A292476
Number of solutions to +-1 +- 3 +- 5 +- 7 +- ... +- (4*n-1) = 0.
Original entry on oeis.org
1, 0, 2, 2, 8, 20, 68, 206, 692, 2306, 7930, 27492, 96792, 343670, 1231932, 4447510, 16164914, 59086618, 217091832, 801247614, 2969432270, 11045446688, 41224168020, 154329373022, 579377940390, 2180684278698, 8227240466520, 31107755899600
Offset: 0
For n=2 the 2 solutions are +1-3-5+7 = 0 and -1+3+5-7 = 0.
For n=3 the 2 solutions are +1+3+5-7+9-11 = 0 and -1-3-5+7-9+11 = 0.
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a[n_] := SeriesCoefficient[Product[x^(2k - 1) + 1/x^(2k - 1), {k, 1, 2n}], {x, 0, 0}];
Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Mar 10 2023 *)
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{a(n) = polcoeff(prod(k=1, 2*n, x^(2*k-1)+1/x^(2*k-1)), 0)}
A292497
Number of solutions to 1^2 +- 3^2 +- 5^2 +- 7^2 +- ... +- (4*n-1)^2 = 0.
Original entry on oeis.org
1, 0, 0, 0, 1, 0, 6, 0, 20, 5, 258, 62, 2510, 914, 24285, 16403, 263945, 222240, 3068971, 3241157, 35286928, 46638022, 426740187, 650095127, 5330192371, 9185814630, 67064945191, 129902075662, 864443143567, 1833530501143, 11336065334984, 25990268638322
Offset: 0
A292522
Number of solutions to +- 1^3 +- 3^3 +- 5^3 +- 7^3 +- ... +- (4*n-1)^3 = 0.
Original entry on oeis.org
1, 0, 0, 0, 0, 0, 0, 0, 2, 6, 2, 10, 118, 88, 254, 3308, 2558, 9578, 84568, 121804, 496396, 3312400, 5755724, 19021024, 116780256, 241754350, 883730786, 4923089216, 11668601596, 42357336066, 205859270250, 538878582526, 1974181071852, 9194146886086, 26277093562150
Offset: 0
For n=8 the 2 solutions are
+1^3-3^3-5^3+7^3-9^3+11^3+13^3-15^3-17^3+19^3+21^3-23^3+25^3-27^3-29^3+31^3 = 0 and
-1^3+3^3+5^3-7^3+9^3-11^3-13^3+15^3+17^3-19^3-21^3+23^3-25^3+27^3+29^3-31^3 = 0.
A292550
a(n) = smallest k >= 1 such that {1, 3^n, 5^n, ... , (4*k-1)^n} can be partitioned into two sets with equal sums.
Original entry on oeis.org
1, 2, 4, 8, 10, 14, 19
Offset: 0
n = 0
1^0 = 3^0.
n = 1
1^1 + 7^1 = 3^1 + 5^1.
n = 2
1^2 + 7^2 + 11^2 + 13^2 = 3^2 + 5^2 + 9^2 + 15^2.
n = 3
1^3 + 7^3 + 11^3 + 13^3 + 19^3 + 21^3 + 25^3 + 31^3 = 3^3 + 5^3 + 9^3 + 15^3 + 17^3 + 23^3 + 27^3 + 29^3.
n = 4
1^4 + 5^4 + 13^4 + 17^4 + 19^4 + 25^4 + 27^4 + 29^4 + 31^4 + 39^4 = 3^4 + 7^4 + 9^4 + 11^4 + 15^4 + 21^4 + 23^4 + 33^4 + 35^4 + 37^4.
n = 5
1^5 + 3^5 + 7^5 + 11^5 + 17^5 + 21^5 + 33^5 + 35^5 + 37^5 + 39^5 + 41^5 + 43^5 + 51^5 + 53^5 = 5^5 + 9^5 + 13^5 + 15^5 + 19^5 + 23^5 + 25^5 + 27^5 + 29^5 + 31^5 + 45^5 + 47^5 + 49^5 + 55^5.
Showing 1-4 of 4 results.