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A035937
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Number of partitions in parts not of the form 7k, 7k+1 or 7k-1. Also number of partitions with no part of size 1 and differences between parts at distance 2 are greater than 1.
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4
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1, 0, 1, 1, 2, 2, 3, 3, 5, 6, 8, 9, 13, 14, 19, 22, 28, 32, 41, 47, 59, 68, 83, 96, 117, 134, 161, 186, 221, 254, 301, 344, 405, 464, 541, 619, 720, 820, 949, 1081, 1245, 1414, 1624, 1840, 2106, 2384, 2717, 3070, 3492, 3936, 4464, 5026, 5684, 6388, 7210
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Case k=3,i=1 of Gordon Theorem.
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REFERENCES
| G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109.
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FORMULA
| Expansion of f(-x, -x^6)/ f(-x, -x^2) in powers of x where f() is Ramanujan's theta function.
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EXAMPLE
| 1 + x^2 + x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 5*x^8 + 6*x^9 + 8*x^10 + ...
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MAPLE
| with (numtheory):
GordonsTheorem := proc(A, n) local L, M, m, i, s, d;
L := []; M := []; m := nops(A);
for i in [$1..n] do
s := add(d*A[((d-1) mod m) + 1], d = divisors(i));
L := [op(L), s];
s := s + add(L[d]*M[i-d], d = [$1..i-1]);
M := [op(M), s/i];
od; M end:
A035937_list := n -> GordonsTheorem([0, 1, 1, 1, 1, 0, 0], n):
A035937_list(40); # Peter Luschny, Jan 22 2012
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PROG
| (Sage)
def GordonsTheorem(A, n) :
L = []; M = [];
m = len(A)
for i in range(n) :
s = sum(d*A[(d-1) % m] for d in divisors(i+1))
L.append(s)
s = s + sum(L[d-1]*M[i-d] for d in (1..i))
M.append(s/(i+1))
return M
def A035937_list(len) : return GordonsTheorem([0, 1, 1, 1, 1, 0, 0], len)
A035937_list(40) # Peter Luschny, Jan 22 2012
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CROSSREFS
| Sequence in context: A035371 A035577 A002723 * A020999 A079955 A192928
Adjacent sequences: A035934 A035935 A035936 * A035938 A035939 A035940
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KEYWORD
| nonn,easy,changed
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AUTHOR
| Olivier Gerard (olivier.gerard(AT)gmail.com)
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