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

Author

Daniel Poveda Parrilla, Sep 22 2016

Keywords

• nonn
• easy

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).

Crossrefs

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

Programs

Magma

[n*(2*n+(-1)^n): n in [0..40]]; // G. C. Greubel, Aug 19 2022

Mathematica

Table[n (2 n + (-1)^n), {n, 0, 48}] (* Michael De Vlieger, Sep 23 2016 *)

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

SageMath

[n*(2*n+(-1)^n) for n in (0..40)] # G. C. Greubel, Aug 19 2022

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