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A172073
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a(n) = (4*n^3 + n^2 - 3*n)/2.
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7
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0, 1, 15, 54, 130, 255, 441, 700, 1044, 1485, 2035, 2706, 3510, 4459, 5565, 6840, 8296, 9945, 11799, 13870, 16170, 18711, 21505, 24564, 27900, 31525, 35451, 39690, 44254, 49155, 54405, 60016, 66000, 72369, 79135, 86310, 93906, 101935, 110409, 119340
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OFFSET
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0,3
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COMMENTS
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14-gonal (or tetradecagonal) pyramidal numbers generated by the formula n*(n+1)*(2*d*n-(2*d-3))/6 for d=6.
In fact, the sequence is related to A000567 by a(n) = n*A000567(n) - Sum_{i=0..n-1} A000567(i) and this is the case d=6 in the identity n*(n*(d*n-d+2)/2) - Sum_{k=0..n-1} k*(d*k-d+2)/2 = n*(n+1)*(2*d*n-2*d+3)/6. - Bruno Berselli, Nov 29 2010
Except for the initial 0, this is the principal diagonal of the convolution array A213761. - Clark Kimberling, Jul 04 2012
Starting (1, 15, 54, ...), this is the binomial transform of (1, 14, 25, 12, 0, 0, 0, ...). - Gary W. Adamson, Jul 29 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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Bruno Berselli, A description of the recursive method in Comments lines: website Matem@ticamente (in Italian), 2008.
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FORMULA
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a(n) = n*(n+1)*(4*n-3)/2.
a(0)=0, a(1)=1, a(2)=15, a(3)=54; for n > 3, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Harvey P. Dale, Jan 29 2013
a(n) = Sum_{i=0..n-1} (n-i)*(12*i+1), with a(0)=0. - Bruno Berselli, Feb 10 2014
Sum_{n>=1} 1/a(n) = 4*Pi/21 + 8*log(2)/7 - 2/7.
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*sqrt(2)*Pi/21 + 8*sqrt(2)*log(sqrt(2)+2)/21 - (20 + 4*sqrt(2))*log(2)/21 + 2/7. (End)
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MAPLE
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MATHEMATICA
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f[n_]:= n(n+1)(4n-3)/2; Array[f, 40, 0]
LinearRecurrence[{4, -6, 4, -1}, {0, 1, 15, 54}, 40] (* Harvey P. Dale, Jan 29 2013 *)
CoefficientList[Series[x (1+11x)/(1-x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Jan 01 2014 *)
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PROG
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(Sage) [n*(n+1)*(4*n-3)/2 for n in (0..40)] # G. C. Greubel, Aug 30 2019
(GAP) List([0..40], n-> n*(n+1)*(4*n-3)/2); # G. C. Greubel, Aug 30 2019
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CROSSREFS
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Cf. similar sequences listed in A237616.
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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