A050410 Truncated square pyramid numbers: a(n) = Sum_{k = n..2*n-1} k^2.

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

Offset

0,3

Comments

Starting with offset 1 = binomial transform of [1, 12, 25, 14, 0, 0, 0, ...]. - 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(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=0, a(1)=1, a(2)=13, a(3)=50. - Harvey P. Dale, Feb 29 2012

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

E.g.f.: x*(6 + 33*x + 14*x^2)*exp(x)/6. - G. C. Greubel, Oct 30 2019

Example

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

Maple

seq(n*(7*n-1)*(2*n-1)/6, n=0..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), ", "))
(PARI) vector(40, n, (n-1)*(7*n-8)*(2*n-3)/6) \\ G. C. Greubel, Oct 30 2019
(Magma) [n*(7*n-1)*(2*n-1)/6: n in [0..40]]; // Vincenzo Librandi, Apr 27 2012
(Sage) [n*(7*n-1)*(2*n-1)/6 for n in (0..40)] # G. C. Greubel, Oct 30 2019
(GAP) List([0..40], n-> n*(7*n-1)*(2*n-1)/6); # G. C. Greubel, Oct 30 2019

Crossrefs

Cf. A072474, A240137.

Sequence in context: A231947 A322615 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