A082660 -id:A082660 - cp's OEIS frontend

A082657 Integers expressible as the sum of a square and a triangular number in just one way.

1, 2, 3, 4, 5, 6, 11, 12, 14, 16, 17, 21, 24, 25, 29, 30, 32, 35, 36, 39, 42, 44, 49, 50, 51, 53, 54, 56, 57, 65, 66, 71, 72, 74, 75, 77, 78, 80, 81, 84, 95, 96, 101, 104, 105, 107, 110, 116, 117, 119, 120, 122, 126, 128, 129, 131, 137, 141, 149, 150, 152, 153, 156, 161

Offset: 1

Jason Earls, May 17 2003

nonn

It is assumed here that 0 is a square but not a triangular number. - Amiram Eldar, Dec 08 2019

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000217, A000290, A082658, A082659, A082660.

aQ[n_] := Length @ Solve[x^2 + y (y + 1)/2 == n && x >= 0 && y > 0, {x, y}, Integers] == 1; Select[Range[161], aQ] (* Amiram Eldar, Dec 08 2019 *)

A082658 Integers expressible as the sum of a square and a triangular number in exactly two distinct ways.

Original entry on oeis.org

7, 15, 22, 26, 28, 31, 40, 45, 52, 55, 59, 61, 67, 79, 85, 87, 92, 94, 100, 102, 103, 106, 114, 115, 124, 130, 140, 142, 147, 155, 157, 159, 166, 175, 178, 180, 184, 187, 189, 190, 191, 197, 202, 205, 206, 210, 211, 214, 220, 224, 231, 232, 240, 241, 246, 247

Offset: 1

Jason Earls, May 17 2003

nonn

It is assumed here that 0 is a square but not a triangular number. - Amiram Eldar, Dec 08 2019

			a(4) = 26 because 26 = 1 + 25; 26 = 10 + 16.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000217, A000290, A082657, A082659, A082660.

aQ[n_] := Length @ Solve[x^2 + y (y + 1)/2 == n && x >= 0 && y > 0, {x, y}, Integers] == 2; Select[Range[250], aQ] (* Amiram Eldar, Dec 08 2019 *)

Name clarified by Amiram Eldar, Dec 08 2019

A082659 Integers expressible as the sum of a square and a triangular number in exactly three distinct ways.

Original entry on oeis.org

10, 19, 46, 64, 82, 109, 121, 127, 154, 169, 217, 253, 257, 262, 271, 316, 352, 361, 379, 397, 400, 451, 460, 478, 487, 496, 505, 514, 529, 586, 620, 640, 649, 667, 694, 721, 757, 767, 856, 865, 910, 937, 961, 964, 991, 1045, 1054, 1072, 1099, 1104, 1135, 1153

Offset: 1

Jason Earls, May 17 2003

nonn

It is assumed here that 0 is a square but not a triangular number. - Amiram Eldar, Dec 08 2019

			a(5) = 82 because 82 = 1 + 81; 82 = 66 + 16; 82 = 78 + 4.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000217, A000290, A082657, A082658, A082660.

aQ[n_] := Length @ Solve[x^2 + y (y + 1)/2 == n && x >= 0 && y > 0, {x, y}, Integers] == 3; Select[Range[1200], aQ] (* Amiram Eldar, Dec 08 2019 *)

Name clarified by Amiram Eldar, Dec 08 2019

A101428 Number of ways to write n as an ordered sum of a triangular number (A000217) and a square (A000290).

Original entry on oeis.org

1, 2, 1, 1, 2, 1, 1, 2, 0, 1, 3, 1, 1, 0, 1, 2, 2, 1, 0, 3, 0, 1, 2, 0, 1, 2, 2, 0, 2, 1, 1, 2, 1, 0, 0, 1, 2, 4, 0, 1, 2, 0, 1, 0, 1, 2, 3, 0, 0, 2, 1, 1, 2, 1, 1, 2, 1, 1, 0, 2, 0, 2, 0, 0, 4, 1, 1, 2, 0, 0, 4, 1, 1, 0, 1, 1, 0, 1, 1, 2, 1, 2, 3, 0, 1, 2, 0, 2, 0, 0, 0, 4, 2, 0, 2, 1, 1, 0, 0, 0, 3, 1, 2, 2, 1

Offset: 0

R. J. Mathar, Jan 03 2009

0 is both a triangular number and a square number.
First occurrence of k beginning at 0: 8, 2, 1, 10, 37, 136, 235, 1549, 631, 2314, 2116, 11026, 3997, 148240, 19045, 20827, 25876, 893116, 67951, ?19?, 35974, 187444, 1542655, 354061, 131905, ?25?, ?26?, 835399, 323767, ?29?, 611560, ?31?, 515629, ?33?, ?34?, ?35?, 1187146, ?37?, ?38?, ?39?, 1474939, ..., . - Robert G. Wilson v, Mar 30 2014
Variant of A082660 (which allows only positive triangular numbers). - R. J. Mathar, Apr 28 2020
a(n) is the number of representations of 8*n + 1 as 2*A^2 + B^2 with A even and B odd and both nonnegative integers. - Vladimir Pletser, Aug 30 2025

			Examples: n=1 gives the a(1)=2 cases 1=1+0=0+1; a(26)=2 because 26=25+1=16+10.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000925, A115171, A115172, A115173, A115174, A115175, A115176, A115177, A144642.

A000217 := proc(n) n*(n+1)/2 ; end:
A101428 := proc(n)
local a,y,t ;
a := 0 ;
for y from 0 do
t := A000217(y) ;
if n-t < 0 then
RETURN(a) ;
else
if issqr(n-t) then
a := a+1 ;
fi;
fi;
od:
end:
for n from 0 to 100 do printf("%a,",A101428(n)) ; od:

t = FoldList[#1 + #2 &, 0, Range@ 15]; s = Range[0, 10]^2, a = Sort@ Flatten@ Table[ s[[j]] + t[[k]], {j, 15}, {k, 11}]; Table[Count[a, n], {n, 0, 104}] (* or *)
triQ[n_] := IntegerQ@ Sqrt[8n + 1]; f[n_] := Block[{c = k = 0, lmt = 2 + Floor[Sqrt[n]]}, While[k < lmt, If[ triQ[n - k^2], c++]; k++]; c]; Array[f, 105, 0] (* Robert G. Wilson v, Mar 30 2014 *)

G.f.: sum(i>=0, x^(i^2) ) * sum(i>=0, x^(i*(i+1)/2) ). - Ralf Stephan, May 17 2014

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A082657 Integers expressible as the sum of a square and a triangular number in just one way.

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A082658 Integers expressible as the sum of a square and a triangular number in exactly two distinct ways.

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A082659 Integers expressible as the sum of a square and a triangular number in exactly three distinct ways.

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Programs

Mathematica

Extensions

A101428 Number of ways to write n as an ordered sum of a triangular number (A000217) and a square (A000290).

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