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A005214 Triangular numbers together with squares (excluding 0). 12

%I

%S 1,3,4,6,9,10,15,16,21,25,28,36,45,49,55,64,66,78,81,91,100,105,120,

%T 121,136,144,153,169,171,190,196,210,225,231,253,256,276,289,300,324,

%U 325,351,361,378,400,406,435,441,465,484,496,528,529,561,576,595,625,630,666,676

%N Triangular numbers together with squares (excluding 0).

%C A010052(a(n)) + A010054(a(n)) > 0, A010052(a(A193714(n))) = 1, A010054(a(A193715(n))) = 1. - _Reinhard Zumkeller_, Aug 03 2011

%D Hofstadter, D. R., 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 Reinhard Zumkeller, <a href="/A005214/b005214.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SquareTriangularNumber.html">Square Triangular Number</a>

%H D. 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>.

%p a := proc(n) floor(sqrt(n)): floor(sqrt(n+n)):

%p `if`(n+n = %*% + % or n = %% * %%, n, NULL) end: # _Peter Luschny_, May 01 2014

%t 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 (Haskell)

%o import Data.List.Ordered (union)

%o a005214 n = a005214_list !! (n-1)

%o a005214_list = tail $ union a000290_list a000217_list

%o -- _Reinhard Zumkeller_, Feb 15 2015, Aug 03 2011

%o (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

%Y Cf. A054686.

%Y Cf. A001110; union of A000290 and A000217; A117704 (first differences), A193711 (partial sums); A193748, A193749 (partitions into).

%Y Cf. A241241 (subsequence).

%Y Cf. A242401 (complement).

%K nonn,easy

%O 1,2

%A _Russ Cox_, Jun 14 1998

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Last modified November 18 04:44 EST 2019. Contains 329248 sequences. (Running on oeis4.)