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A005214 Triangular numbers together with squares (excluding 0).

%I A005214 #65 Apr 04 2025 04:18:51
%S A005214 1,3,4,6,9,10,15,16,21,25,28,36,45,49,55,64,66,78,81,91,100,105,120,
%T A005214 121,136,144,153,169,171,190,196,210,225,231,253,256,276,289,300,324,
%U A005214 325,351,361,378,400,406,435,441,465,484,496,528,529,561,576,595,625,630,666,676
%N A005214 Triangular numbers together with squares (excluding 0).
%D A005214 Douglas R. Hofstadter, Fluid Concepts and Creative Analogies: Computer Models of the Fundamental Mechanisms of Thought, (together with the Fluid Analogies Research Group), NY: Basic Books, 1995, p. 15.
%H A005214 Reinhard Zumkeller, <a href="/A005214/b005214.txt">Table of n, a(n) for n = 1..10000</a>
%H A005214 Douglas R. Hofstadter, Analogies and Sequences: Intertwined Patterns of Integers and Patterns of Thought Processes, DIMACS Conference on Challenges of Identifying Integer Sequences, Rutgers University, October 10 2014; <a href="http://vimeo.com/109139374">Part 1</a>, <a href="http://vimeo.com/109139377">Part 2</a>.
%H A005214 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SquareTriangularNumber.html">Square Triangular Number</a>.
%F A005214 From _Reinhard Zumkeller_, Aug 03 2011: (Start)
%F A005214 A010052(a(n)) + A010054(a(n)) > 0.
%F A005214 A010052(a(A193714(n))) = 1.
%F A005214 A010054(a(A193715(n))) = 1. (End)
%F A005214 a(n) ~ c * n^2, where c = 3 - 2*sqrt(2) = A157259 - 4 = 0.171572... . - _Amiram Eldar_, Apr 04 2025
%p A005214 a := proc(n) floor(sqrt(n)): floor(sqrt(n+n)):
%p A005214 `if`(n+n = %*% + % or n = %% * %%, n, NULL) end: # _Peter Luschny_, May 01 2014
%t A005214 With[{upto=700},Module[{maxs=Floor[Sqrt[upto]], maxt=Floor[(Sqrt[8upto+1]- 1)/2]},Union[Join[Range[maxs]^2, Table[(n(n+1))/2,{n,maxt}]]]]] (* _Harvey P. Dale_, Sep 17 2011 *)
%o A005214 (Haskell)
%o A005214 import Data.List.Ordered (union)
%o A005214 a005214 n = a005214_list !! (n-1)
%o A005214 a005214_list = tail $ union a000290_list a000217_list
%o A005214 -- _Reinhard Zumkeller_, Feb 15 2015, Aug 03 2011
%o A005214 (PARI) upTo(lim)=vecsort(concat(vector(sqrtint(lim\1),n,n^2), vector(floor(sqrt(2+2*lim)-1/2),n,n*(n+1)/2)),,8) \\ _Charles R Greathouse IV_, Aug 04 2011
%o A005214 (PARI) isok(m) = ispolygonal(m,3) || ispolygonal(m,4); \\ _Michel Marcus_, Mar 13 2021
%Y A005214 Union of A000290 and A000217.
%Y A005214 Cf. A001110, A054686, A157259, A117704 (first differences), A193711 (partial sums), A193748, A193749 (partitions into).
%Y A005214 Cf. A010052, A010054, A193714, A193715.
%Y A005214 Cf. A241241 (subsequence), A242401 (complement).
%K A005214 nonn,easy
%O A005214 1,2
%A A005214 _Russ Cox_, Jun 14 1998