A193714 -id:A193714 - cp's OEIS frontend

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

Russ Cox, Jun 14 1998

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.

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.

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).

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

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

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 *)

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

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

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

A193715 Positions of triangular numbers (A000217) in the union of squares and triangular numbers (A005214).

Original entry on oeis.org

1, 2, 4, 6, 7, 9, 11, 12, 13, 15, 17, 18, 20, 22, 23, 25, 27, 29, 30, 32, 34, 35, 37, 39, 41, 42, 44, 46, 47, 49, 51, 52, 54, 56, 58, 59, 61, 63, 64, 66, 68, 70, 71, 73, 75, 76, 78, 80, 81, 82, 84, 86, 87, 89, 91, 92, 94, 96, 98, 99, 101, 103, 104, 106, 108

Offset: 1

Reinhard Zumkeller, Aug 03 2011

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000217, A005214, A010054, A193714.

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a193715 n = a193715_list !! (n-1)
a193715_list =
   map ((+ 1) . fromJust . (`elemIndex` a005214_list)) $ tail a000217_list

With[{max = 2500}, Position[Union[Join[Accumulate[Range[Floor[(Sqrt[8*max + 1] - 1)/2]]], Range[Floor[Sqrt[max]]]^2]], ?(IntegerQ[Sqrt[8*# + 1]] &)] // Flatten] (* _Amiram Eldar, Apr 04 2025 *)

A010054(A005214(a(n))) = 1.

a(n) ~ c * n, where c = 1 + sqrt(2)/2 = 1.707106... . - Amiram Eldar, Apr 04 2025

cp's OEIS Frontend

A005214 Triangular numbers together with squares (excluding 0).

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