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A050441 Partial sums of A051865. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

This sequence is related to A180223 by 2*a(n) = n*A180223(n) - sum(A180223(i), i=0..n-1). Also, 13-gonal (or tridecagonal) pyramidal numbers. - Bruno Berselli, Dec 14 2010

REFERENCES

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.

LINKS

Bruno Berselli, Table of n, a(n) for n = 0..1000

B. 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).

FORMULA

a(n) = n(n+1)(11n-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( (n-i)*(11*i+1), i=0..n-1 ), with a(0)=0. - Bruno Berselli, Feb 10 2014

EXAMPLE

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

MATHEMATICA

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 *)

PROG

(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

CROSSREFS

Cf. A051865, A180223.

Cf. similar sequences listed in A237616.

Sequence in context: A005914 A009960 A009928 * A225921 A205354 A281699

Adjacent sequences:  A050438 A050439 A050440 * A050442 A050443 A050444

KEYWORD

nonn,easy,nice

AUTHOR

Barry E. Williams, Dec 23 1999

STATUS

approved

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Last modified July 23 08:53 EDT 2017. Contains 289686 sequences.