A112355 -id:A112355 - cp's OEIS frontend

A112353 Triangular numbers that are the sum of three distinct positive triangular numbers.

10, 28, 45, 55, 66, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275, 1326, 1378, 1431, 1485

Offset: 1

Rick L. Shepherd, Sep 05 2005

nonn

Subsequence of A112355: it doesn't require the three positive triangular numbers to be distinct.

			45 is a term because 45 = 3 + 6 + 36 and these four numbers are distinct triangular numbers (A000217(9) = A000217(2) + A000217(3) + A000217(8)).

Cf. A000217 (triangular numbers), A112352 (triangular numbers that are the sum of two distinct positive triangular numbers), A112355.

trnos=Accumulate[Range[200]];
Take[Union[Select[Total/@Subsets[trnos,{3}],MemberQ[trnos,#]&]],50]  (* Harvey P. Dale, Jan 15 2011 *)

A350295 2nd subdiagonal of the triangle A350292.

Original entry on oeis.org

6, 8, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275, 1326, 1378, 1431, 1485, 1540

Offset: 3

Stefano Spezia, Dec 23 2021

Harvey P. Dale, Table of n, a(n) for n = 3..1000
Heiko Harborth and Hauke Nienborg, Saturated vertex Turán numbers for cube graphs, Congr. Num. 208 (2011), 183-188.
Mathonline, Cube Graphs
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A112355, A161680, A350292.

Join[{6,8},Table[Binomial[n,2],{n,5,56}]]
LinearRecurrence[{3,-3,1},{6,8,10,15,21},60] (* Harvey P. Dale, Jul 01 2022 *)

a(n) = binomial(n, 2) = A000217(n-1) for n > 4 with a(3) = 6 and a(4) = 8 (see Theorem 3 in Harborth and Nienborg).
O.g.f.: x^3*(2*x^4 - 3*x^3 - 4*x^2 + 10*x - 6)/(x - 1)^3.
E.g.f.: x^2*(x^2 + 6*x + 6*exp(x) - 6)/12.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 7.

cp's OEIS Frontend

A112353 Triangular numbers that are the sum of three distinct positive triangular numbers.

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