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A050410 Truncated square pyramid numbers: a(n) = sum( k^2, k=n..2*n-1 ). 7
0, 1, 13, 50, 126, 255, 451, 728, 1100, 1581, 2185, 2926, 3818, 4875, 6111, 7540, 9176, 11033, 13125, 15466, 18070, 20951, 24123, 27600, 31396, 35525, 40001, 44838, 50050, 55651, 61655, 68076, 74928, 82225, 89981, 98210, 106926, 116143 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Starting with offset 1 = binomial transform of [1, 12, 25, 14, 0, 0, 0,...]. [From Gary W. Adamson, Jan 09 2009]

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

FORMULA

a(n)=n*(7*n-1)*(2*n-1)/6.

a(0)=0, a(1)=1, a(2)=13, a(3)=50, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). [Harvey P. Dale, Feb 29 2012]

G.f.: x*(1+9*x+4*x^2)/(1-x)^4. [Colin Barker, Mar 23 2012]

EXAMPLE

1^2 + 1; 2^2 + 3^2 = 13; 3^2 + 4^2 + 5^2 = 50; ...

MAPLE

seq(add((n+k+1)^2, k=0..n), n=-1..36); - Zerinvary Lajos, Dec 01 2006

MATHEMATICA

Table[Sum[k^2, {k, n, 2n-1}], {n, 0, 40}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {0, 1, 13, 50}, 40] (* Harvey P. Dale, Feb 29 2012 *)

PROG

(PARI) for(n=1, 100, print1(sum(i=0, n-1, (n+i)^2), ", "))

(MAGMA) [n*(7*n-1)*(2*n-1)/6: n in [0..40]]; // Vincenzo Librandi, Apr 27 2012

CROSSREFS

Cf. A072474, A240137.

Sequence in context: A189054 A231947 A209995 * A121991 A121990 A050491

Adjacent sequences:  A050407 A050408 A050409 * A050411 A050412 A050413

KEYWORD

nonn,easy,nice

AUTHOR

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 22 1999

STATUS

approved

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Last modified January 20 17:31 EST 2018. Contains 297960 sequences.