A005892 Truncated square numbers: 7*n^2 + 4*n + 1.

1, 12, 37, 76, 129, 196, 277, 372, 481, 604, 741, 892, 1057, 1236, 1429, 1636, 1857, 2092, 2341, 2604, 2881, 3172, 3477, 3796, 4129, 4476, 4837, 5212, 5601, 6004, 6421, 6852, 7297, 7756, 8229, 8716, 9217, 9732, 10261, 10804, 11361, 11932

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• L. Hogben, Choice and Chance by Cardpack and Chessboard, Vol. 1, Max Parrish and Co, London, 1950, p. 36.
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A135704.

Programs

Magma

[7*n^2 + 4*n + 1: n in [0..50]]; // G. C. Greubel, Nov 30 2017

Maple

A005892:=-(1+9*z+4*z**2)/(z-1)**3; # Simon Plouffe in his 1992 dissertation

Mathematica

Table[7n^2+4n+1,{n,0,50}]  (* Harvey P. Dale, Mar 24 2011 *)

PARI

a(n)=7*n^2+4*n+1 \\ Charles R Greathouse IV, Jun 17 2017

Sage

[7*n^2+4*n+1 for n in (0..50)] # G. C. Greubel, Apr 19 2019

Formula

a(n) = a(n-1) + 14*n - 3 (with a(0)=1). - Vincenzo Librandi, Nov 18 2010

From G. C. Greubel, Nov 30 2017: (Start)

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

G.f.: (1 + 9*x + 4*x^2)/(1 - x)^3.

E.g.f.: (1 + 11*x + 7*x^2)*exp(x). (End)

Extensions

More terms from Frank Ellermann, Jan 18 2002