A133216 Integers that are simultaneously triangular (A000217) and decagonal (A001107).

0, 1, 10, 1540, 11935, 1777555, 13773376, 2051297326, 15894464365, 2367195337045, 18342198104230, 2731741367653000, 21166880717817451, 3152427171076225351, 24426562006163234620, 3637898223680596402450, 28188231388231654934425, 4198131397700237172202345

Offset

1,3

Comments

Positive terms are of the form (m^2-9)/16 where m runs over the elements of A077443 that are congruent to 5 modulo 8. Correspondingly, for n>1, sqrt(16*a(n)+9) form a subsequence of A077443, while sqrt(8*a(n)+1) form a subsequence of A077442 with indices congruent to 2,3 modulo 4. [Max Alekseyev]

Links

Table of n, a(n) for n=1..18.

Index entries for linear recurrences with constant coefficients, signature (1, 1154, -1154, -1, 1).

Formula

a(n) = A000217(A133218(n)) = A001107(A133217(n)).

For n>5, a(n) = 1154*a(n-2) - a(n-4) + 396.

For n>6, a(n) = a(n-1) + 1154*a(n-2) - 1154*a(n-3) - a(n-4) + a(n-5).

For n>1, a(n) = 1/64 * ( (9 + 4* sqrt(2)*(-1)^n)*(1+sqrt(2))^(4*n-6) + (9 - 4* sqrt(2)*(-1)^n)*(1-sqrt(2))^(4*n-6) - 22).

a(n) = floor ( 1/64 * (9 + 4*sqrt(2)*(-1)^n) * (1+sqrt(2))^(4*n-6) ).

G.f.: (x^5 + 9*x^4 + 376*x^3 + 9*x^2 + x)/((1 - x)*(x^2 - 34*x + 1)*(x^2 + 34*x + 1)). [corrected by Peter Luschny, Apr 04 2019]

Lim (n -> Infinity, a(2n+1)/a(2n)) = (1/49)*(3649+2580*sqrt(2)).

Lim (n -> Infinity, a(2n)/a(2n-1)) = (1/49)*(193+132*sqrt(2)).

Example

The initial terms of the sequences of triangular (A000217) and decagonal (A001107) numbers are 0, 1, 3, 6, 10, 15, ... and 0, 1, 10, 27, ... respectively. As the third number which is common to both sequences is 10, we have a(3) = 10.

Mathematica

LinearRecurrence[{1, 1154, -1154, -1, 1} , {0, 1, 10, 1540, 11935, 1777555}, 17] (* first term 0 corrected by Georg Fischer, Apr 02 2019 *)

Crossrefs

Cf. A000217, A001107, A133217, A133218, A077443, A077442.

Sequence in context: A370678 A211914 A194792 * A099128 A172958 A286397

Adjacent sequences: A133213 A133214 A133215 * A133217 A133218 A133219

Keyword

nonn

Author

Richard Choulet, Oct 11 2007; Ant King, Nov 04 2011

Extensions

Entry revised by N. J. A. Sloane, Nov 06 2011

Term 0 prepended and entry revised accordingly by Max Alekseyev, Nov 06 2011

Status

approved