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A074377
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Generalized 10-gonal numbers: k*(4*k-3) for k = 0, +- 1, +- 2, +- 3,...
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64
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0, 1, 7, 10, 22, 27, 45, 52, 76, 85, 115, 126, 162, 175, 217, 232, 280, 297, 351, 370, 430, 451, 517, 540, 612, 637, 715, 742, 826, 855, 945, 976, 1072, 1105, 1207, 1242, 1350, 1387, 1501, 1540, 1660, 1701, 1827, 1870, 2002, 2047, 2185, 2232, 2376, 2425
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OFFSET
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0,3
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COMMENTS
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Also called generalized decagonal numbers.
Odd triangular numbers decremented and halved.
It appears that this is zero together with the partial sums of A165998. - Omar E. Pol, Sep 10 2011 [this is correct, see the g.f., Joerg Arndt, Sep 29 2013]
Also, A033954 and positive members of A001107 interleaved. - Omar E. Pol, Aug 04 2012
Also, numbers m such that 16*m+9 is a square. After 1, therefore, there are no squares in this sequence. - Bruno Berselli, Jan 07 2016
Convolution of the sequences A047522 and A059841. - Ilya Gutkovskiy, Mar 16 2017
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Neville Holmes, More Gemometric Integer Sequences
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).
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FORMULA
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(n(n+1)-2)/4 where n(n+1)/2 is odd.
G.f.: x(1+6x+x^2)/((1-x)(1-x^2)^2). - Michael Somos, Mar 04 2003
a(2*k) = k*(4*k+3); a(2*k+1) = (2*k+1)^2+k. [Benoit Jubin, Feb 05 2009]
a(n) = n^2+n-1/4+(-1)^n/4+n*(-1)^n/2. - R. J. Mathar, Oct 08 2011
Sum_{n>=1} 1/a(n) = (4 + 3*Pi)/9. - Vaclav Kotesovec, Oct 05 2016
E.g.f.: exp(x)*x^2 + (2*exp(x) - exp(-x)/2)*x - sinh(x)/2. - Ilya Gutkovskiy, Mar 16 2017
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MATHEMATICA
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f[n_]:=n(n+3)/4; Select[Table[f[n], {n, 0, 6!}], IntegerQ] (* Vladimir Joseph Stephan Orlovsky, Apr 29 2010 *)
CoefficientList[Series[ x (1 + 6 x + x^2)/((1 - x) (1 - x^2)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Sep 29 2013 *)
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PROG
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(PARI) a(n)=(2*n+3-4*(n%2))*(n-n\2)
(PARI) concat([0], Vec(x*(1 + 6*x + x^2)/((1 - x)*(1 - x^2)^2) +O(x^50))) \\ Indranil Ghosh, Mar 16 2017
(MAGMA) [n^2+n-1/4+(-1)^n/4+n*(-1)^n/2: n in [0..50]]; // Vincenzo Librandi, Sep 29 2013
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CROSSREFS
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Cf. A011848, A014493, A074378, A118277.
Cf. A001107 (10-gonal numbers).
Column 6 of A195152.
Generalized k-gonal numbers, k>=5: A001318, A000217, A085787, A001082, A118277, this sequence, A195160, A195162, A195313, A195818.
Cf. sequences of the form m*(m+k)/(k+1) listed in A274978. [Bruno Berselli, Jul 25 2016]
Sequence in context: A300021 A097634 A120312 * A103119 A054224 A183330
Adjacent sequences: A074374 A074375 A074376 * A074378 A074379 A074380
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KEYWORD
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nonn,easy
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AUTHOR
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W. Neville Holmes, Sep 04 2002
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EXTENSIONS
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New name from T. D. Noe, Apr 21 2006
Formula in sequence name from Omar E. Pol, May 28 2012
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STATUS
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approved
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