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A172073 a(n) = (4*n^3 + n^2 - 3*n)/2. 7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

REFERENCES

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93. - Bruno Berselli, Feb 13 2014

LINKS

Vincenzo Librandi, 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)*(4*n-3)/2.

G.f.: x*(1+11*x)/(1-x)^4. - Bruno Berselli, Dec 15 2010

a(n) = Sum_{i=0..n} A051866(i). - Bruno Berselli, Dec 15 2010

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

MATHEMATICA

f[n_] := n (n + 1) (4 n - 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 + 11 x)/(1 - x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Jan 01 2014 *)

PROG

(MAGMA) [(4*n^3+n^2-3*n)/2: n in [0..50]]; // Vincenzo Librandi, Jan 01 2014

(PARI) a(n)=(4*n^3+n^2-3*n)/2 \\ Charles R Greathouse IV, Oct 07 2015

CROSSREFS

Cf. similar sequences listed in A237616.

Cf. A194454.

Sequence in context: A198955 A063436 A010004 * A059145 A086643 A240814

Adjacent sequences:  A172070 A172071 A172072 * A172074 A172075 A172076

KEYWORD

nonn,easy

AUTHOR

Vincenzo Librandi, Jan 25 2010

EXTENSIONS

Edited by Bruno Berselli, Dec 14 2010

STATUS

approved

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Last modified June 28 05:07 EDT 2017. Contains 288813 sequences.