A051843 Partial sums of A002419.

0, 1, 11, 51, 161, 406, 882, 1722, 3102, 5247, 8437, 13013, 19383, 28028, 39508, 54468, 73644, 97869, 128079, 165319, 210749, 265650, 331430, 409630, 501930, 610155, 736281, 882441, 1050931, 1244216, 1464936, 1715912, 2000152, 2320857, 2681427, 3085467, 3536793

Offset

0,3

Comments

5-dimensional form of octagonal-based pyramidal numbers. - Derek I. Thomas (dithom02(AT)louisville.edu), Jun 30 2007

Convolution of triangular numbers (A000217) and octagonal numbers (A000567). [Bruno Berselli, Jul 21 2015]

Also the number of 4-cycles in the (n+2)-triangular honeycomb bishop graph. - Eric W. Weisstein, Aug 10 2017

References

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 215.

H. J. Ryser, Combinatorial Mathematics, Carus Mathematical Monographs No. 14, John Wiley and Sons, 1963, pp. 1-8.

Links

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Luis Verde-Star, A Matrix Approach to Generalized Delannoy and Schröder Arrays, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.

Eric Weisstein's World of Mathematics, Graph Cycle

Eric Weisstein's World of Mathematics, Octagonal Number

Eric Weisstein's World of Mathematics, Pyramidal Number

Index to sequences related to pyramidal numbers

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Formula

a(n) = C(n+3,4) * (6*n-1)/5

G.f.: x*(1+5*x)/(1-x)^6.

a(n) = n*(n+1)*(n+2)*(n+3)*(6n-1)/120. - Derek I. Thomas (dithom02(AT)louisville.edu), Jun 30 2007

E.g.f.: exp(x)*x*(120 + 540*x + 420*x^2 + 95*x^3 + 6*x^4)/120. - Stefano Spezia, Oct 29 2025

Mathematica

Join[{0}, Accumulate[LinearRecurrence[{5, -10, 10, -5, 1}, {1, 10, 40, 110, 245}, 40]]] (* Harvey P. Dale, Nov 30 2014 *)
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 1, 11, 51, 161, 406}, 40] (* Harvey P. Dale, Nov 30 2014 *)
Table[(6 n - 1) Binomial[n + 3, 4]/5, {n, 0, 20}] (* Eric W. Weisstein, Aug 10 2017 *)

Crossrefs

Cf. A002419; A000217, A000567.

Cf. A093563 ((6, 1) Pascal, column m=5).

Cf. A034827 (3-cycles in the triangular honeycomb bishop graph), A290775 (5-cycles), A290779 (6-cycles).

Sequence in context: A175360 A226451 A185505 * A107464 A027942 A168214

Adjacent sequences: A051840 A051841 A051842 * A051844 A051845 A051846

Keyword

easy,nonn

Author

Barry E. Williams, Dec 13 1999

Extensions

a(1) corrected by Gael Linder (linder.gael(AT)wanadoo.fr), Oct 31 2007

a(0) prepended by Joerg Arndt, Jun 26 2013

Status

approved