A005214 Triangular numbers together with squares (excluding 0).

1, 3, 4, 6, 9, 10, 15, 16, 21, 25, 28, 36, 45, 49, 55, 64, 66, 78, 81, 91, 100, 105, 120, 121, 136, 144, 153, 169, 171, 190, 196, 210, 225, 231, 253, 256, 276, 289, 300, 324, 325, 351, 361, 378, 400, 406, 435, 441, 465, 484, 496, 528, 529, 561, 576, 595, 625, 630, 666, 676

Offset: 1

Author

Russ Cox, Jun 14 1998

Keywords

• nonn
• easy

References

• 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.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• 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; Part 1, Part 2.
• Eric Weisstein's World of Mathematics, Square Triangular Number.

Crossrefs

Union of A000290 and A000217.

Cf. A001110, A054686, A157259, A117704 (first differences), A193711 (partial sums), A193748, A193749 (partitions into).

Cf. A010052, A010054, A193714, A193715.

Cf. A241241 (subsequence), A242401 (complement).

Programs

Haskell

import Data.List.Ordered (union)
a005214 n = a005214_list !! (n-1)
a005214_list = tail $ union a000290_list a000217_list
-- Reinhard Zumkeller, Feb 15 2015, Aug 03 2011

Maple

a := proc(n) floor(sqrt(n)): floor(sqrt(n+n)):
`if`(n+n = %*% + % or n = %% * %%, n, NULL) end: # Peter Luschny, May 01 2014

Mathematica

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 *)

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

PARI

isok(m) = ispolygonal(m,3) || ispolygonal(m,4); \\ Michel Marcus, Mar 13 2021

Formula

From Reinhard Zumkeller, Aug 03 2011: (Start)

A010052(a(n)) + A010054(a(n)) > 0.

A010052(a(A193714(n))) = 1.

A010054(a(A193715(n))) = 1. (End)

a(n) ~ c * n^2, where c = 3 - 2*sqrt(2) = A157259 - 4 = 0.171572... . - Amiram Eldar, Apr 04 2025