A082658 -id:A082658 - cp's OEIS frontend

A082660 Number of ways n can be expressed as the sum of a square and a triangular number.

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

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

The greedy inverse (positions of the first occurrence of n) is 1, 7, 10, 37, 136, 235, 1225, 631, 2116, 4789, 11026, 3997, 148240, 19045, 20827, 25876, ... - R. J. Mathar, Apr 28 2020

			a(631) = 8 because:
1. 631 = 6 + 625
2. 631 = 55 + 576
3. 631 = 190 + 441
4. 631 = 231 + 400
5. 631 = 406 + 225
6. 631 = 435 + 196
7. 631 = 595 + 36
8. 631 = 630 + 1

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

Cf. A000217, A000290, A082657, A082658, A082659.

A082660 := proc(n)
    local a,tidx,t;
    a := 0 ;
    for tidx from 1 do
        t := A000217(tidx) ;
        if t > n then
            break;
        end if;
        if issqr(n-t) then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Apr 28 2020

a[n_] := Length @ Solve[x^2 + y (y + 1)/2 == n && x >= 0 && y > 0, {x, y}, Integers]; Array[a, 100] (* Amiram Eldar, Dec 08 2019 *)

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

Original entry on oeis.org

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 *)

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

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A082660 Number of ways n can be expressed as the sum of a square and a triangular number.

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

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