A124978 -id:A124978 - cp's OEIS frontend

A002635 Number of partitions of n into 4 squares.

1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 2, 3, 2, 2, 2, 2, 1, 1, 3, 3, 3, 3, 2, 2, 2, 1, 3, 4, 2, 4, 3, 3, 2, 2, 3, 4, 3, 2, 4, 2, 2, 2, 4, 5, 3, 5, 3, 5, 3, 1, 4, 5, 3, 3, 4, 3, 4, 2, 4, 6, 4, 4, 4, 5, 2, 3, 5, 5, 5, 5, 4, 4, 3, 2, 6, 7, 4, 5, 5, 5, 4, 2, 5, 9, 5, 3, 5, 4, 3, 1, 6, 7, 6, 7, 5, 7, 5, 3, 6, 7, 4

Offset: 0

N. J. A. Sloane

a(A124978(n)) = n; a(A006431(n)) = 1; a(A180149(n)) = 2; a(A245022(n)) = 3. - Reinhard Zumkeller, Jul 13 2014

			1: 1000; 2: 1100; 3:1110; 4: 2000 and 1111; 5: 2100; 6: 2110; 7: 2111; 8: 2200; 9: 3000 and 2210; 10: 3100 and 2211; etc.

G. Loria, Sulla scomposizione di un intero nella somma di numeri poligonali. (Italian) Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (8) 1, (1946). 7-15.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..10000
E. Grosswald, The Problem of the Uniqueness of Essentially Distinct Representations, in Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 84.
D. H. Lehmer, Review of Loria article, Math. Comp. 2 (1947), 301-302.
Gino Loria, Sulla scomposizione di un intero nella somma di numeri poligonali. (Italian). Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (8) 1, (1946). 7-15. Also D. H. Lehmer, Review of Loria article, Math. Comp. 2 (1947), 301-302. [Annotated scanned copies]
M. D. Hirschhorn, Some formulas for partitions into squares, Discrete Math, 211 (2000), pp. 225-228.
James A. Sellers, Partitions Excluding Specific Polygonal Numbers As Parts, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.4.
Index entries for sequences related to sums of squares

Cf. A000290, A124978, A006431, A180149, A245022.

Equivalent sequences for other numbers of squares: A010052 (1), A000161 (2), A000164 (3), A000174 (5), A000177 (6), A025422 (7), A025423 (8), A025424 (9), A025425 (10).

a002635 = p (tail a000290_list) 4 where
p ks'@(k:ks) c m = if m == 0 then 1 else
if c == 0 || m < k then 0 else p ks' (c - 1) (m - k) + p ks c m
-- Reinhard Zumkeller, Jul 13 2014

Length[PowersRepresentations[ #, 4, 2]] & /@ Range[0, 107] (* Ant King, Oct 19 2010 *)

for(n=1,100,print1(sum(a=0,n,sum(b=0,a,sum(c=0,b,sum(d=0,c,if(a^2+b^2+c^2+d^2-n,0,1))))),","))

a(n)=local(c=0); if(n>=0, forvec(x=vector(4,k,[0,sqrtint(n)]),c+=norml2(x)==n,1)); c

A061262 Smallest number representable as the sum of 3 triangular numbers in exactly n ways.

Original entry on oeis.org

0, 3, 12, 21, 52, 57, 91, 121, 136, 211, 192, 226, 409, 331, 367, 406, 511, 507, 886, 637, 772, 721, 871, 952, 1102, 1066, 1227, 1192, 1641, 1621, 1396, 1381, 1501, 1732, 1792, 1927, 1942, 2401, 2611, 2551, 2422, 2557, 2887, 2821, 3136, 3271, 3607, 3376

Offset: 1

Ed Pegg Jr, Apr 24 2001

Fermat claimed, Euler tried, Gauss proved (July 10, 1796) that every number can be represented as a sum of three triangular numbers. I'm considering 0 as a triangular number here. If at first you do not succeed, tri + tri + tri again.
Conjecture: for n large enough, 1 < a(n)/n^2 < 2. - Benoit Cloitre, May 10 2003
Conjecture: No term a(n) with n > 2 is congruent to 0 or 3 modulo 5. - Zhi-Wei Sun, Feb 25 2015

			57 is the smallest number that can be represented by exactly 6 different triangular triple sums: {6, 6, 5}, {7, 7, 1}, {8, 5, 3}, {8, 6, 0}, {9, 3, 3}, {10, 1, 1}.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A000217, A053614, A060773, A002636.

Cf. A124978.

a = Table[ n(n + 1)/2, {n, 0, 85} ]; b = {0}; c = Table[0, {3655} ]; Do[ b = Append[b, a[[i] ] + a[[j]] + a[[k]]], {k, 1, 85}, {j, 1, k}, {i, 1, j} ]; b = Delete[b, 1]; b = Sort[b]; l = Length[b]; Do[ If[b[[n]] < 3655, c[[b[[n]] + 1]]++ ], {n, 1, l} ]; Do[ k = 1; While[ c[[k]] != n, k++ ]; Print[k - 1], {n, 1, 48} ]

A176661 Partial sums of A061262.

Original entry on oeis.org

0, 3, 15, 36, 88, 145, 236, 357, 493, 704, 896, 1122, 1531, 1862, 2229, 2635, 3146, 3653, 4539, 5176, 5948, 6669, 7540, 8492, 9594, 10660, 11887, 13079, 14720, 16341, 17737, 19118, 20619, 22351, 24143, 26070, 28012, 30413, 33024, 35575, 37997

Offset: 0

Jonathan Vos Post, Apr 23 2010

nonn

Partial sums of smallest number representable as the sum of 3 triangular numbers in exactly n ways. The subsequence of triangular numbers in the partial sum begins: 3, 15, 36. The subsequence of primes in the partial sum begins: 3, 1531, 11887, 17737, 37997, 43441.

			a(13) = 0 + 3 + 12 + 21 + 52 + 57 + 91 + 121 + 136 + 211 + 192 + 226 + 409 = 1531 is prime.

Cf. A061262, A000217, A053614, A060773, A002636, A124978.

a(n) = SUM[i=0..n] A061262(i).

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A002635 Number of partitions of n into 4 squares.

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A061262 Smallest number representable as the sum of 3 triangular numbers in exactly n ways.

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A176661 Partial sums of A061262.

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