A061336 -id:A061336 - cp's OEIS frontend

A175270 Numbers k such that the least number of squares that add up to k equals the least number of triangular numbers that add up to k. Equivalently, A002828(k) = A061336(k).

0, 1, 2, 13, 14, 18, 19, 20, 29, 33, 34, 35, 36, 37, 44, 54, 58, 59, 61, 62, 65, 72, 73, 75, 77, 86, 90, 96, 97, 101, 106, 107, 118, 129, 130, 131, 134, 137, 138, 140, 146, 147, 148, 152, 155, 157, 158, 160, 161, 164, 166, 176, 179, 181, 184, 187, 193, 195, 200

Offset: 1

Ctibor O. Zizka, Mar 19 2010

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000217, A000290, A002828, A061336.

is2s(n)=my(f=factor(n>>valuation(n, 2))); for(i=1, #f~, if(bitand(f[i, 2], 1) && bitand(f[i, 1], 3)==3, return(0))); 1
is2t(n)=my(m9=n%9,f); if(m9==5 || m9==8, return(0)); is2s(4*n+1)
is(n)=my(o2=valuation(n, 2),f); if(n==0, return(1)); if(bitand(o2, 1)==0 && bitand(n>>o2, 7)==7, return(0)); if(issquare(n), return(ispolygonal(n,3))); if(ispolygonal(n,3), return(0)); is2t(n)==is2s(n) \\ Charles R Greathouse IV, Mar 17 2022

Data corrected and extended by Mohammed Yaseen, Mar 17 2022

A104246 Minimal number of tetrahedral numbers (A000292(k) = k(k+1)(k+2)/6) needed to sum to n.

Original entry on oeis.org

1, 2, 3, 1, 2, 3, 4, 2, 3, 1, 2, 3, 4, 2, 3, 4, 5, 3, 4, 1, 2, 3, 4, 2, 3, 4, 5, 3, 4, 2, 3, 4, 5, 3, 1, 2, 3, 4, 2, 2, 3, 4, 3, 3, 2, 3, 4, 4, 3, 3, 4, 5, 4, 4, 2, 1, 2, 3, 3, 2, 3, 4, 4, 3, 3, 2, 3, 4, 4, 2, 3, 4, 5, 3, 3, 2, 3, 4, 4, 3, 4, 5, 5, 1, 2, 3, 4, 2, 3, 3, 2, 3, 4, 2, 3, 3, 4, 3, 4, 4, 3, 4

Offset: 1

Eric W. Weisstein, Feb 26 2005

nonn

According to Dickson, Pollock conjectures that a(n) <= 5 for all n. Watson shows that a(n) <= 8 for all n, and Salzer and Levine show that a(n) <= 5 for n <= 452479659. - N. J. A. Sloane, Jul 15 2011
Possible correction of the first comment by Sloane 2011: it appears to me from the linked reference by Salzer and Levine 1968 that 452479659 is instead the upper limit for sums of five Qx = Tx + x, where Tx are the tetrahedral numbers we want. They also mention an upper limit for sums of five Tx, which is: a(n) <= 5 for n <= 276976383. - Ewoud Dronkert, May 30 2024
If we use the greedy algorithm for this, we get A281367. - N. J. A. Sloane, Jan 30 2017
Could be extended with a(0) = 0, in analogy to A061336. Kim (2003, first row of table "d = 3" on p. 73) gives max {a(n)} = 5 as a "numerical result", but the value has no "* denoting exact values" (see Remark at end of paper), which means this could be incorrect. - M. F. Hasler, Mar 06 2017, edited Sep 22 2022

Dickson, L. E., History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Dover, 1952, see p. 13.

Lars Blomberg, Table of n, a(n) for n = 1..10000
Hyun Kwang Kim, On regular polytope numbers, Proc. Amer. Math. Soc. 131 (2003), pp. 65-75.
F. Pollock, On the Extension of the Principle of Fermat's Theorem of the Polygonal Numbers to the Higher Orders of Series Whose Ultimate Differences Are Constant. With a New Theorem Proposed, Applicable to All the Orders, Abs. Papers Commun. Roy. Soc. London 5, 922-924, 1843-1850.
H. E. Salzer and N. Levine, Table of Integers Not Exceeding 1000000 that are Not Expressible as the Sum of Four Tetrahedral Numbers, Math. Comp. 12, 141-144, 1958.
H. E. Salzer and N. Levine, Proof that every integer <= 452,479,659 is a sum of five numbers of the form Q_x = (x^3+5x)/6, x >= 0, Math. Comp., (1968), 191-192.
N. J. A. Sloane, Transforms.
G. L. Watson, Sums of eight values of a cubic polynomial, J. London Math. Soc., 27 (1952), 217-224.
Eric Weisstein's World of Mathematics, Tetrahedral Number.

Cf. A000292 (tetrahedral numbers, indices of 1s), A102795 (indices of 2s), A102796 (indices of 3s), A102797 (indices of 4s), A000797 (numbers that need 5 tetrahedral numbers).
See also A102798-A102806, A102855-A102858, A193101, A193105, A281367 (the "triangular nachos" numbers).
Cf. A061336 (analog for triangular numbers).

tet:=[seq((n^3-n)/6,n=1..20)];
LAGRANGE(tet,8, 120); # the LAGRANGE transform of a sequence is defined in A193101. - N. J. A. Sloane, Jul 15 2011
# alternative
N := 10000:
L := [seq(0,i=1..N)] :
# put 1's where tetrahedral numbers reside
for i from 1 to N do
    Aj := A000292(i) ;
    if Aj <= N then
        L := subsop(Aj=1,L) ;
    end if;
end do:
for a from 1 do
    # select positions of a's, skip forward by all available Aj and
    # if that addresses a not-yet-set position in the array put a+1 there.
    for i from 1 to N do
        if op(i,L) =a then
            for j from 1 do
                Aj := A000292(j) ;
                if i+Aj <=N and op(i+Aj,L) = 0 then
                    L := subsop(i+Aj=a+1,L) ;
                end if;
                if i +Aj > N then
                    break ;
                end if;
            end do:
        end if;
    end do:
    # if all L[] are non-zero, terminate the loop
    allset := true;
    for i from 1 to N do
        if op(i,L) = 0 then
            allset := false ;
            break ;
        end if;
    end do:
    if allset then
        break ;
    end if;
end do:
seq( L[i],i=1..N) ; # R. J. Mathar, Jun 06 2025

\\ available on request. - M. F. Hasler, Mar 06 2017

seq(N) = {
  my(a = vector(N, k, 8), T = k->(k*(k+1)*(k+2))\6);
  for (n = 1, N,
    my (k1 = sqrtnint((6*n)\8, 3), k2 = sqrtnint(6*n, 3));
    while(n < T(k2), k2--); if (n == T(k2), a[n] = 1; next());
    for (k = k1, k2, a[n] = min(a[n], a[n - T(k)] + 1))); a;
};
seq(102)  \\ Gheorghe Coserea, Mar 14 2017

Edited by N. J. A. Sloane, Jul 15 2011

Edited by M. F. Hasler, Mar 06 2017

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

Original entry on oeis.org

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

A020757 Numbers that are not the sum of two triangular numbers.

Original entry on oeis.org

5, 8, 14, 17, 19, 23, 26, 32, 33, 35, 40, 41, 44, 47, 50, 52, 53, 54, 59, 62, 63, 68, 71, 74, 75, 77, 80, 82, 85, 86, 89, 95, 96, 98, 103, 104, 107, 109, 113, 116, 117, 118, 122, 124, 125, 128, 129, 131, 134, 138, 140, 143, 145, 147, 149, 152, 155, 158, 161, 162, 166, 167

Offset: 1

David W. Wilson

nonn

A052343(a(n)) = 0. - Reinhard Zumkeller, May 15 2006
Numbers of the form (p^(2k+1)s-1)/4, where p is a prime number of the form 4n+3, and s is a number of the form 4m+3 and prime to p, are not expressible as the sum of two triangular numbers. See Satyanarayana (1961), Theorem 2. - Hans J. H. Tuenter, Oct 11 2009
An integer n is in this sequence if and only if at least one 4k+3 prime factor in the canonical form of 4n+1 occurs with an odd exponent. - Ant King, Dec 02 2010
A nonnegative integer n is in this sequence if and only if A000729(n) = 0. - Michael Somos, Feb 13 2011
4*a(n) + 1 are terms of A022544. - XU Pingya, Aug 05 2018 [Actually, k is here if and only if 4*k + 1 is in A022544. - Jianing Song, Feb 09 2021]
Integers m such that the smallest number of triangular numbers which sum to m is 3, hence A061336(a(n)) = 3. - Bernard Schott, Jul 21 2022

			3 = 0 + 3 and 7 = 1 + 6 are not terms, but 8 = 1 + 1 + 6 is a term.

T. D. Noe, Table of n, a(n) for n = 1..10000
John A. Ewell, On Sums of Triangular Numbers and Sums of Squares, The American Mathematical Monthly, Vol. 99, No. 8 (October 1992), pp. 752-757. [From _Ant King_, Dec 02 2010]
U. V. Satyanarayana, On the representation of numbers as sums of triangular numbers, The Mathematical Gazette, 45(351):40-43, February 1961. [From _Hans J. H. Tuenter_, Oct 11 2009]

Complement of A020756.
Cf. A000217, A052343, A022544, A061336.
Numbers k such that the coefficient of x^k in the expansion of Product_{j>=1} (1 - x^j)^m is zero: A090864 (m=1), A213250 (m=2), A014132 (m=3), A302056 (m=4), A302057 (m=5), this sequence (m=6), A322430 (m=8), A322431 (m=10), A322432 (m=14), A322043 (m=15), A322433 (m=26).

a020757 n = a020757_list !! (n-1)
a020757_list = filter ((== 0) . a052343) [0..]
-- Reinhard Zumkeller, Jul 25 2014

data = Reduce[m (m + 1) + n (n + 1) == 2 # && 0 <= m && 0 <= n, {m, n}, Integers] & /@ Range[167]; Position[data, False] // Flatten  (* Ant King, Dec 05 2010 *)
t = Array[PolygonalNumber, 18, 0]; Complement[Range@ 169, Flatten[ Outer[ Plus, t, t]]] (* Robert G. Wilson v, Aug 07 2024 *)

is(n)=my(m9=n%9,f); if(m9==5 || m9==8, return(1)); f=factor(4*n+1); for(i=1,#f~, if(f[i,1]%4==3 && f[i,2]%2, return(1))); 0 \\ Charles R Greathouse IV, Mar 17 2022

A100878 Smallest number of pentagonal numbers which sum to n.

Original entry on oeis.org

0, 1, 2, 3, 4, 1, 2, 3, 4, 5, 2, 3, 1, 2, 3, 3, 4, 2, 3, 4, 4, 5, 1, 2, 2, 3, 4, 2, 3, 3, 4, 5, 3, 4, 2, 1, 2, 3, 4, 3, 2, 3, 4, 5, 2, 3, 3, 2, 3, 3, 4, 1, 2, 3, 4, 5, 2, 2, 3, 3, 4, 3, 3, 2, 3, 4, 3, 4, 3, 3, 1, 2, 3, 2, 3, 2, 3, 4, 3, 3, 3, 4, 2, 3, 4, 3, 2, 3, 4, 5, 4, 3, 1, 2, 3, 3, 4, 2, 3, 4, 4, 4, 2, 3, 2

Offset: 0

Franz Vrabec, Jan 09 2005

nonn

From Bernard Schott, Jul 15 2022: (Start)
In September 1636, Fermat, in a letter to Mersenne, made the statement that every number is a sum of at most three triangular numbers, four squares, five pentagonal numbers, and so on.
The square case was proved by Lagrange in 1770; it is known as Lagrange's four squares theorem (see A002828). Then Gauss proved the triangular case in 1796 (see A061336).
In 1813, Cauchy proved this polygonal number theorem: for m >= 3, every positive integer N can be represented as a sum of m+2 (m+2)-gonal numbers, at most four of which are different from 0 and 1 (Deza reference). Hence every number is expressible as the sum of at most five positive pentagonal numbers (A000326). (End)

			a(5)=1 since 5=5, a(6)=2 since 6=1+5, a(7)=3 since 7=1+1+5, a(10)=2 since 10=5+5 with 1 and 5 pentagonal numbers.

Elena Deza and Michel Marie Deza, Fermat's polygonal number theorem, Figurate numbers, World Scientific Publishing (2012), Chapter 5, pp. 313-377.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section D3, Figurate numbers, pp. 222-228.

Augustin-Louis Cauchy, Démonstration du théorème général de Fermat sur les nombres polygones, Extrait des Mémoires de l'Institut, 1813-15.
Eric Weisstein's World of Mathematics, Fermat's Polygonal Number Theorem.
Wikipedia, Fermat polygonal number theorem.

Cf. A002828, A061336, A355717.

Cf. A000326 (a(n) = 1), A003679 (a(n) = 4 or 5), A355660 (a(n) = 4), A133929 (a(n) = 5).

a(n) = my(nb=oo); forpart(vp=n, if (vecsum(apply(x->ispolygonal(x, 5), Vec(vp))) == #vp, nb = min(nb, #vp)),,5); nb; \\ Michel Marcus, Jul 15 2022

a(n) = for(i = 1, oo, p = partitions(n, , [i,i]); for(j = 1, #p, if(sum(k = 1, i, ispolygonal(p[j][k],5)) == i, return(i)))) \\ David A. Corneth, Jul 15 2022

a(n) <= 5 (inequality proposed by Fermat and proved by Cauchy). - Bernard Schott, Jul 13 2022

More terms from David Wasserman, Mar 04 2008

A234533 Smallest number of hex numbers summing to n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 2, 3, 4, 5, 6, 7, 8, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 2, 3, 4, 5, 6, 7, 8, 3, 4, 5, 6, 7, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 2, 3, 4, 5, 6, 7, 2, 3, 4, 5, 6, 7, 2, 3, 4, 5, 6, 7, 8, 3

Offset: 1

Allan C. Wechsler, Dec 27 2013

nonn

This sequence is to A003215 as A002828 is to A000290, and as A061336 is to A000217.

			a(21) = 3, because 21 = 19 + 1 + 1 = 7 + 7 + 7, so there are two ways to express 21 as the sum of 3 hex numbers, but no way to express 21 as the sum of 2 hex numbers.

Lars Blomberg, Table of n, a(n) for n = 1..10000

a(27)-a(87) from Lars Blomberg, Jan 07 2014

A258257 The number of representations of n as a minimal number of triangular numbers, A000217(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 1, 1, 1, 3, 2, 2, 1, 2, 1, 2, 2, 1, 1, 1, 2, 5, 1, 2, 1, 2, 1, 1, 3, 1, 1, 2, 3, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 2, 4, 2, 3, 1, 1, 2, 3, 6, 1, 1, 3, 3, 1

Offset: 1

Martin Renner, May 24 2015

nonn

			a(5) = 1 since 5 = 1 + 1 + 3 is the only representation as a minimal number of three triangular numbers.
a(16) = 2 since 16 = 1 + 15 = 6 + 10 has two representations as a minimal number of two triangular numbers.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A061336, A180466.

t[n_] := n (n + 1)/2; a[n_] := Block[{k = 1, t, tt = t /@ Range[ Sqrt[2*n]]}, While[{} == (r = IntegerPartitions[n, {k}, tt]), k++]; Length@r]; Array[a, 100] (* Giovanni Resta, Jun 09 2015 *)

A283365 Minimal number of numbers in A000332 = { C(k,4); k=1,2,3,... } whose sum equals n.

Original entry on oeis.org

0, 1, 2, 3, 4, 1, 2, 3, 4, 5, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 2, 3, 4, 5, 6, 3, 4, 5, 6, 7, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 2, 3, 4, 5, 6, 3, 4, 5, 6, 7, 2, 3, 4, 5, 6, 3, 4, 5, 6, 7, 4, 5, 6, 7, 8, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 2, 3, 4, 5, 6, 3, 4, 5, 6, 7, 2, 3

Offset: 0

M. F. Hasler, Mar 06 2017

nonn

Analog, for A000332 = {C(n,4)}, of A061336 (for triangular numbers A000217) and A104246 (for tetrahedral numbers A000292).

R. J. Mathar, Table of n, a(n) for n = 0..10000
Hyun Kwang Kim, On regular polytope numbers, Proc. Amer. Math. Soc. 131 (2003), p. 65-75. DOI:10.1090/S0002-9939-02-06710-2.

Cf. A000332 = {C(n,4)}; A061336 (analog for triangular numbers A000217), A104246 (analog for tetrahedral numbers A000292).

{a(n,k=4,M=9e9,N=n) = (n <= k || M <= k+1) && return(n); for(m=k,M,binomial(m,k)>n && (M=m) && break); M-- <= k && return(n); my(b=binomial(M,k),c=binomial(M-1,k),NN); forstep( nn=n\b,0,-1, if(N>NN=nn+g(n-nn*b,k,M,N,d),N=NN); n-(nn-1)*b >= (N-nn+1)*c && break); N}

a(n) <= 8 = a(64) for all n, according to Kim (2003, first row of table "d = 4", p. 74), but this "numerical result" has no "* denoting exact values" (see Remark at end of paper), so it could be incorrect. [Disclaimer added by M. F. Hasler, Sep 22 2022]

A338480 Least number of heptagonal numbers needed to represent n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 2, 3, 4, 5, 1, 2, 3, 3, 4, 5, 6, 2, 3, 4, 4, 5, 6, 7, 3, 4, 1, 2, 2, 3, 4, 4, 5, 2, 3, 3, 4, 5, 5, 6, 3, 4, 4, 5, 2, 3, 3, 1, 2, 3, 4, 3, 4, 4, 2, 3, 4, 5, 4, 5, 2, 3, 3, 4, 4, 2, 3, 3, 4, 4, 5, 5, 3, 1, 2, 3, 4, 5, 3, 4, 2, 2, 3, 3, 4, 4, 5, 3, 3, 4, 4, 2, 3, 4

Offset: 1

Ilya Gutkovskiy, Oct 29 2020

nonn

Eric Weisstein's World of Mathematics, Heptagonal Number
Index to sequences related to polygonal numbers

Cf. A000566, A002828, A061336, A100878, A338479, A338481.

A338482 Least number of centered triangular numbers that sum to n.

Original entry on oeis.org

1, 2, 3, 1, 2, 3, 4, 2, 3, 1, 2, 3, 4, 2, 3, 4, 5, 3, 1, 2, 3, 4, 2, 3, 4, 5, 3, 4, 2, 3, 1, 2, 3, 4, 2, 3, 4, 2, 3, 4, 2, 3, 4, 5, 3, 1, 2, 3, 4, 2, 3, 4, 5, 3, 4, 2, 3, 4, 5, 3, 4, 2, 3, 1, 2, 3, 4, 2, 3, 4, 5, 3, 4, 2, 3, 4, 2, 3, 4, 5, 3, 4, 2, 3, 1, 2, 3, 4, 2, 3, 4, 2, 3, 4, 2, 3, 4, 5, 3, 4, 5

Offset: 1

Ilya Gutkovskiy, Oct 29 2020

nonn

It appears that a(n) = 3 for n == 0 (mod 3), 1 <= a(n) <= 4 for n == 1 (mod 3), and 2 <= a(n) <= 5 for n == 2 (mod 3). - Robert Israel, Nov 13 2020

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Centered Triangular Number
Index entries for sequences related to centered polygonal numbers

Cf. A005448, A061336, A101412, A104246, A234533, A338484.

f:= proc(n) option remember; local r,i;
    r:= sqrt(24*n-15)/6+1/2;
    if r::integer then return 1 fi;
    1+min(seq(procname(n-(3*i*(i-1)/2+1)),i=1..floor(r)))
end proc:
map(f, [$1..200]); # Robert Israel, Nov 13 2020

f[n_] := f[n] = Module[{r}, r = Sqrt[24n-15]/6+1/2; If[IntegerQ[r], Return[1]]; 1+Min[Table[f[n-(3i*(i-1)/2+1)], {i, 1, Floor[r]}]]];
Map[f, Range[200]] (* Jean-François Alcover, Sep 16 2022, after Robert Israel *)

cp's OEIS Frontend

A175270 Numbers k such that the least number of squares that add up to k equals the least number of triangular numbers that add up to k. Equivalently, A002828(k) = A061336(k).

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A104246 Minimal number of tetrahedral numbers (A000292(k) = k(k+1)(k+2)/6) needed to sum to n.

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Maple

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

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Haskell

Maple

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A020757 Numbers that are not the sum of two triangular numbers.

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Haskell

Mathematica

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A100878 Smallest number of pentagonal numbers which sum to n.

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PARI

PARI

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A234533 Smallest number of hex numbers summing to n.

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A258257 The number of representations of n as a minimal number of triangular numbers, A000217(n).

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Mathematica