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Subsequence of triangular numbers obtained by adding a square and two smaller triangles, a(n) = n^2 + 2*A000217(A052928(n)).
3

%I #32 Aug 21 2022 04:19:05

%S 0,1,10,15,36,45,78,91,136,153,210,231,300,325,406,435,528,561,666,

%T 703,820,861,990,1035,1176,1225,1378,1431,1596,1653,1830,1891,2080,

%U 2145,2346,2415,2628,2701,2926,3003,3240,3321,3570,3655,3916,4005,4278,4371,4656

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

%C 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.

%C a(A276915(n)) is a triangular pentagonal number.

%C a(A079291(n)) is a triangular square number, as A275496 is a subsequence of this.

%H Daniel Poveda Parrilla, <a href="/A276914/b276914.txt">Table of n, a(n) for n = 0..10000</a>

%H Daniel Poveda Parrilla, <a href="/A276914/a276914.gif">Illustration of initial terms</a>.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).

%F a(n) = n^2 + 2*A000217(A052928(n)).

%F a(n) = A000217(A042948(n)).

%F a(n) = n*(2*n + (-1)^n).

%F a(n) = n*A168277(n + 1).

%F a(n) = n*A016813(A004526(n)).

%F From _Colin Barker_, Sep 23 2016: (Start)

%F G.f.: x*(1 + 9*x + 3*x^2 + 3*x^3) / ((1 - x)^3*(1 + x)^2).

%F a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4.

%F a(n) = n*(2*n+1) for n even.

%F a(n) = n*(2*n-1) for n odd. (End)

%F E.g.f.: x*( 2*(1+x)*exp(x) - exp(-x) ). - _G. C. Greubel_, Aug 19 2022

%F Sum_{n>=1} 1/a(n) = 2 - log(2). - _Amiram Eldar_, Aug 21 2022

%t Table[n (2 n + (-1)^n), {n, 0, 48}] (* _Michael De Vlieger_, Sep 23 2016 *)

%o (PARI) 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

%o (Magma) [n*(2*n+(-1)^n): n in [0..40]]; // _G. C. Greubel_, Aug 19 2022

%o (SageMath) [n*(2*n+(-1)^n) for n in (0..40)] # _G. C. Greubel_, Aug 19 2022

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

%K nonn,easy

%O 0,3

%A _Daniel Poveda Parrilla_, Sep 22 2016