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A254407
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a(n) = n*(n+1)*(11*n +10)/6.
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7
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0, 7, 32, 86, 180, 325, 532, 812, 1176, 1635, 2200, 2882, 3692, 4641, 5740, 7000, 8432, 10047, 11856, 13870, 16100, 18557, 21252, 24196, 27400, 30875, 34632, 38682, 43036, 47705, 52700, 58032, 63712, 69751, 76160, 82950, 90132, 97717, 105716, 114140, 123000
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OFFSET
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0,2
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COMMENTS
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Similar sequences of the type m*P(s,m) - Sum_{i=1..m} P(s-1,i), where P(s,m) is the m-th s-gonal number:
s=3: A027480(n) = (n+1)*A000217(n+1) - Sum_{i=1..n+1} i;
s=4: A162148(n) = (n+1)*A000290(n+1) - Sum_{i=1..n+1} A000217(i);
s=5: A245301(n) = (n+1)*A000326(n+1) - Sum_{i=1..n+1} A000290(i);
s=6: A085788(n) = (n+1)*A000384(n+1) - Sum_{i=1..n+1} A000326(i);
s=7: a(n) = (n+1)*A000566(n+1) - Sum_{i=1..n+1} A000384(i).
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Wikipedia, Polygonal numbers: Table of values.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
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FORMULA
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G.f.: x*(7 + 4*x)/(1 - x)^4.
a(-n) = -A132112(n-1).
a(n) = Sum_{k=0..n} A011875(11*k+2).
Equivalently, partial sums of A254963.
E.g.f.: x*(42 + 54*x + 11*x^2)*exp(x)/6. - G. C. Greubel, Mar 31 2021
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EXAMPLE
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532 is the 7th term because A000566(7)=112 and Sum_{i=1..7} A000384(i)=252, therefore 7*112-252 = 532.
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MAPLE
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A254407:= n-> n*(n+1)*(11*n+10)/6; seq(A254407(n), n=0..50); # G. C. Greubel, Mar 31 2021
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MATHEMATICA
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Table[n (n + 1) (11 n + 10)/6, {n, 0, 40}]
Column[CoefficientList[Series[x (7 + 4 x) / (1 - x)^4, {x, 0, 60}], x]] (* Vincenzo Librandi, Jan 31 2015 *)
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PROG
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(PARI) vector(40, n, n--; n*(n+1)*(11*n+10)/6)
(Sage) [n*(n+1)*(11*n+10)/6 for n in (0..40)]
(Magma) [n*(n+1)*(11*n+10)/6: n in [0..40]];
(Maxima) makelist(n*(n+1)*(11*n+10)/6, n, 0, 40);
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CROSSREFS
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Cf. A011875, A027480, A057145, A085788, A132112, A162148, A245301, A254963.
Sequence in context: A067982 A126562 A190096 * A219510 A164270 A182820
Adjacent sequences: A254404 A254405 A254406 * A254408 A254409 A254410
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KEYWORD
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nonn,easy
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AUTHOR
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Bruno Berselli, Jan 30 2015
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STATUS
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approved
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