A276915 Indices of triangular numbers in A276914 which are also pentagonal.

0, 1, 10, 143, 1988, 27693, 385710, 5372251, 74825800, 1042188953, 14515819538, 202179284583, 2815994164620, 39221739020101, 546288352116790, 7608815190614963, 105977124316492688, 1476070925240282673, 20559015829047464730, 286350150681424223551

Offset: 0

Author

Daniel Poveda Parrilla, Sep 22 2016

Keywords

• nonn
• easy

Comments

A276914(a(n)) = A014979(n + 1). All numbers which are both triangular and pentagonal can be found in sequence A276914.

Links

• Daniel Poveda Parrilla, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (13,13,-1).

Crossrefs

Cf. A014979, A046175, A276914.

Programs

Mathematica

RecurrenceTable[{a[n] == 14 a[n - 1] - a[n - 2] - 4 (-1)^n, a[0] == 0, a[1] == 1}, a, {n, 19}] (* Michael De Vlieger, Sep 23 2016 *)

PARI

concat(0, Vec(x*(1-3*x)/((1+x)*(1-14*x+x^2)) + O(x^30))) \\ Colin Barker, Sep 23 2016

Formula

a(n) = 14*a(n-1) - a(n-2) - 4*(-1)^n for n>1, a(0)=0, a(1)=1.

a(n) = (A046175(n) + (A046175(n) mod 2))/2.

From Colin Barker, Sep 23 2016: (Start)

G.f.: x*(1 - 3*x) / ((1 + x)*(1 - 14*x + x^2)).

a(n) = 13*a(n-1) + 13*a(n-2) - a(n-3) for n>2.

a(n) = ( -6*(-1)^n + (3+sqrt(3))*(7-4*sqrt(3))^n - (-3+sqrt(3))*(7+4*sqrt(3))^n )/24. (End)