The OEIS is supported by the many generous donors to the OEIS Foundation.

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 60th year, we have over 367,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A306471 Number of ways to write n as C(2w+1,2) + C(x+2,3) + C(y+3,4) + C(z+4,5) with w,x,y,z nonnegative integers, where C(n,k) denotes the binomial coefficient n!/(k!*(n-k)!). 6
 1, 3, 3, 2, 4, 6, 5, 4, 4, 5, 7, 8, 6, 4, 5, 8, 8, 5, 4, 6, 7, 10, 10, 6, 6, 12, 13, 8, 7, 7, 6, 11, 9, 4, 3, 8, 16, 12, 8, 9, 9, 13, 14, 10, 7, 9, 18, 12, 6, 5, 4, 11, 10, 4, 2, 5, 19, 21, 11, 9, 13, 20, 16, 9, 6, 8, 17, 17, 4, 2, 9, 20, 17, 6, 9, 9, 15, 23, 14, 9, 15 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Conjecture 1: a(n) > 1 for all n > 0. We have verified a(n) > 0 for all n = 0..5*10^6. Conjecture 2: For each r = 0, 1, any positive integer can be written as w^2 + C(x,3) + C(y,4) + C(z,5), where w,x,y,z are nonnegative integers with w - r even. See also A306462 and A306477 for similar conjectures. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 0..10000 EXAMPLE a(0) = 1 with 0 = C(1,2) + C(2,3) + C(3,4) + C(4,5). a(3) = 2 with 3 = C(3,2) + C(2,3) + C(3,4) + C(4,5) = C(1,2) + C(3,3) + C(4,4) + C(5,5). a(54) = 2 with 54 = C(3,2) + C(7,3) + C(6,4) + C(5,5) = C(3,2) + C(5,3) + C(7,4) + C(6,5). a(69) = 1 with 69 = C(3,2) + C(5,3) + C(7,4) + C(7,5) = C(3,2) + C(5,3) + C(3,4) + C(8,5). MATHEMATICA f[m_, n_]:=f[m, n]=Binomial[m+n-1, m]; HQ[n_]:=HQ[n]=IntegerQ[Sqrt[8n+1]]&&Mod[Sqrt[8n+1], 4]==1; tab={}; Do[r=0; Do[If[f[5, z]>n, Goto[cc]]; Do[If[f[4, y]>n-f[5, z], Goto[bb]]; Do[If[f[3, x]>n-f[5, z]-f[4, y], Goto[aa]]; If[HQ[n-f[5, z]-f[4, y]-f[3, x]], r=r+1], {x, 0, n-f[5, z]-f[4, y]}]; Label[aa], {y, 0, n-f[5, z]}]; Label[bb], {z, 0, n}]; Label[cc]; tab=Append[tab, r], {n, 0, 80}]; Print[tab] CROSSREFS Cf. A000217, A000290, A000292, A000332, A000389, A000797, A014105, A262813, A306460, A306462, A306477. Sequence in context: A271510 A349661 A282545 * A354698 A050610 A151848 Adjacent sequences: A306468 A306469 A306470 * A306472 A306473 A306474 KEYWORD nonn AUTHOR Zhi-Wei Sun, Feb 17 2019 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 7 01:11 EST 2023. Contains 367616 sequences. (Running on oeis4.)