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A172117
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a(n) = n*(n+1)*(20*n-17)/6.
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4
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0, 1, 23, 86, 210, 415, 721, 1148, 1716, 2445, 3355, 4466, 5798, 7371, 9205, 11320, 13736, 16473, 19551, 22990, 26810, 31031, 35673, 40756, 46300, 52325, 58851, 65898, 73486, 81635, 90365, 99696, 109648, 120241, 131495, 143430, 156066
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OFFSET
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0,3
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COMMENTS
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Generated by the formula n*(n+1)*(2*d*n-2*d+3)/6 for d=10.
This sequence is related to A051624 by a(n) = n*A051624(n) - Sum_{i=0..n-1} A051624(i) = n*(n+1)*(20*n-17)/2; in fact, this is the case d=10 in the identity n*(n*(d*n-d+2)/2) - Sum_{i=0..n-1} i*(d*i-d+2)/2 = n*(n+1)*(2*d*n-2*d+3)/6. - Bruno Berselli, Aug 26 2010
Also, a(n) = n*A190816(n) - Sum_{i=0..n-1} A190816(i) for n>0. - Bruno Berselli, Dec 18 2013
Starting with offset 1, the sequence is the binomial transform of (1, 22, 41, 20, 0, 0, 0, ...). - Gary W. Adamson, Jul 31 2015
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REFERENCES
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E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93. [Bruno Berselli, Feb 13 2014]
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Bruno Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian), 2008.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
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FORMULA
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G.f.: x*(1+19*x)/(1-x)^4. - Bruno Berselli, Aug 26 2010
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 3. - Harvey P. Dale, May 15 2011
a(n) = Sum_{i=0..n-1} (n-i)*(20*i+1), with a(0)=0. - Bruno Berselli, Feb 11 2014
E.g.f.: (1/6)*x*(6 + 63*x + 20*x^2)*exp(x). - G. C. Greubel, Apr 15 2022
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MATHEMATICA
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Table[(20n^3+3n^2-17n)/6, {n, 0, 50}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {0, 1, 23, 86}, 40] (* Harvey P. Dale, May 15 2011 *)
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PROG
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(PARI) a(n)=n*(20*n^2+3*n-17)/6 \\ Charles R Greathouse IV, Jan 11 2012
(Magma) [n*(n+1)*(20*n-17)/6: n in [0..50]]; // Vincenzo Librandi, Aug 01 2015
(SageMath) [sum( (-1)^j*(20-j)*binomial(n+2-j, 3-j) for j in (0..1)) for n in (0..50)] # G. C. Greubel, Apr 15 2022
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CROSSREFS
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Cf. A051624.
Cf. similar sequences listed in A237616.
Sequence in context: A060456 A056580 A010011 * A217529 A284711 A193018
Adjacent sequences: A172114 A172115 A172116 * A172118 A172119 A172120
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KEYWORD
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nonn,easy
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AUTHOR
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Vincenzo Librandi, Jan 26 2010
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STATUS
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approved
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