[seqfan] Request for references
Charles Marion
charliemath at optonline.net
Wed Oct 6 17:26:48 CEST 2010
Greetings,
I've just submitted the following two comments for the triangular numbers in
A000217:
It is well known that a(n) - a(n-1) = n. Less well known is that\Q
a(n) - 2a(n-1) + a(n-2) = 1, a(n) - 3a(n-1) + 3a(n-2) - a(n-3) = 0 and\Q
a(n) - 4a(n-1) + 6a(n-2) - 4(a-3) + a(n-4) = 0.\Q
In general, for n>=m>2, sum_{k=0,...,m}(-1)^k*binomial(m,m-k)*a(n-k)=0.\Q
For example, 1*28 - 5*21 + 10*15 - 10*10 + 5*6 - 1*3 = 0.\Q
It is well known that a(n) + a(n-1) = n^2. Less well known is that\Q
a(n)+2a(n-1)+a(n-2) = n^2+(n-1)^2; e.g., 10+2*6+3=25=4^2+3^2 and\Q
a(n)+3a(n-1)+3a(n-2)+a(n-3)= n^2+2*(n-1)^2+(n-2)^2;\Q
e.g., 15+3*10+3*6+3=66=5^2+2*4^2+3^2.\Q
In general, for n>=m>2,sum_{k=0,...,m}binomial(m,m-k)*a(n-k)=\Q
sum_{k=0,...,m-1}binomial(m-1,m-1-k)*(n-k)^2 For example,\Q
1*28+5*21+10*15+10*10+5*6+1*3=416=1*7^2+4*6^2+6*5^2+4*4^2+1*3^2.\Q
Can anyone supply a reference in the literature for these results?
Thanks.
Charlie Marion
Yorktown Heights NY
PS Any seqfans near Yorktown Heights?
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