A053612 -id:A053612 - cp's OEIS frontend

A039596 Numbers that are simultaneously triangular and square pyramidal.

0, 1, 55, 91, 208335

Offset: 1

Equivalent to 0^2 + 1^2 + 2^2 + 3^2 + ... + r^2 = 0 + 1 + 2 + 3 + ... + s = n for some r and s.

			1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 1 + 2 + 3 + ... + 10 = 55, so 55 is in the sequence.

Joe Roberts, Lure of the Integers, page 245 (entry for 645).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, p. 108.

R. Finkelstein and H. London, On triangular numbers which are sums of consecutive squares, J. Number Theory 4 (1972), 455-462.
M. Gardner, Letter to N. J. A. Sloane, circa Aug 11 1980, concerning A001110, A027568, A039596, etc.
H. E. Thomas Jr., Problem 5634, Amer. Math. Monthly, 75 (1968), p. 1018.

Intersection of A000217 and A000330.

Cf. A053611, A053612.

q:= n-> issqr(8*n+1):
select(q, [sum(j^2, j=1..n)$n=0..100])[];  # Alois P. Heinz, Oct 17 2024

Additional comments from Jud McCranie, Mar 19 2000

Zero inserted by Daniel Mondot, Sep 07 2023

A053611 Numbers k such that 1 + 4 + 9 + ... + k^2 = 1 + 2 + 3 + ... + m for some m.

Original entry on oeis.org

1, 5, 6, 85

Offset: 1

Jud McCranie, Mar 19 2000

These are the only possibilities for a sum of the first n squares to equal a triangular number.
From Seiichi Manyama, Aug 25 2019: (Start)
The complete list of solutions to k*(k+1)*(2*k+1)/6 = m*(m+1)/2 is as follows.
(k,m) = (-1, 0), (0, 0), (1, 1), (5, 10), (6, 13), (85, 645),
        (-1,-1), (0,-1), (1,-2), (5,-11), (6,-14), (85,-646). (End)

			1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 1 + 2 + 3 + ... + 10, so 5 is in the sequence.

E. T. Avanesov, The Diophantine equation 3y(y+1) = x(x+1)(2x+1), Volz. Mat. Sb. Vyp., 8 (1971), 3-6.
R. K. Guy, Unsolved Problems in Number Theory, Section D3.
Joe Roberts, Lure of the Integers, page 245 (entry for 645).

R. Finkelstein, H. London, On triangular numbers which are sums of consecutive squares, J. Number Theory 4 (1972), 455-462.
Eric Weisstein's World of Mathematics, Square Pyramidal Number

Cf. A039596, A053612 (values of m).

Cf. A000217, A000330.

istriangular:=proc(n) local t1; t1:=floor(sqrt(2*n)); if n = t1*(t1+1)/2 then RETURN(true) else RETURN(false); fi; end;
M:=1000; for n from 1 to M do if istriangular(n*(n+1)*(2*n+1)/6) then lprint(n,n*(n+1)*(2*n+1)/6); fi; od: # N. J. A. Sloane
# second Maple program:
q:= n-> issqr(8*sum(j^2, j=1..n)+1):
select(q, [$1..100])[];  # Alois P. Heinz, Oct 10 2024

Select[Range[90], IntegerQ[(Sqrt[(4/3) * (# + 3 * #^2 + 2 * #^3) + 1] - 1)/2] &] (* Harvey P. Dale, Sep 22 2014 *)

Edited by N. J. A. Sloane, May 25 2008

A126973 a(n+1) is the smallest integer greater than a(n) such that the sum of the squares of its decimal digits is equal to a(n).

Original entry on oeis.org

1, 10, 13, 23, 1233, 33999999999999999

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Mar 20 2007; corrected Mar 23 2007

			10 --> 1^2+0^2 = 1+0 =1
13 --> 1^2+3^2 = 1+9 = 10
23 --> 2^2+3^2 = 4+9 =13
1233 --> 1^2+2^2+3^3+3^2 = 1+4+9+9 = 23
33999999999999999 = 3^2*2 + 9^2*15 = 1233

Cf. A001273, A053612.

Next term is greater than 10^419753086419753. [From Charles R Greathouse IV, Nov 13 2010]

A136276 Consider pairs of nonnegative integers (m,k) such that 2^2 + 4^2 + 6^2 + ... + (2m)^2 = k(k+1); sequence gives k values.

Original entry on oeis.org

0, 4, 7, 84

Offset: 1

Ken Knowlton (www.KnowltonMosaics.com), Mar 29 2008

The problem arises when trying to build a square pyramid out of dominoes. The solution (m,k) = (3,7) for example corresponds to building a pyramid with layers of sizes 2 X 2, 4 X 4 and 6 X 6 from one set of double-6 dominoes.
The three nonzero solutions use one double-3 set, one double-6 set and one double-83 set. (The sequence 3,6,83 is too short to warrant a separate entry.)
The problem is equivalent to finding integers (m,k) such that 2m(m+1)(m+2)/3 = k*(k+1). This is a nonsingular cubic, so by Siegel's theorem, there are only finitely many solutions. - N. J. A. Sloane, May 25 2008. See also the articles by Stroeker and Tzanakis and Stroeker and de Weger.

			The known solutions are (m,k) = (0,0), (2,4), (3,7) and (17,84). There are no other solutions.

John Cannon, Using MAGMA to prove there are no other solutions
J. H. Silverman and J. Tate, Rational Points on Elliptic Curves, Springer, 1992,
R. J. Stroeker and B. M. M. de Weger, Solving elliptic Diophantine equations: the general cubic case, Acta Arith. 87 (1999), 339-365.
R. J. Stroeker and N. Tzanakis, Solving elliptic Diophantine equations by estimating linear forms in elliptic logarithms, Acta Arith. 67 (1994), 177-196.

Cf. A039596, A053611, A053612.

Simple-minded Maple program from N. J. A. Sloane:
f1:=m-> 1+8*m*(m+1)*(2*m+1)/3;
for m from 0 to 10^8 do if issqr(f1(m)) then lprint( m, (-1+sqrt(f1(m)))/2); fi; od;

Edited by N. J. A. Sloane, May 25 2008, Aug 17 2008

May 26 2008: John Cannon used MAGMA to show there are no further solutions (see link)

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A039596 Numbers that are simultaneously triangular and square pyramidal.

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A053611 Numbers k such that 1 + 4 + 9 + ... + k^2 = 1 + 2 + 3 + ... + m for some m.

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A126973 a(n+1) is the smallest integer greater than a(n) such that the sum of the squares of its decimal digits is equal to a(n).

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Author

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Examples

Crossrefs

Extensions

A136276 Consider pairs of nonnegative integers (m,k) such that 2^2 + 4^2 + 6^2 + ... + (2m)^2 = k(k+1); sequence gives k values.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Extensions