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A276914
Subsequence of triangular numbers obtained by adding a square and two smaller triangles, a(n) = n^2 + 2*A000217(A052928(n)).
3
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
0,3
COMMENTS
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.
LINKS
FORMULA
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
MATHEMATICA
Table[n (2 n + (-1)^n), {n, 0, 48}] (* Michael De Vlieger, Sep 23 2016 *)
PROG
(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
(Magma) [n*(2*n+(-1)^n): n in [0..40]]; // G. C. Greubel, Aug 19 2022
(SageMath) [n*(2*n+(-1)^n) for n in (0..40)] # G. C. Greubel, Aug 19 2022
KEYWORD
nonn,easy
AUTHOR
STATUS
approved