A085989 -id:A085989 - cp's OEIS frontend

A051533 Numbers that are the sum of two positive triangular numbers.

2, 4, 6, 7, 9, 11, 12, 13, 16, 18, 20, 21, 22, 24, 25, 27, 29, 30, 31, 34, 36, 37, 38, 39, 42, 43, 46, 48, 49, 51, 55, 56, 57, 58, 60, 61, 64, 65, 66, 67, 69, 70, 72, 73, 76, 79, 81, 83, 84, 87, 88, 90, 91, 92, 93, 94, 97, 99, 100, 101, 102, 106, 108

Offset: 1

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)

Numbers n such that 8n+2 is in A085989. - Robert Israel, Mar 06 2017

			666 is in the sequence because we can write 666 = 435 + 231 = binomial(22,2) + binomial(30,2).

T. D. Noe, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Fermat's Polygonal Number Theorem

Cf. A000217, A020756 (sums of two triangular numbers), A001481 (sums of two squares), A007294, A051611 (complement).
Cf. A061336: minimal number of triangular numbers that sum up to n.
Cf. A085989.

a051533 n = a051533_list !! (n-1)
a051533_list = filter ((> 0) . a053603) [1..]
-- Reinhard Zumkeller, Jun 28 2013

isA051533 := proc(n)
    local a,ta;
    for a from 1 do
        ta := A000217(a) ;
        if 2*ta > n then
            return false;
        end if;
        if isA000217(n-ta) then
            return true;
        end if;
    end do:
end proc:
for n from 1 to 200 do
    if isA051533(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Dec 16 2015

f[k_] := If[!
   Head[Reduce[m (m + 1) + n (n + 1) == 2 k && 0 < m && 0 < n, {m, n},
       Integers]] === Symbol, k, 0]; DeleteCases[Table[f[k], {k, 1, 108}], 0] (* Ant King, Nov 22 2010 *)
nn=50; tri=Table[n(n+1)/2, {n,nn}]; Select[Union[Flatten[Table[tri[[i]]+tri[[j]], {i,nn}, {j,i,nn}]]], #<=tri[[-1]] &]
With[{nn=70},Take[Union[Total/@Tuples[Accumulate[Range[nn]],2]],nn]] (* Harvey P. Dale, Jul 16 2015 *)

is(n)=for(k=ceil((sqrt(4*n+1)-1)/2),(sqrt(8*n-7)-1)\2, if(ispolygonal(n-k*(k+1)/2, 3), return(1))); 0 \\ Charles R Greathouse IV, Jun 09 2015

A053603(a(n)) > 0. - Reinhard Zumkeller, Jun 28 2013

A061336(a(n)) = 2. - M. F. Hasler, Mar 06 2017

A294073 Numbers that are the sum of 2 cubes > 1.

Original entry on oeis.org

16, 35, 54, 72, 91, 128, 133, 152, 189, 224, 243, 250, 280, 341, 351, 370, 407, 432, 468, 520, 539, 559, 576, 637, 686, 728, 737, 756, 793, 854, 855, 945, 1008, 1024, 1027, 1064, 1072, 1125, 1216, 1241, 1339, 1343, 1358, 1395, 1456, 1458, 1512, 1547, 1674, 1729, 1736, 1755, 1792, 1843, 1853, 1944, 2000, 2060

Offset: 1

Ilya Gutkovskiy, Feb 07 2018

nonn

			133 is in the sequence because 133 = 2^3 + 5^3.

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sums of cubes

Cf. A000578, A003325, A004999, A024670, A085989, A086119.

N:= 3000: # to get all terms <= N
M := floor(N^(1/3)):
sort(convert(select(`<=`, {seq(seq(i^3+j^3, j = 2 .. i), i = 2 .. M)}, N), list)) # Robert Israel, Feb 08 2018

nmax = 2060; f[x_] := Sum[x^k^3, {k, 2, 14}]^2; Exponent[#, x] & /@ List @@ Normal[Series[f[x], {x, 0, nmax}]]

A281154 Expansion of (Sum_{k>=2} x^(k^2))^2.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Jan 16 2017

nonn

Number of ways to write n as an ordered sum of 2 squares > 1.

			G.f. = x^8 + 2*x^13 + x^18 + 2*x^20 + 2*x^25 + 2*x^29 + x^32 + 2*x^34 + 2*x^40 + ...
a(13) = 2 because we have [9, 4] and [4, 9].

Index entries for sequences related to sums of squares

Cf. A000290, A000925, A004018, A006456, A063725, A078134, A085989, A280542.

nmax = 105; CoefficientList[Series[Sum[x^k^2, {k, 2, nmax}]^2, {x, 0, nmax}], x]
CoefficientList[Series[(1 + 2 x - EllipticTheta[3, 0, x])^2/4, {x, 0, 105}], x]

G.f.: (Sum_{k>=2} x^(k^2))^2.

G.f.: (1/4)*(1 + 2*x - theta_3(0,x))^2, where theta_3 is the 3rd Jacobi theta function.

A302359 Numbers that are the sum of 3 squares > 1.

Original entry on oeis.org

12, 17, 22, 24, 27, 29, 33, 34, 36, 38, 41, 43, 44, 45, 48, 49, 50, 54, 56, 57, 59, 61, 62, 65, 66, 67, 68, 69, 70, 72, 74, 75, 76, 77, 78, 81, 82, 83, 84, 86, 88, 89, 90, 93, 94, 96, 97, 98, 99, 101, 102, 104, 105, 106, 107, 108, 109, 110, 113, 114, 115, 116, 117, 118, 120, 121, 122, 123, 125, 126, 129

Offset: 1

Ilya Gutkovskiy, Apr 06 2018

nonn

			33 is in the sequence because 33 = 2^2 + 2^2 + 5^2.

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sums of squares

Cf. A000378, A000408, A085989, A214514, A281155.

max = 130; f[x_] := Sum[x^(k^2), {k, 2, 20}]^3; Exponent[#, x] & /@ List @@ Normal[Series[f[x], {x, 0, max}]]
With[{nn=15},Select[Union[Total/@Tuples[Range[2,nn]^2,3]],#<=nn^2+8&]] (* Harvey P. Dale, Jul 05 2021 *)

from itertools import count, takewhile, combinations_with_replacement as mc
def aupto(N):
    sqrs = list(takewhile(lambda x: x<=N, (i**2 for i in count(2))))
    sum3 = set(sum(c) for c in mc(sqrs, 3) if sum(c) <= N)
    return sorted(sum3)
print(aupto(129)) # Michael S. Branicky, Dec 17 2021

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A051533 Numbers that are the sum of two positive triangular numbers.

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A294073 Numbers that are the sum of 2 cubes > 1.

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A281154 Expansion of (Sum_{k>=2} x^(k^2))^2.

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A302359 Numbers that are the sum of 3 squares > 1.

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