A081436 Fifth subdiagonal in array of n-gonal numbers A081422.

1, 7, 24, 58, 115, 201, 322, 484, 693, 955, 1276, 1662, 2119, 2653, 3270, 3976, 4777, 5679, 6688, 7810, 9051, 10417, 11914, 13548, 15325, 17251, 19332, 21574, 23983, 26565, 29326, 32272, 35409, 38743, 42280, 46026, 49987, 54169, 58578, 63220

Offset: 0

Author

Paul Barry, Mar 21 2003

Keywords

• easy
• nonn

Comments

One of a family of sequences with palindromic generators.

Also as A(n) = (1/6)*(6*n^3 - 3*n^2 + 3*n), n>0: structured pentagonal diamond numbers (vertex structure 5). (Cf. A004068 = alternate vertex; A000447 = structured diamonds; A100145 for more on structured numbers.) - James A. Record (james.record(AT)gmail.com), Nov 07 2004

Sequence of the absolute values of the z^1 coefficients of the polynomials in the GF4 denominators of A156933. See A157705 for background information. - Johannes W. Meijer, Mar 07 2009

Row 1 of the convolution arrays A213831 and A213833. - Clark Kimberling, Jul 04 2012

Partial sums of A056109. - J. M. Bergot, Jun 22 2013

Number of ordered pairs of intersecting multisets of size 2, each chosen with repetition from {1,...,n}. - Robin Whitty, Feb 12 2014

Row sums of A244418. - L. Edson Jeffery, Jan 10 2015

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..5000
• Christian Barrientos, The number of spanning trees of cyclic snakes, Indones. J. Comb. (2025) Vol. 9, No. 1, 21-30. See p. 29.
• J. A. Dias da Silva and Pedro J. Freitas, Counting Spectral Radii of Matrices with Positive Entries, arXiv:1305.1139 [math.CO], 2013.
• Theorem of the Day, Lovász Local Lemma example involving intersecting pairs of multisets
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A081434, A081435, A081437, A156933, A157705, A244418.

Programs

GAP

List([0..40], n-> (n+1)*(2*(n+1)^2-n)/2); # G. C. Greubel, Aug 14 2019

Magma

[(2*n^3+5*n^2+5*n+2)/2: n in [0..40]]; // Vincenzo Librandi, Jul 19 2011

Maple

A081436 := proc(n)
    (n+1)*(2*n^2+3*n+2)/2 ;
end proc:
seq(A081436(n),n=0..60) ; # R. J. Mathar, Jun 26 2013

Mathematica

LinearRecurrence[{4, -6, 4, -1}, {1, 7, 24, 58}, 40] (* Jean-François Alcover, Sep 21 2017 *)

PARI

a(n)=n^3+5/2*n*(n+1)+1 \\ Charles R Greathouse IV, Jun 20 2013

Sage

[(n+1)*(2*(n+1)^2-n)/2 for n in (0..40)] # G. C. Greubel, Aug 14 2019

Formula

a(n) = (n+1)*(2*n^2 + 3*n + 2)/2.

G.f.: (1+x)*(1+2*x)/(1-x)^4. (Convolution of A005408 and A016777.)

a(n) = A110449(n, n-1), for n>1.

a(n) = (n+1)*T(n+1) + n*T(n), where T( ) are triangular numbers. Binomial transform of [1, 6, 11, 6, 0, 0, 0, ...]. - Gary W. Adamson, Dec 28 2007

E.g.f.: exp(x)*(2 + 12*x + 11*x^2 + 2*x^3)/2. - Stefano Spezia, Apr 13 2021

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Wesley Ivan Hurt, Apr 14 2021

Extensions

G.f. simplified and crossrefs added by Johannes W. Meijer, Mar 07 2009