A005214 -id:A005214 - cp's OEIS frontend

A193714 Positions of squares (A000290) in the union of squares and triangular numbers (A005214).

1, 3, 5, 8, 10, 12, 14, 16, 19, 21, 24, 26, 28, 31, 33, 36, 38, 40, 43, 45, 48, 50, 53, 55, 57, 60, 62, 65, 67, 69, 72, 74, 77, 79, 81, 83, 85, 88, 90, 93, 95, 97, 100, 102, 105, 107, 109, 112, 114, 117, 119, 122, 124, 126, 129, 131, 134, 136, 138, 141, 143

Offset: 1

Reinhard Zumkeller, Aug 03 2011

nonn

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

Cf. A000290, A005214, A010052, A014176, A193715.

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a193714 n = a193714_list !! (n-1)
a193714_list =
   map ((+ 1) . fromJust . (`elemIndex` a005214_list)) $ tail a000290_list

With[{max = 4000}, Position[Union[Join[Accumulate[Range[Floor[(Sqrt[8*max + 1] - 1)/2]]], Range[Floor[Sqrt[max]]]^2]], ?(IntegerQ[Sqrt[#]] &)] // Flatten] (* _Amiram Eldar, Apr 04 2025 *)

A010052(A005214(a(n))) = 1.

a(n) ~ c * n, where c = 1 + sqrt(2) = A014176 = 2.414213... . - 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

A193748 Number of partitions of n into parts that are squares or triangular numbers (A005214).

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 5, 6, 7, 10, 13, 14, 19, 23, 26, 33, 41, 45, 56, 67, 76, 91, 108, 120, 143, 168, 188, 218, 255, 284, 328, 378, 422, 480, 551, 612, 695, 789, 876, 984, 1115, 1233, 1380, 1549, 1713, 1907, 2135, 2349, 2607, 2902, 3192, 3524, 3911, 4287, 4722

Offset: 0

Reinhard Zumkeller, Aug 03 2011

nonn

			a(10) = #{10, 9+1, 6+4, 6+3+1, 6+1+1+1+1, 4+4+1+1, 4+3+3, 4+3+1+1+1, 4+6x1, 3+3+3+1, 3+3+1+1+1+1, 3+7x1, 10x1} = 13;
a(11) = #{10+1, 9+1+1, 6+4+1, 6+3+1+1, 6+1+1+1+1+1, 4+4+3, 4+4+1+1+1, 4+3+1+1+1+1, 4+7x1, 3+3+3+1+1, 3+3+5x1, 3+8x1, 11x1} = 14;
a(12) = #{10+1+1, 9+3, 9+1+1+1, 6+6, 6+4+1+1, 6+3+3, 6+3+1+1+1, 6+6x1, 4+4+4, 4+4+3+1, 4+4+1+1+1+1, 4+3+3+1+1, 4+3+5x1, 4+8x1, 3+3+3+3, 3+3+3+1+1+1, 3+3+6x1, 3+9x1, 12x1} = 19.

Reinhard Zumkeller, Table of n, a(n) for n = 0..100

Cf. A000290, A000217, A001156, A007294, A193749, A000041.

a193748 = p a005214_list where
   p _          0 = 1
   p ks'@(k:ks) m
     | m < k      = 0
     | otherwise  = p ks' (m - k) + p ks m

A193749 Number of partitions of n into distinct parts that are squares or triangular numbers (A005214).

Original entry on oeis.org

1, 1, 0, 1, 2, 1, 1, 2, 1, 2, 4, 2, 1, 4, 4, 3, 5, 4, 3, 7, 7, 4, 6, 7, 6, 10, 10, 5, 10, 14, 10, 12, 14, 9, 14, 20, 13, 13, 20, 17, 18, 24, 17, 16, 27, 26, 22, 27, 25, 26, 35, 31, 26, 35, 37, 36, 42, 37, 35, 48, 47, 40, 49, 49, 48, 60, 58, 49, 61, 66, 61

Offset: 0

Reinhard Zumkeller, Aug 03 2011

nonn

			a(10) = #{10, 9+1, 6+4, 6+3+1} = 4;
a(11) = #{10+1, 6+4+1} = 2;
a(12) = #{9+3} = 1;
a(13) = #{10+3, 9+4, 9+3+1, 6+4+3} = 4.

Reinhard Zumkeller, Table of n, a(n) for n = 0..250

Cf. A000290, A000217, A033461, A024940, A193748, A000009.

a193749 = p a005214_list where
   p _      0    = 1
   p (k:ks) m
     | m < k     = 0
     | otherwise = p ks (m - k) + p ks m

A001110 Square triangular numbers: numbers that are both triangular and square.

Original entry on oeis.org

0, 1, 36, 1225, 41616, 1413721, 48024900, 1631432881, 55420693056, 1882672131025, 63955431761796, 2172602007770041, 73804512832419600, 2507180834294496361, 85170343853180456676, 2893284510173841030625, 98286503002057414584576, 3338847817559778254844961, 113422539294030403250144100

Offset: 0

N. J. A. Sloane

Satisfies a recurrence of S_r type for r=36: 0, 1, 36 and a(n-1)*a(n+1)=(a(n)-1)^2. First observed by Colin Dickson in alt.math.recreational, Mar 07 2004. - Rainer Rosenthal, Mar 14 2004
For every n, a(n) is the first of three triangular numbers in geometric progression. The third number in the progression is a(n+1). The middle triangular number is sqrt(a(n)*a(n+1)). Chen and Fang prove that four distinct triangular numbers are never in geometric progression. - T. D. Noe, Apr 30 2007
The sum of any two terms is never equal to a Fermat number. - Arkadiusz Wesolowski, Feb 14 2012
Conjecture: No a(2^k), where k is a nonnegative integer, can be expressed as a sum of a positive square number and a positive triangular number. - Ivan N. Ianakiev, Sep 19 2012
For n=2k+1, A010888(a(n))=1 and for n=2k, k > 0, A010888(a(n))=9. - Ivan N. Ianakiev, Oct 12 2013
For n > 0, these are the triangular numbers which are the sum of two consecutive triangular numbers, for instance 36 = 15 + 21 and 1225 = 595 + 630. - Michel Marcus, Feb 18 2014
The sequence is the case P1 = 36, P2 = 68, Q = 1 of the 3-parameter family of 4th order linear divisibility sequences found by Williams and Guy. - Peter Bala, Apr 03 2014
For n=2k, k > 0, a(n) is divisible by 12 and is therefore abundant. I conjecture that for n=2k+1 a(n) is deficient [true for k up to 43 incl.]. - Ivan N. Ianakiev, Sep 30 2014
The conjecture is true for all k > 0 because: For n=2k+1, k > 0, a(n) is odd. If a(n) is a prime number, it is deficient; otherwise a(n) has one or two distinct prime factors and is therefore deficient again. So for n=2k+1, k > 0, a(n) is deficient. - Muniru A Asiru, Apr 13 2016
Numbers k for which A139275(k) is a perfect square. - Bruno Berselli, Jan 16 2018

			a(2) = ((17 + 12*sqrt(2))^2 + (17 - 12*sqrt(2))^2 - 2)/32 = (289 + 24*sqrt(2) + 288 + 289 - 24*sqrt(2) + 288 - 2)/32 = (578 + 576 - 2)/32 = 1152/32 = 36 and 6^2 = 36 = 8*9/2 => a(2) is both the 6th square and the 8th triangular number.

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 193.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 38, 204.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923; see Vol. 2, p. 10.
Martin Gardner, Time Travel and other Mathematical Bewilderments, Freeman & Co., 1988, pp. 16-17.
Miodrag S. Petković, Famous Puzzles of Great Mathematicians, Amer. Math. Soc. (AMS), 2009, p. 64.
J. H. Silverman, A Friendly Introduction to Number Theory, Prentice Hall, 2001, p. 196.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 257-259.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 93.

Kade Robertson, Table of n, a(n) for n = 0..600 [This replaces an earlier b-file computed by T. D. Noe]
Nikola Adžaga, Andrej Dujella, Dijana Kreso, and Petra Tadić, On Diophantine m-tuples and D(n)-sets, 2018.
Muniru A. Asiru, All square chiliagonal numbers, International Journal of Mathematical Education in Science and Technology, Volume 47, 2016 - Issue 7.
Tom Beldon and Tony Gardiner, Triangular numbers and perfect squares, The Mathematical Gazette, 2002, pp. 423-431, esp pp. 424-426.
K. S. Brown, Square Triangular Numbers.
P. Catarino, H. Campos, and P. Vasco, On some identities for balancing and cobalancing numbers, Annales Mathematicae et Informaticae, 45 (2015) pp. 11-24.
Kwang-Wu Chen and Yu-Ren Pan, Greatest Common Divisors of Shifted Horadam Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.5.8.
Yong-Gao Chen and Jin-Hui Fang, Triangular numbers in geometric progression, INTEGERS 7 (2007), #A19.
F. Javier de Vega, On the parabolic partitions of a number, J. Alg., Num. Theor., and Appl. (2023) Vol. 61, No. 2, 135-169.
Colin Dickson et al., ratio of integers = sqrt(2), thread in newsgroup alt.math.recreational, March 7, 2004.
H. G. Forder, A Simple Proof of a Result on Diophantine Approximation, Math. Gaz., 47 (1963), 237-238.
M. Gardner, Letter to N. J. A. Sloane, circa Aug 11 1980, concerning A001110, A027568, A039596, etc.
Bill Gosper, The Triangular Squares, 2014.
Jon Grantham and Hester Graves, The abc Conjecture Implies That Only Finitely Many Cullen Numbers Are Repunits, arXiv:2009.04052 [math.NT], 2020. Mentions this sequence.
Gillian Hatch, Pythagorean Triples and Triangular Square Numbers, Mathematical Gazette 79 (1995), 51-55.
P. Lafer, Discovering the square-triangular numbers, Fib. Quart., 9 (1971), 93-105.
Roger B. Nelsen, Multi-Polygonal Numbers, Mathematics Magazine, Vol. 89, No. 3 (June 2016), pp. 159-164.
A. Nowicki, The numbers a^2+b^2-dc^2, J. Int. Seq. 18 (2015) # 15.2.3.
Matthew Parker, The first 5000 square triangular numbers (7-Zip compressed file).
J. L. Pietenpol, A. V. Sylwester, E. Just, and R. M. Warten, Problem E1473, Amer. Math. Monthly, Vol. 69, No. 2 (Feb. 1962), pp. 168-169. (From the editorial note on p. 169 of this source, we learn that the question about the existence of perfect squares in the sequence of triangular numbers cropped up in the Euler-Goldbach Briefwechsel of 1730; the translation into English of the relevant letters can be found at Correspondence of Leonhard Euler with Christian Goldbach (part II), pp. 614-615.) - _José Hernández_, May 24 2022
Vladimir Pletser, Recurrent Relations for Multiple of Triangular Numbers being Triangular Numbers, arXiv:2101.00998 [math.NT], 2020.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
D. A. Q., Triangular square numbers: a postscript, Math. Gaz., 56 (1972), 311-314.
K. Ramsey, Re_Generalized_Proof_re_Square_Triangular_Numbers. [Broken link]
K. Ramsey, Generalized Proof re Square Triangular Numbers, digest of 2 messages in Triangular_and_Fibonacci_Numbers Yahoo group, May 27, 2005 - Oct 10, 2011.
Fabio Roman, On certain ratios regarding integer numbers which are both triangulars and squares, arXiv:1703.06701 [math.NT], 2017.
Jon E. Schoenfield, Prime factorization of a(n) for n = 1..300.
Jaap Spies, A Bit of Math, The Art of Problem Solving, Jaap Spies Publishers (2019).
Ralf Stephan, Boring proof of a nonlinearity.
UWC, Problem A, Nieuw Archief voor Wiskunde, Dec 2004; Jaap Spies, Solution.
Michel Waldschmidt, Continued fractions, École de recherche CIMPA-Oujda, Théorie des Nombres et ses Applications, 18 - 29 mai 2015: Oujda (Maroc).
Eric Weisstein's World of Mathematics, Square Triangular Number.
Eric Weisstein's World of Mathematics, Triangular Number.
Wikipedia, Square triangular number.
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences, Integers, Volume 12A (2012) The John Selfridge Memorial Volume.
Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

Cf. A001108, A001109.
Other S_r type sequences are S_4=A000290, S_5=A004146, S_7=A054493, S_8=A001108, S_9=A049684, S_20=A049683, S_36=this sequence, S_49=A049682, S_144=A004191^2.
Cf. A001014; intersection of A000217 and A000290; A010052(a(n))*A010054(a(n)) = 1.
Cf. A005214, A054686, A232847 and also A233267 (reveals an interesting divisibility pattern for this sequence).
Cf. A240129 (triangular numbers that are squares of triangular numbers), A100047.
See A229131, A182334, A299921 for near-misses.

a001110 n = a001110_list !! n
a001110_list = 0 : 1 : (map (+ 2) $
   zipWith (-) (map (* 34) (tail a001110_list)) a001110_list)
-- Reinhard Zumkeller, Oct 12 2011

[n le 2 select n-1 else Floor((6*Sqrt(Self(n-1)) - Sqrt(Self(n-2)))^2): n in [1..20]]; // Vincenzo Librandi, Jul 22 2015

a:=17+12*sqrt(2); b:=17-12*sqrt(2); A001110:=n -> expand((a^n + b^n - 2)/32); seq(A001110(n), n=0..20); # Jaap Spies, Dec 12 2004
A001110:=-(1+z)/((z-1)*(z**2-34*z+1)); # Simon Plouffe in his 1992 dissertation

f[n_]:=n*(n+1)/2; lst={}; Do[If[IntegerQ[Sqrt[f[n]]],AppendTo[lst,f[n]]],{n,0,10!}]; lst (* Vladimir Joseph Stephan Orlovsky, Feb 12 2010 *)
Table[(1/8) Round[N[Sinh[2 n ArcSinh[1]]^2, 100]], {n, 0, 20}] (* Artur Jasinski, Feb 10 2010 *)
Transpose[NestList[Flatten[{Rest[#],34Last[#]-First[#]+2}]&, {0,1},20]][[1]]  (* Harvey P. Dale, Mar 25 2011 *)
LinearRecurrence[{35, -35, 1}, {0, 1, 36}, 20] (* T. D. Noe, Mar 25 2011 *)
LinearRecurrence[{6,-1},{0,1},20]^2 (* Harvey P. Dale, Oct 22 2012 *)
(* Square = Triangular = Triangular = A001110 *)
ChebyshevU[#-1,3]^2==Binomial[ChebyshevT[#/2,3]^2,2]==Binomial[(1+ChebyshevT[#,3])/2,2]=={1,36,1225,41616,1413721}[[#]]&@Range[5]
True (* Bill Gosper, Jul 20 2015 *)
L=0;r={};Do[AppendTo[r,L];L=1+17*L+6*Sqrt[L+8*L^2],{i,1,19}];r (* Kebbaj Mohamed Reda, Aug 02 2023 *)

a=vector(100);a[1]=1;a[2]=36;for(n=3,#a,a[n]=34*a[n-1]-a[n-2]+2);a \\ Charles R Greathouse IV, Jul 25 2011

;; With memoizing definec-macro from Antti Karttunen's IntSeq-library.
(definec (A001110 n) (if (< n 2) n (+ 2 (- (* 34 (A001110 (- n 1))) (A001110 (- n 2))))))
;; Antti Karttunen, Dec 06 2013

;; For testing whether n is in this sequence:
(define (inA001110? n) (and (zero? (A068527 n)) (inA001109? (floor->exact (sqrt n)))))
(define (inA001109? n) (= (* 8 n n) (floor->exact (* (sqrt 8) n (ceiling->exact (* (sqrt 8) n))))))
;; Antti Karttunen, Dec 06 2013

a(0) = 0, a(1) = 1; for n >= 2, a(n) = 34 * a(n-1) - a(n-2) + 2.
G.f.: x*(1 + x) / (( 1 - x )*( 1 - 34*x + x^2 )).
a(n-1) * a(n+1) = (a(n)-1)^2. - Colin Dickson, posting to alt.math.recreational, Mar 07 2004
If L is a square-triangular number, then the next one is 1 + 17*L + 6*sqrt(L + 8*L^2). - Lekraj Beedassy, Jun 27 2001
a(n) - a(n-1) = A046176(n). - Sophie Kuo (ejiqj_6(AT)yahoo.com.tw), May 27 2006
a(n) = A001109(n)^2 = A001108(n)*(A001108(n)+1)/2 = (A000129(n)*A001333(n))^2 = (A000129(n)*(A000129(n) + A000129(n-1)))^2. - Henry Bottomley, Apr 19 2000
a(n) = (((17+12*sqrt(2))^n) + ((17-12*sqrt(2))^n)-2)/32. - Bruce Corrigan (scentman(AT)myfamily.com), Oct 26 2002
Limit_{n->oo} a(n+1)/a(n) = 17 + 12*sqrt(2). See UWC problem link and solution. - Jaap Spies, Dec 12 2004
From Antonio Alberto Olivares, Nov 07 2003: (Start)
a(n) = 35*(a(n-1) - a(n-2)) + a(n-3);
a(n) = -1/16 + ((-24 + 17*sqrt(2))/2^(11/2))*(17 - 12*sqrt(2))^(n-1) + ((24 + 17*sqrt(2))/2^(11/2))*(17 + 12*sqrt(2))^(n-1). (End)
a(n+1) = (17*A029547(n) - A091761(n) - 1)/16. - R. J. Mathar, Nov 16 2007
a(n) = A001333^2 * A000129^2 = A000129(2*n)^2/4 = binomial(A001108,2). - Bill Gosper, Jul 28 2008
Closed form (as square = triangular): ( (sqrt(2)+1)^(2*n)/(4*sqrt(2)) - (1-sqrt(2))^(2*n)/(4*sqrt(2)) )^2 = (1/2) * ( ( (sqrt(2)+1)^n / 2 - (sqrt(2)-1)^n / 2 )^2 + 1 )*( (sqrt(2)+1)^n / 2 - (sqrt(2)-1)^n / 2 )^2. - Bill Gosper, Jul 25 2008
a(n) = (1/8)*(sinh(2*n*arcsinh(1)))^2. - Artur Jasinski, Feb 10 2010
a(n) = floor((17 + 12*sqrt(2))*a(n-1)) + 3 = floor(3*sqrt(2)/4 + (17 + 12*sqrt(2))*a(n-1) + 1). - Manuel Valdivia, Aug 15 2011
a(n) = (A011900(n) + A001652(n))^2; see the link about the generalized proof of square triangular numbers. - Kenneth J Ramsey, Oct 10 2011
a(2*n+1) = A002315(n)^2*(A002315(n)^2 + 1)/2. - Ivan N. Ianakiev, Oct 10 2012
a(2*n+1) = ((sqrt(t^2 + (t+1)^2))*(2*t+1))^2, where t = (A002315(n) - 1)/2. - Ivan N. Ianakiev, Nov 01 2012
a(2*n) = A001333(2*n)^2 * (A001333(2*n)^2 - 1)/2, and a(2*n+1) = A001333(2*n+1)^2 * (A001333(2*n+1)^2 + 1)/2. The latter is equivalent to the comment above from Ivan using A002315, which is a bisection of A001333. Using A001333 shows symmetry and helps show that a(n) are both "squares of triangular" and "triangular of squares". - Richard R. Forberg, Aug 30 2013
a(n) = (A001542(n)/2)^2.
From Peter Bala, Apr 03 2014: (Start)
a(n) = (T(n,17) - 1)/16, where T(n,x) denotes the Chebyshev polynomial of the first kind.
a(n) = U(n-1,3)^2, for n >= 1, where U(n,x) denotes the Chebyshev polynomial of the second kind.
a(n) = the bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, -17; 1, 18].
See the remarks in A100047 for the general connection between Chebyshev polynomials of the first kind and 4th-order linear divisibility sequences. (End)
a(n) = A096979(2*n-1) for n > 0. - Ivan N. Ianakiev, Jun 21 2014
a(n) = (6*sqrt(a(n-1)) - sqrt(a(n-2)))^2. - Arkadiusz Wesolowski, Apr 06 2015
From Daniel Poveda Parrilla, Jul 16 2016 and Sep 21 2016: (Start)
a(n) = A000290(A002965(2*n)*A002965(2*n + 1)) (after Hugh Darwen).
a(n) = A000217(2*(A000129(n))^2 - (A000129(n) mod 2)).
a(n) = A000129(n)^4 + Sum_{k=0..(A000129(n)^2 - (A000129(n) mod 2))} 2*k. This formula can be proved graphically by taking the corresponding triangle of a square triangular number and cutting both acute angles, one level at a time (sum of consecutive even numbers), resulting in a square of squares (4th powers).
a(n) = A002965(2*n)^4 + Sum_{k=A002965(2*n)^2..A002965(2*n)*A002965(2*n + 1) - 1} 2*k + 1. This formula takes an equivalent sum of consecutives, but odd numbers. (End)
E.g.f.: (exp((17-12*sqrt(2))*x) + exp((17+12*sqrt(2))*x) - 2*exp(x))/32. - Ilya Gutkovskiy, Jul 16 2016

A054686 Multiset consisting of squares and triangular numbers.

Original entry on oeis.org

0, 0, 1, 1, 3, 4, 6, 9, 10, 15, 16, 21, 25, 28, 36, 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

Offset: 1

Michael Somos, Apr 19 2000

Terms of A001110 occur twice. [Reinhard Zumkeller, Aug 03 2011]

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.

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

Cf. A005214, A000217, A000290.

a054686_list = merge a000290_list a000217_list where
   merge xs'@(x:xs) ys'@(y:ys)
     | x <= y    = x : merge xs ys'
     | otherwise = y : merge xs' ys
-- Reinhard Zumkeller, Aug 03 2011

stnos[max_]:=Module[{sqmax=Floor[Sqrt[max]],trmax=Floor[(Sqrt[ 8max+1]- 1)/2]}, Sort[Join[Range[0,sqmax]^2,Accumulate[Range[0,trmax]]]]]; stnos[ 528] (* Harvey P. Dale, Feb 06 2012 *)

upTo(lim)=vecsort(concat(vector(sqrtint(lim\1)+1,n,(n-1)^2),vector(floor(sqrt(2+2*lim)+1/2),n,n*(n-1)/2))) \\ Charles R Greathouse IV, Aug 04 2011

Offset fixed by Reinhard Zumkeller, Aug 04 2011

A242401 Numbers that are neither triangular nor square.

Original entry on oeis.org

2, 5, 7, 8, 11, 12, 13, 14, 17, 18, 19, 20, 22, 23, 24, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 79, 80, 82, 83, 84

Offset: 1

J. Lowell, May 13 2014

			6 is not a term because it's triangular;
9 is not a term because it's square.

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

Cf. A000217, A000290.

Cf. A005214 (complement); intersection of A000037 and A014132.

a242401 n = a242401_list !! (n-1)
a242401_list = filter ((== 0) . a010054) a000037_list
-- Reinhard Zumkeller, May 13 2014

With[{nn=90},Complement[Range[nn],Range[Floor[Sqrt[nn]]]^2,Accumulate[ Range[ Floor[(Sqrt[1+8nn]-1)/2]]]]] (* Harvey P. Dale, May 03 2022 *)

[x for x in [1..100] if not(is_square(x)) and not(is_triangular_number(x))] # Tom Edgar, May 13 2014

A010052(a(n)) + A010054(a(n)) = 0. - Reinhard Zumkeller, May 13 2014

A117704 Least refined sequence that can be grouped to sum to either natural numbers or odd numbers.

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 5, 1, 5, 4, 3, 8, 9, 4, 6, 9, 2, 12, 3, 10, 9, 5, 15, 1, 15, 8, 9, 16, 2, 19, 6, 14, 15, 6, 22, 3, 20, 13, 11, 24, 1, 26, 10, 17, 22, 6, 29, 6, 24, 19, 12, 32, 1, 32, 15, 19, 30, 5, 36, 10, 27, 26, 12, 39, 4, 36, 21, 20, 39, 3, 43, 15, 29, 34, 11, 46, 8, 39, 28, 20, 49, 50

Offset: 1

Franklin T. Adams-Watters, Apr 27 2006

In other words, least common refinement of the natural numbers and the odd numbers.

			As natural numbers: 1,2,1+2,3+1,5,1+5,4+3,8,...
As odd numbers: 1,2+1,2+3,1+5+1,5+4,3+8,...

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

Cf. A118382, A005214, A000027, A005408.

a117704 n = a117704_list !! (n-1)
a117704_list = 1 : zipWith (-) (tail a005214_list) a005214_list
-- Reinhard Zumkeller, Aug 03 2011

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

A193711 Partial sums of the union of squares and triangular numbers.

Original entry on oeis.org

1, 4, 8, 14, 23, 33, 48, 64, 85, 110, 138, 174, 219, 268, 323, 387, 453, 531, 612, 703, 803, 908, 1028, 1149, 1285, 1429, 1582, 1751, 1922, 2112, 2308, 2518, 2743, 2974, 3227, 3483, 3759, 4048, 4348, 4672, 4997, 5348, 5709, 6087, 6487, 6893, 7328, 7769, 8234

Offset: 1

Reinhard Zumkeller, Aug 03 2011

nonn

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

Cf. A000217, A000290, A000292, A000330, A005214.

a193711 n = a193711_list !! (n-1)
a193711_list = scanl1 (+) a005214_list :: [Integer]

Module[{upto=500,sq,tr},sq=Range[Floor[Sqrt[upto]]]^2;tr=Accumulate[ Range[ Floor[ (Sqrt[1+8*upto]-1)/2]]];Accumulate[Union[Join[tr,sq]]]] (* Harvey P. Dale, Jun 24 2018 *)

a(n) = Sum_{k=1..n} A005214(k).

a(n) ~ c * n^3, where c = 1 - 2*sqrt(2)/3 = 0.0571909... . - Amiram Eldar, Apr 04 2025

Program and b-file corrected by Reinhard Zumkeller, Aug 04 2011

A241241 If x is in the sequence then so are x^2 and x(x+1)/2.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 9, 10, 16, 21, 36, 45, 55, 81, 100, 136, 231, 256, 441, 666, 1035, 1296, 1540, 2025, 3025, 3321, 5050, 6561, 9316, 10000, 18496, 26796, 32896, 53361, 65536, 97461, 194481, 222111, 443556, 536130, 840456, 1071225, 1186570, 1679616, 2051325

Offset: 1

Reinhard Zumkeller, Apr 17 2014

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A000290, A000217.

Subsequence of A005214; some subsequences: A001146, A007501, A011764, A176594, A173501, A050909.

import Data.Set (singleton, deleteFindMin, insert)
a241241 n = a241241_list !! (n-1)
a241241_list = 0 : 1 : f (singleton 2) where
   f s = m : f (insert (a000290 m) $ insert (a000217 m) s')
         where (m, s') = deleteFindMin s

Nest[Flatten[{#,#^2,(#(#+1))/2}]&,{0,1,2},5]//Union (* Harvey P. Dale, Aug 12 2016 *)

Initial 0 and 1 prepended by Jon Perry, Apr 17 2014

cp's OEIS Frontend

A193714 Positions of squares (A000290) in the union of squares and triangular numbers (A005214).

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

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A193748 Number of partitions of n into parts that are squares or triangular numbers (A005214).

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A193749 Number of partitions of n into distinct parts that are squares or triangular numbers (A005214).

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A001110 Square triangular numbers: numbers that are both triangular and square.

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A054686 Multiset consisting of squares and triangular numbers.

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A242401 Numbers that are neither triangular nor square.

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