login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

A251914
Numbers n such that the sum of the triangular numbers T(n) and T(n+1) is equal to the pentagonal number P(m) for some m.
2
0, 98, 9700, 950598, 93149000, 9127651498, 894416697900, 87643708742798, 8588189040096400, 841554882220704498, 82463790268588944500, 8080609891439495856598, 791817305570802005002200, 77590015336047156994359098, 7603029685627050583442189500
OFFSET
1,2
COMMENTS
Also nonnegative integers x in the solutions to 2*x^2-3*y^2+4*x+y+2 = 0, the corresponding values of y being A046172.
FORMULA
a(n) = A046173(n)-1.
a(n) = 99*a(n-1)-99*a(n-2)+a(n-3).
G.f.: 2*x^2*(x-49) / ((x-1)*(x^2-98*x+1)).
a(n) = 2*(-1/2+1/48*(49+20*sqrt(6))^(-n)*(-12-5*sqrt(6)+(-12+5*sqrt(6))*(49+20*sqrt(6))^(2*n))). - Colin Barker, Mar 03 2016
EXAMPLE
98 is in the sequence because T(98)+T(99) = 4851+4950 = 9801 = P(81).
MATHEMATICA
LinearRecurrence[{99, -99, 1}, {0, 98, 9700}, 20] (* Vincenzo Librandi, Mar 03 2016 *)
PROG
(PARI) concat(0, Vec(2*x^2*(x-49) / ((x-1)*(x^2-98*x+1)) + O(x^100)))
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Colin Barker, Dec 11 2014
STATUS
approved

pFad - Phonifier reborn

Pfad - The Proxy pFad of © 2024 Garber Painting. All rights reserved.

Note: This service is not intended for secure transactions such as banking, social media, email, or purchasing. Use at your own risk. We assume no liability whatsoever for broken pages.


Alternative Proxies:

Alternative Proxy

pFad Proxy

pFad v3 Proxy

pFad v4 Proxy