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1, 13, 36, 60, 84, 108, 132, 156, 180, 204, 228, 252, 276, 300, 324, 348, 372, 396, 420, 444, 468, 492, 516, 540, 564, 588, 612, 636, 660, 684, 708, 732, 756, 780, 804, 828, 852, 876, 900, 924, 948, 972, 996, 1020, 1044, 1068, 1092, 1116, 1140, 1164, 1188, 1212, 1236, 1260
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OFFSET
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1,2
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COMMENTS
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For more information, cross-references etc., see A101104.
For n >= 3, a(n) is equal to the number of functions f:{1,2,3,4}->{1,2,...,n} such that Im(f) contains 3 fixed elements. - Aleksandar M. Janjic and Milan Janjic, Mar 08 2007
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LINKS
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FORMULA
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a(n) = 2*a(n-1) - a(n-2), n > 4.
G.f.: x*(1+x)*(1 + 10*x + x^2)/(1-x)^2.
a(n) = 24*n - 36, n >= 3.
a(n) = Sum_{j=0..n} (-1)^j*binomial(3, j)*(n - j)^4. [Indices shifted, Nov 01 2010]
a(n) = Sum_{i=1..4} A008292(4,i)*binomial(n-i+1,1). [Indices shifted, Nov 01 2010]
Sum_{n>=1} (-1)^(n+1)/a(n) = 157/156 - Pi/48. - Amiram Eldar, Jan 26 2022
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MAPLE
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seq(coeff(series(x*(1+x)*(1+10*x+x^2)/(1-x)^2, x, n+1), x, n), n = 1 .. 60); # Muniru A Asiru, Dec 02 2018
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MATHEMATICA
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MagicNKZ=Sum[(-1)^j*Binomial[n+1-z, j]*(k-j+1)^n, {j, 0, k+1}]; Table[MagicNKZ, {n, 4, 4}, {z, 2, 2}, {k, 0, 34}] OR SeriesAtLevelR = Sum[Eulerian[n, i - 1]*Binomial[n + x - i + r, n + r], {i, 1, n}]; Table[SeriesAtLevelR, {n, 4, 4}, {r, -3, -3}, {x, 3, 35}]
Join[{1, 13}, LinearRecurrence[{2, -1}, {36, 60}, 33]] (* Ray Chandler, Sep 23 2015 *)
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PROG
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(PARI) my(x='x+O('x^60)); Vec(x*(1+x)*(1+10*x+x^2)/(1-x)^2) \\ G. C. Greubel, Dec 01 2018
(Magma) I:=[36, 60]; [1, 13] cat [n le 2 select I[n] else 2*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Dec 01 2018
(Sage) s=(x*(1+x)*(1+10*x+x^2)/(1-x)^2).series(x, 30); s.coefficients(x, sparse=False) # G. C. Greubel, Dec 01 2018
(GAP) Concatenation([1, 13], List([3..60], n->24*n-36)); # Muniru A Asiru, Dec 02 2018
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 15 2004
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EXTENSIONS
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Removed redundant information already in A101104. Reduced formulas by expansion of constants - R. J. Mathar, Nov 01 2010
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STATUS
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approved
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