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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 (list; graph; refs; listen; history; text; internal format)
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)

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

CROSSREFS

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

Sequence in context: A037379 A055049 A128703 * A078818 A194267 A261853

Adjacent sequences:  A276911 A276912 A276913 * A276915 A276916 A276917

KEYWORD

nonn,easy

AUTHOR

Daniel Poveda Parrilla, Sep 22 2016

STATUS

approved

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Last modified August 21 06:54 EDT 2019. Contains 326162 sequences. (Running on oeis4.)