A048902 -id:A048902 - cp's OEIS frontend

A048903 Heptagonal hexagonal numbers.

1, 121771, 12625478965, 1309034909945503, 135723357520344181225, 14072069153115290487843091, 1459020273797576190840203197981, 151274140013808225465578657485241095, 15684405383452644158924550174544564031953, 1626190518815862911671806985731550830475727995

Offset: 1

Eric W. Weisstein

As n increases, this sequence is approximately geometric with common ratio r = lim(n->Infinity,a(n)/a(n-1)) = (2+sqrt(5))^8 = 51841+23184*sqrt(5). - Ant King, Dec 24 2011

Colin Barker, Table of n, a(n) for n = 1..200
Eric Weisstein's World of Mathematics, Heptagonal Hexagonal Number
Index entries for linear recurrences with constant coefficients, signature (103683,-103683,1).

Cf. A048901, A048902.

LinearRecurrence[{103683, -103683, 1}, {1, 121771, 12625478965}, 8]; (* Ant King, Dec 24 2011 *)

Vec(-x*(55*x^2+18088*x+1)/((x-1)*(x^2-103682*x+1)) + O(x^20)) \\ Colin Barker, Jun 23 2015

From Ant King, Dec 24 2011: (Start)
G.f.: x*(1+18088*x+55*x^2)/((1-x)*(1-103682*x+x^2)).
a(n) = 103683*a(n-1)-103683*a(n-2)+a(n-3).
a(n) = 103682*a(n-1)-a(n-2)+18144.
a(n) = 1/80*((sqrt(5)-1)*(2+sqrt(5))^(8n-5)- (sqrt(5)+1)*(2-sqrt(5))^(8n-5)-14).
a(n) = floor(1/80*(sqrt(5)-1)*(2+sqrt(5))^(8n-5)).
(End)

A048901 Indices of hexagonal numbers which are also heptagonal.

Original entry on oeis.org

1, 247, 79453, 25583539, 8237820025, 2652552464431, 854113655726677, 275021944591525483, 88556212044815478769, 28514825256485992638055, 9181685176376444813974861, 2956474111967958744107267107

Offset: 1

Eric W. Weisstein

As n increases, this sequence is approximately geometric with common ratio r = lim_{n->infinity} a(n)/a(n-1) = (2 + sqrt(5))^4 = 161 + 72*sqrt(5). - Ant King, Dec 24 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Eric Weisstein's World of Mathematics, Heptagonal hexagonal number.
Index entries for linear recurrences with constant coefficients, signature (323,-323,1).

Cf. A048902, A048903.

I:=[1, 247, 79453]; [n le 3 select I[n] else 323*Self(n-1)-323*Self(n-2)+Self(n-3): n in [1..20]]; // Vincenzo Librandi, Dec 28 2011

LinearRecurrence[{323, -323, 1}, {1, 247, 79453}, 12]; (* Ant King, Dec 24 2011 *)

G.f.: x*(-1 + 76*x + 5*x^2) / ( (x-1)*(x^2 - 322*x + 1) ). - R. J. Mathar, Dec 21 2011
From Ant King, Dec 24 2011: (Start)
a(n) = 322*a(n-1) - a(n-2) - 80.
a(n) = (1/40)*sqrt(5)*((1+sqrt(5))*(sqrt(5)+2)^(4*n-3) + (1-sqrt(5))*(sqrt(5)-2)^(4*n-3) + 2*sqrt(5)).
a(n) = ceiling((1/40)*sqrt(5)*(1+sqrt(5))*(sqrt(5)+2)^(4*n-3)).
(End)

cp's OEIS Frontend

A048903 Heptagonal hexagonal numbers.

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