A276914 Subsequence of triangular numbers obtained by adding a square and two smaller triangles, a(n) = n^2 + 2*A000217(A052928(n)).

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

Daniel Poveda Parrilla, Table of n, a(n) for n = 0..10000

Daniel Poveda Parrilla, Illustration of initial terms.

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

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

Crossrefs

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

Sequence in context: A037379 A055049 A128703 * A078818 A194267 A372044

Adjacent sequences: A276911 A276912 A276913 * A276915 A276916 A276917

Keyword

nonn,easy

Author

Daniel Poveda Parrilla, Sep 22 2016

Status

approved