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A101428
Number of ways to write n as an ordered sum of a triangular number (A000217) and a square (A000290).
4
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
0,2
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
LINKS
FORMULA
G.f.: sum(i>=0, x^(i^2) ) * sum(i>=0, x^(i*(i+1)/2) ). - Ralf Stephan, May 17 2014
EXAMPLE
Examples: n=1 gives the a(1)=2 cases 1=1+0=0+1; a(26)=2 because 26=25+1=16+10.
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 *)
KEYWORD
nonn
AUTHOR
R. J. Mathar, Jan 03 2009
STATUS
approved