A005920 Tricapped prism numbers.

1, 9, 33, 82, 165, 291, 469, 708, 1017, 1405, 1881, 2454, 3133, 3927, 4845, 5896, 7089, 8433, 9937, 11610, 13461, 15499, 17733, 20172, 22825, 25701, 28809, 32158, 35757, 39615, 43741, 48144, 52833, 57817, 63105, 68706, 74629, 80883, 87477, 94420

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

a(n) = (n+1)*A000326(n+1) - Sum_{i=0...n} A001477(i) = (n+1)*((n+1)*(3*n+2)/2) - A000217(n) = (n+1)*(3*n^2+4n+2)/2. - Bruno Berselli, Apr 25 2010

Also central terms of triangle A093445: a(n) = A093445(2*n+1,n+1). - Reinhard Zumkeller, Oct 03 2012

References

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

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• 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 (4,-6,4,-1).

Crossrefs

Cf. for recursive method [Ar(m) is the m-th term of a sequence in the OEIS] a(n) = n*Ar(n) - A000217(n-1) or a(n) = (n+1)*Ar(n+1) - A000217(n) or similar: A081436, A005945, A006003 and the terms T(2, n) or T(3, n) in the sequence A125860. - Bruno Berselli, Apr 25 2010

Cf. A000326, A001477, A093445.

Programs

Haskell

a005920 n = (n * (n * (3 * n + 7) + 6) + 2) `div` 2
-- Reinhard Zumkeller, Oct 03 2012

Magma

[(3*n^3+7*n^2+6*n+2)/2 : n in [0..50]]; // Wesley Ivan Hurt, May 05 2021

Maple

a:=n->(3*n^3+7*n^2+6*n+2)/2: seq(a(n),n=0..60);
A005920:=(1+5*z+3*z**2)/(z-1)**4; # conjectured by Simon Plouffe in his 1992 dissertation

Mathematica

CoefficientList[ Series[(1+5x+3x^2)/(1-x)^4, {x, 0, 39}], x] (* Jean-François Alcover, Dec 02 2011, after Simon Plouffe *)
LinearRecurrence[{4,-6,4,-1},{1,9,33,82},40] (* Harvey P. Dale, Sep 25 2012 *)

PARI

a(n)=n*(3*n^2+7*n+6)/2+1 \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = (1/2) * (3*n^3 + 7*n^2 + 6*n + 2). - Ralf Stephan, Apr 20 2004

a(0)=1, a(1)=9, a(2)=33, a(3)=82, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Harvey P. Dale, Sep 25 2012

E.g.f.: exp(x)*(2 + 16*x + 16*x^2 + 3*x^3)/2. - Stefano Spezia, Jun 10 2022

Extensions

More terms from Emeric Deutsch, May 09 2004