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A373711 Numbers that are simultaneously k-gonal and k-gonal pyramidal for some k >= 3.

%I A373711 #33 Jan 09 2025 11:47:09
%S A373711 0,1,10,120,175,441,946,1045,1540,4900,5985,7140,23001,23725,48280,
%T A373711 195661,245905,314755,801801,975061,1169686,3578401,10680265,27453385,
%U A373711 55202400,63016921,101337426,132361021,197427385,258815701,432684460,477132085,837244045
%N A373711 Numbers that are simultaneously k-gonal and k-gonal pyramidal for some k >= 3.
%C A373711 Matt Parker calls these numbers cannonball numbers, after the cannonball problem involving finding a number both square and square pyramidal.
%C A373711 If m==2 (mod 3), the m-gonal number A057145(m,(m^3-6*m^2+3*m+19)/9) = (m^2-4*m-2)*(m^2-4*m+1)*(m^3-6*m^2+3*m+19)/162 = A344410((m-2)/3) is a term. See comment in A027696. - _Pontus von Brömssen_, Dec 09 2024
%H A373711 Pontus von Brömssen, <a href="/A373711/b373711.txt">Table of n, a(n) for n = 1..46</a>
%H A373711 Brady Haran and Matt Parker, <a href="https://www.numberphile.com/videos/90525801730-cannon-balls">90,525,801,730 Cannon Balls</a>, Numberphile video (2019).
%H A373711 Brady Haran and Matt Parker, <a href="https://www.numberphile.com/cannon-ball-numbers">Cannon Ball Numbers</a>, Numberphile (2019).
%H A373711 <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>.
%F A373711 a(n) = A057145(A379973(n),A379974(n)) = A080851(A379973(n)-2,A379975(n)-1). - _Pontus von Brömssen_, Jan 09 2025
%e A373711 4900 is a term because it is both the 70th square and the 24th square pyramidal number.
%Y A373711 Subsequences: A027568, A344376, A344410.
%Y A373711 Cf. A027669, A027696, A057145, A080851, A379973, A379974, A379975.
%K A373711 nonn
%O A373711 1,3
%A A373711 _Kelvin Voskuijl_, Jun 14 2024
%E A373711 a(13)-a(33) from _Pontus von Brömssen_, Dec 08 2024