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A099137
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Iterated pentagonal numbers starting with 1.
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0
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1, 5, 35, 1820, 4967690, 37016913420305, 2055377818749516111922729385, 6336866966711278058881411525857304983854251425952352645, 60233824430694891039657533544539595428316639098287449541196288430502921814267035
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Because of the identity linking triangular numbers T(n) and pentagonal numbers P(n), namely: P(n)= [T(3n-1)]/3, we may write the Iterated pentagonal numbers in terms of triangular numbers: P(2,n) = (1/6){[T(3n-1)] 2} - [(1/6)T(3n-1)]; P(3,n) = (1/24)[T(3n-1)] 4 - (1/12)[T(3n-1)] 3 - (1/24)[T(3n-1)] 2 - (1/12)T(3n-1); and so forth.
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REFERENCES
| J. V. Post, "Iterated Polygonal Numbers", preprint.
J. V. Post, "Iterated Triangular Numbers", preprint.
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LINKS
| Eric Weisstein's World of Mathematics, "Pentagonal Number."
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FORMULA
| Where P(n) = n*(3*n-1)/2 = the n-th pentagonal number, we define: a(1, n) = P(n) = n*(3*n-1)/2 = (1/2)*[3*(n^2)-n]; a(2, n) = P(P(n)) = (1/2)*{(1/2)*[3*(n^2)-n]}*{(3/2)[3*(n^2)-n] - 1} = (27/8)*(n^4) - (9/4)*(n^3)- (3/2)*(n^2) + (1/4)*n; a(3, n) = P(P(P(n))) = (2187/128)*(n8) - (729/32)*(n^7)- (243/64)*(n^6) + (81/16)*(n^5) - (621/128)*(n^4) + (63/32)*(n^3) + (15/32)*(n^2) - (1/4)*n; P(4, n) = P(P(P(P(n)))) = (14348907/32768)*(n^16) - (4782969/4096)*(n^15)- (1594323/4096)*(n^14) + (4782969/4096)*(n^13) - (621/128)*(n^12) + (295245/1024)*(n^11) + (1161297/5096)*(n^10) - (32805/512)*(n^9) - (32805/1024)*(n^8) + (64881/1024)*(n^7) + (1215/256)*(n^6) - (2025/128)*(n^5) - (405/2048)*(n^4) + (135/256)*(n^3) - (15/32)*(n^2) + (1/16)*n and in general: a(k+1, n) = P(a(k, n), n) = a(k, n)*(3*a(k, n)-1)/2.
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EXAMPLE
| a(3) = 1820 because a(1) = the first pentagonal number = 5, a(2) = the 5th pentagonal number = 35 and a(3) = the 35th pentagonal number = 1820.
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CROSSREFS
| Cf. A007501, A000326.
Sequence in context: A001802 A122590 A057991 * A000871 A035416 A042029
Adjacent sequences: A099134 A099135 A099136 * A099138 A099139 A099140
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KEYWORD
| easy,nonn,uned
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AUTHOR
| Jonathan Vos Post (jvospost3(AT)gmail.com), Nov 14 2004
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