A225549 -id:A225549 - cp's OEIS frontend

A221991 a(n) is the number of terms in the expansion of (x-y)(x^2-y^2)(x^3-y^3)(x^5-y^5)...(x^p_i-y^p_i), where p_i is the i-th prime.

2, 4, 6, 8, 10, 20, 22, 36, 42, 66, 90, 110, 142, 184, 232, 284, 342, 400, 458, 532, 604, 678, 756, 838, 928, 1026, 1126, 1230, 1336, 1446, 1558, 1686, 1816, 1954, 2092, 2242, 2392, 2550, 2712, 2880, 3052, 3232, 3412, 3604, 3796, 3994, 4192, 4404, 4626, 4854, 5082

Offset: 1

Robert G. Wilson v, May 12 2013

nonn

In the definition one can take y=1. Thus the sequence becomes the number of terms in the polynomial of the product{k=0..n} (1-x^p_i), where p_i is the i-th prime and p_0 = 1.

Offset is 1 to keep it parallel to other like sequences.

Cf. A000124, A086781, A086795, A086796, A086817, A225549.

f[n_] := Length@ ExpandAll[(1 - x) Product[(1 - x^Prime[k]), {k, n}]]; Array[f, 51, 0]

A222028 a(n) is the number of terms in the expansion of (x-y)(x^3-y^3)(x^6-y^6)(x^10-y^10)...(x^T_i-y^T_i), where T_i is the i-th triangular number.

Original entry on oeis.org

2, 4, 8, 15, 28, 41, 66, 92, 132, 175, 232, 287, 360, 475, 570, 727, 852, 1009, 1220, 1397, 1646, 1891, 2154, 2441, 2772, 3121, 3508, 3891, 4334, 4791, 5282, 5797, 6376, 6983, 7618, 8285, 8984, 9713, 10500, 11319, 12182, 13093, 14028, 15023, 16064, 17157, 18276, 19447, 20680, 21953

Offset: 1

Robert G. Wilson v, May 12 2013

nonn

In the definition one can take y=1. Thus the sequence becomes the number of terms in the polynomial of the product{k=0..n} (1-x^T_i), where G_i is the i-th triangular number.

Cf. A000124, A086781, A086795, A086796, A086817, A221991, A225549.

f[n_] := Length@ ExpandAll@ Product[1 - x^(k (k + 1)/2), {k, n}]; Array[f, 50]

cp's OEIS Frontend

A221991 a(n) is the number of terms in the expansion of (x-y)(x^2-y^2)(x^3-y^3)(x^5-y^5)...(x^p_i-y^p_i), where p_i is the i-th prime.

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A222028 a(n) is the number of terms in the expansion of (x-y)(x^3-y^3)(x^6-y^6)(x^10-y^10)...(x^T_i-y^T_i), where T_i is the i-th triangular number.

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cp's OEIS Frontend

A221991 a(n) is the number of terms in the expansion of (x-y)(x^2-y^2)*(x^3-y^3)*(x^5-y^5)*...*(x^p_i-y^p_i), where p_i is the i-th prime.

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A222028 a(n) is the number of terms in the expansion of (x-y)(x^3-y^3)*(x^6-y^6)*(x^10-y^10)*...*(x^T_i-y^T_i), where T_i is the i-th triangular number.

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A221991 a(n) is the number of terms in the expansion of (x-y)(x^2-y^2)(x^3-y^3)(x^5-y^5)...(x^p_i-y^p_i), where p_i is the i-th prime.

A222028 a(n) is the number of terms in the expansion of (x-y)(x^3-y^3)(x^6-y^6)(x^10-y^10)...(x^T_i-y^T_i), where T_i is the i-th triangular number.