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0, 1, 14, 50, 120, 235, 406, 644, 960, 1365, 1870, 2486, 3224, 4095, 5110, 6280, 7616, 9129, 10830, 12730, 14840, 17171, 19734, 22540, 25600, 28925, 32526, 36414, 40600, 45095, 49910, 55056, 60544, 66385, 72590, 79170, 86136, 93499, 101270
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OFFSET
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0,3
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COMMENTS
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This sequence is related to A180223 by 2*a(n) = n*A180223(n) - Sum_{i=0..n-1} A180223(i). Also, 13-gonal (or tridecagonal) pyramidal numbers. - Bruno Berselli, Dec 14 2010
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REFERENCES
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A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 189-196.
E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93.
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LINKS
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Bruno Berselli, 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 to sequences related to pyramidal numbers
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
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FORMULA
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a(n) = n*(n+1)*(11*n-8)/6.
G.f.: x*(1+10*x)/(1-x)^4. - Bruno Berselli, Aug 19 2010
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Bruno Berselli, Aug 19 2010
a(n) = Sum_{i=0..n-1} (n-i)*(11*i+1), with a(0)=0. - Bruno Berselli, Feb 10 2014
E.g.f.: exp(x)*x*(6 + 36*x + 11*x^2)/6. - Stefano Spezia, May 04 2022
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EXAMPLE
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After 0, the sequence is provided by the row sums of the triangle (see above, fourth formula):
1;
2, 12;
3, 24, 23;
4, 36, 46, 34;
5, 48, 69, 68, 45; ... - Vincenzo Librandi, Feb 12 2014
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MAPLE
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seq(n*(n+1)*(11*n-8)/6, n=0..40); # G. C. Greubel, Aug 30 2019
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MATHEMATICA
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Accumulate[Table[n (11n-9)/2, {n, 0, 40}]] (* or *) LinearRecurrence[ {4, -6, 4, -1}, {0, 1, 14, 50}, 40] (* Harvey P. Dale, Nov 14 2011 *)
CoefficientList[Series[x (1 + 10 x)/(1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Feb 12 2014 *)
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PROG
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(Magma) I:=[0, 1, 14, 50]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4) : n in [1..50]]; // Vincenzo Librandi, Feb 12 2014
(PARI) a(n)=n*(n+1)*(11*n-8)/6 \\ Charles R Greathouse IV, Oct 07 2015
(Sage) [n*(n+1)*(11*n-8)/6 for n in (0..40)] # G. C. Greubel, Aug 30 2019
(GAP) List([0..40], n-> n*(n+1)*(11*n-8)/6); # G. C. Greubel, Aug 30 2019
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CROSSREFS
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Cf. A051865, A180223.
Similar sequences are listed in A237616.
Sequence in context: A005914 A009960 A009928 * A225921 A350850 A205354
Adjacent sequences: A050438 A050439 A050440 * A050442 A050443 A050444
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KEYWORD
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nonn,easy,nice
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AUTHOR
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Barry E. Williams, Dec 23 1999
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STATUS
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approved
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