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A074377 Generalized 10-gonal numbers: m*(4*m - 3) for m = 0, +- 1, +- 2, +- 3, ... 80
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
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
Numbers k such that the concatenation k5625 is a square. - Bruno Berselli, Nov 07 2018
Exponents in expansion of Product_{n >= 1} (1 + x^(8*n-7))*(1 + x^(8*n-1))*(1 - x^(8*n)) = 1 + x + x^7 + x^10 + x^22 + .... - Peter Bala, Dec 10 2020
LINKS
Neville Holmes, More Gemometric Integer Sequences. [Wayback Machine copy]
FORMULA
(n(n+1)-2)/4 where n(n+1)/2 is odd.
G.f.: x*(1+6*x+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
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2) - 4/9. - Amiram Eldar, Feb 28 2022
MATHEMATICA
CoefficientList[Series[x(1 +6x +x^2)/((1-x)(1-x^2)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Sep 29 2013 *)
LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 7, 10, 22}, 50] (* G. C. Greubel, Nov 07 2018 *)
PROG
(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
CROSSREFS
Cf. A001107 (10-gonal numbers).
Column 6 of A195152.
Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), A085787 (k=7), A001082 (k=8), A118277 (k=9), this sequence (k=10), A195160 (k=11), A195162 (k=12), A195313 (k=13), A195818 (k=14), A277082 (k=15), A274978 (k=16), A303305 (k=17), A274979 (k=18), A303813 (k=19), A218864 (k=20), A303298 (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30).
Cf. sequences of the form m*(m+k)/(k+1) listed in A274978. [Bruno Berselli, Jul 25 2016]
Sequence in context: A327977 A097634 A120312 * A103119 A054224 A183330
KEYWORD
nonn,easy
AUTHOR
W. Neville Holmes, Sep 04 2002
EXTENSIONS
New name from T. D. Noe, Apr 21 2006
Formula in sequence name from Omar E. Pol, May 28 2012
STATUS
approved

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Last modified June 21 06:38 EDT 2024. Contains 373540 sequences. (Running on oeis4.)