A276915 -id:A276915 - cp's OEIS frontend

A046179 Indices of hexagonal numbers that are also pentagonal.

1, 143, 27693, 5372251, 1042188953, 202179284583, 39221739020101, 7608815190614963, 1476070925240282673, 286350150681424223551, 55550453161271059086173, 10776501563135904038493963, 2090585752795204112408742601, 405562859540706461903257570583

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(3))^4 = 97 + 56*sqrt(3). - Ant King, Dec 14 2011

Colin Barker, Table of n, a(n) for n = 1..438
Eric Weisstein's World of Mathematics, Hexagonal Pentagonal Number.
Index entries for linear recurrences with constant coefficients, signature (195,-195,1).

Cf. A046178, A046180, A276915.

LinearRecurrence[{195, -195, 1}, {1, 143, 27693}, 11] (* Ant King, Dec 14 2011 *)

Vec(-x*(3*x^2-52*x+1)/((x-1)*(x^2-194*x+1)) + O(x^20)) \\ Colin Barker, Jun 21 2015

From Warut Roonguthai, Jan 08 2001: (Start)
a(n) = 194*a(n-1) - a(n-2) - 48.
G.f.: x*(1-52*x+3*x^2)/((1-x)*(1-194*x+x^2)). (End)
From Ant King, Dec 14 2011: (Start)
a(n) = 195*a(n-1) - 195*a(n-2) + a(n-3).
a(n) = (1/24)*sqrt(3)*((sqrt(3)-1)*(2+sqrt(3))^(4n-2)+(sqrt(3)+1)* (2-sqrt(3))^(4n-2)+2*sqrt(3)).
a(n) = ceiling((1/24)*sqrt(3)*(sqrt(3)-1)*(2+sqrt(3))^(4n-2)).
(End)
a(n) = A276915(2n-1). - Daniel Poveda Parrilla, Dec 03 2016

A276914 Subsequence of triangular numbers obtained by adding a square and two smaller triangles, a(n) = n^2 + 2*A000217(A052928(n)).

Original entry on oeis.org

0, 1, 10, 15, 36, 45, 78, 91, 136, 153, 210, 231, 300, 325, 406, 435, 528, 561, 666, 703, 820, 861, 990, 1035, 1176, 1225, 1378, 1431, 1596, 1653, 1830, 1891, 2080, 2145, 2346, 2415, 2628, 2701, 2926, 3003, 3240, 3321, 3570, 3655, 3916, 4005, 4278, 4371, 4656

Offset: 0

Daniel Poveda Parrilla, Sep 22 2016

All terms of this sequence are triangular numbers. Graphically, for each term of the sequence, one corner of the square will be part of the corresponding triangle's hypotenuse if the term is an odd number. Otherwise, it will not be part of it.
a(A276915(n)) is a triangular pentagonal number.
a(A079291(n)) is a triangular square number, as A275496 is a subsequence of this.

Daniel Poveda Parrilla, Table of n, a(n) for n = 0..10000
Daniel Poveda Parrilla, Illustration of initial terms.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000217, A004526, A016813, A042948, A052928, A079291, A168277, A275496, A276915.

[n*(2*n+(-1)^n): n in [0..40]]; // G. C. Greubel, Aug 19 2022

Table[n (2 n + (-1)^n), {n, 0, 48}] (* Michael De Vlieger, Sep 23 2016 *)

concat(0, Vec(x*(1+9*x+3*x^2+3*x^3)/((1-x)^3*(1+x)^2) + O(x^50))) \\ Colin Barker, Sep 23 2016

[n*(2*n+(-1)^n) for n in (0..40)] # G. C. Greubel, Aug 19 2022

a(n) = n^2 + 2*A000217(A052928(n)).
a(n) = A000217(A042948(n)).
a(n) = n*(2*n + (-1)^n).
a(n) = n*A168277(n + 1).
a(n) = n*A016813(A004526(n)).
From Colin Barker, Sep 23 2016: (Start)
G.f.: x*(1 + 9*x + 3*x^2 + 3*x^3) / ((1 - x)^3*(1 + x)^2).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4.
a(n) = n*(2*n+1) for n even.
a(n) = n*(2*n-1) for n odd. (End)
E.g.f.: x*( 2*(1+x)*exp(x) - exp(-x) ). - G. C. Greubel, Aug 19 2022
Sum_{n>=1} 1/a(n) = 2 - log(2). - Amiram Eldar, Aug 21 2022

cp's OEIS Frontend

A046179 Indices of hexagonal numbers that are also pentagonal.

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