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 A101428 Number of ways to write n as an ordered sum of a triangular number (A000217) and a square (A000290). 3
 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 (list; graph; refs; listen; history; text; internal format)
 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 G. C. Greubel, Table of n, a(n) for n = 0..1000 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 *) CROSSREFS Cf. A000925, A115171, A115172, A115173, A115174, A115175, A115176, A115177, A144642. Sequence in context: A140218 A193805 A159704 * A307223 A321787 A331596 Adjacent sequences:  A101425 A101426 A101427 * A101429 A101430 A101431 KEYWORD nonn AUTHOR R. J. Mathar, Jan 03 2009 STATUS approved

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Last modified June 20 10:03 EDT 2021. Contains 345162 sequences. (Running on oeis4.)