This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A276914 #32 Aug 21 2022 04:19:05
%S A276914 0,1,10,15,36,45,78,91,136,153,210,231,300,325,406,435,528,561,666,
%T A276914 703,820,861,990,1035,1176,1225,1378,1431,1596,1653,1830,1891,2080,
%U A276914 2145,2346,2415,2628,2701,2926,3003,3240,3321,3570,3655,3916,4005,4278,4371,4656
%N A276914 Subsequence of triangular numbers obtained by adding a square and two smaller triangles, a(n) = n^2 + 2*A000217(A052928(n)).
%C A276914 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 A276914 a(A276915(n)) is a triangular pentagonal number.
%C A276914 a(A079291(n)) is a triangular square number, as A275496 is a subsequence of this.
%H A276914 Daniel Poveda Parrilla, <a href="/A276914/b276914.txt">Table of n, a(n) for n = 0..10000</a>
%H A276914 Daniel Poveda Parrilla, <a href="/A276914/a276914.gif">Illustration of initial terms</a>.
%H A276914 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A276914 a(n) = n^2 + 2*A000217(A052928(n)).
%F A276914 a(n) = A000217(A042948(n)).
%F A276914 a(n) = n*(2*n + (-1)^n).
%F A276914 a(n) = n*A168277(n + 1).
%F A276914 a(n) = n*A016813(A004526(n)).
%F A276914 From _Colin Barker_, Sep 23 2016: (Start)
%F A276914 G.f.: x*(1 + 9*x + 3*x^2 + 3*x^3) / ((1 - x)^3*(1 + x)^2).
%F A276914 a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>4.
%F A276914 a(n) = n*(2*n+1) for n even.
%F A276914 a(n) = n*(2*n-1) for n odd. (End)
%F A276914 E.g.f.: x*( 2*(1+x)*exp(x) - exp(-x) ). - _G. C. Greubel_, Aug 19 2022
%F A276914 Sum_{n>=1} 1/a(n) = 2 - log(2). - _Amiram Eldar_, Aug 21 2022
%t A276914 Table[n (2 n + (-1)^n), {n, 0, 48}] (* _Michael De Vlieger_, Sep 23 2016 *)
%o A276914 (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 A276914 (Magma) [n*(2*n+(-1)^n): n in [0..40]]; // _G. C. Greubel_, Aug 19 2022
%o A276914 (SageMath) [n*(2*n+(-1)^n) for n in (0..40)] # _G. C. Greubel_, Aug 19 2022
%Y A276914 Cf. A000217, A004526, A016813, A042948, A052928, A079291, A168277, A275496, A276915.
%K A276914 nonn,easy
%O A276914 0,3
%A A276914 _Daniel Poveda Parrilla_, Sep 22 2016