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1, 3, 5, 8, 10, 15, 16, 23, 25, 31, 34, 45, 42, 55, 60, 67, 69, 86, 84, 103, 102, 113, 122, 145, 134, 154, 165, 180, 181, 210, 199, 230, 232, 251, 266, 289, 271, 308, 325, 348, 339, 380, 369, 412, 417, 430, 451, 498, 471, 513, 521, 552, 559, 612, 601, 640, 633
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internal format)
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OFFSET
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1,2
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COMMENTS
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Another definition for this sequence: Let M be the matrix defined in A111490. Sequence gives M(1,1), M(1,2) + M(2,2), M(1,3) + M(2,3) + M(3,3), etc., i.e. a(n)= Sum_{i=1..n} M(i,n).
For all n>=1 the following simplifications hold for the partial sums of the two sequences:
An inductive argument then shows that the two definitions agree.
(End)
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LINKS
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FORMULA
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EXAMPLE
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1(=1+0), 3(=3+0), 5(=4+1), 8(=7+1), 10(=6+4), 15(=12+3), 16(=8+8), etc.
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PROG
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(Python)
from math import isqrt
def A123327(n): return n**2+((s:=isqrt(n-1))**2*(s+1)-sum((q:=(n-1)//k)*((k<<1)+q+1) for k in range(1, s+1))>>1) # Chai Wah Wu, Oct 22 2023
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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