3*n^2+n^3*(r-2)-n*(r-5))/6
%o A254749 vector(100, n, ppg(1, n))
%o A254749 (Magma) [(n*(4+3*n-n^2))/6: n in [1..60]]; // _G. C. Greubel_, Aug 03 2018
%Y A254749 Cf. A005581, A005586, A129936.
%K A254749 sign,easy
%O A254749 1,2
%A A254749 _Colin Barker_, Feb 07 2015