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

1,2

References

D. 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

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; Part 1, Part 2.

Eric Weisstein's World of Mathematics, Square Triangular Number

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)

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 *)

Prog

(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
(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

Crossrefs

Cf. A054686.

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

Cf. A010052, A010054, A193714, A193715.

Cf. A241241 (subsequence).

Cf. A242401 (complement).

Sequence in context: A034706 A245810 A054686 * A268110 A124093 A025061

Adjacent sequences: A005211 A005212 A005213 * A005215 A005216 A005217

Keyword

nonn,easy

Author

Russ Cox, Jun 14 1998

Status

approved