A101428 Number of ways to write n as an ordered sum of a triangular number (A000217) and a square (A000290).

1, 2, 1, 1, 2, 1, 1, 2, 0, 1, 3, 1, 1, 0, 1, 2, 2, 1, 0, 3, 0, 1, 2, 0, 1, 2, 2, 0, 2, 1, 1, 2, 1, 0, 0, 1, 2, 4, 0, 1, 2, 0, 1, 0, 1, 2, 3, 0, 0, 2, 1, 1, 2, 1, 1, 2, 1, 1, 0, 2, 0, 2, 0, 0, 4, 1, 1, 2, 0, 0, 4, 1, 1, 0, 1, 1, 0, 1, 1, 2, 1, 2, 3, 0, 1, 2, 0, 2, 0, 0, 0, 4, 2, 0, 2, 1, 1, 0, 0, 0, 3, 1, 2, 2, 1

Offset: 0

Author

R. J. Mathar, Jan 03 2009

Keywords

• nonn
• changed

Comments

0 is both a triangular number and a square number.

First occurrence of k beginning at 0: 8, 2, 1, 10, 37, 136, 235, 1549, 631, 2314, 2116, 11026, 3997, 148240, 19045, 20827, 25876, 893116, 67951, ?19?, 35974, 187444, 1542655, 354061, 131905, ?25?, ?26?, 835399, 323767, ?29?, 611560, ?31?, 515629, ?33?, ?34?, ?35?, 1187146, ?37?, ?38?, ?39?, 1474939, ..., . - Robert G. Wilson v, Mar 30 2014

Variant of A082660 (which allows only positive triangular numbers). - R. J. Mathar, Apr 28 2020

a(n) is the number of representations of 8*n + 1 as 2*A^2 + B^2 with A even and B odd and both nonnegative integers. - Vladimir Pletser, Aug 30 2025

Examples

			Examples: n=1 gives the a(1)=2 cases 1=1+0=0+1; a(26)=2 because 26=25+1=16+10.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000925, A115171, A115172, A115173, A115174, A115175, A115176, A115177, A144642.

Programs

Maple

A000217 := proc(n) n*(n+1)/2 ; end:
A101428 := proc(n)
local a,y,t ;
a := 0 ;
for y from 0 do
t := A000217(y) ;
if n-t < 0 then
RETURN(a) ;
else
if issqr(n-t) then
a := a+1 ;
fi;
fi;
od:
end:
for n from 0 to 100 do printf("%a,",A101428(n)) ; od:

Mathematica

t = FoldList[#1 + #2 &, 0, Range@ 15]; s = Range[0, 10]^2, a = Sort@ Flatten@ Table[ s[[j]] + t[[k]], {j, 15}, {k, 11}]; Table[Count[a, n], {n, 0, 104}] (* or *)
triQ[n_] := IntegerQ@ Sqrt[8n + 1]; f[n_] := Block[{c = k = 0, lmt = 2 + Floor[Sqrt[n]]}, While[k < lmt, If[ triQ[n - k^2], c++]; k++]; c]; Array[f, 105, 0] (* Robert G. Wilson v, Mar 30 2014 *)

Formula

G.f.: sum(i>=0, x^(i^2) ) * sum(i>=0, x^(i*(i+1)/2) ). - Ralf Stephan, May 17 2014