A074377 Generalized 10-gonal numbers: m*(4*m - 3) for m = 0, +- 1, +- 2, +- 3, ...

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

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Neville Holmes, More Gemometric Integer Sequences. [Wayback Machine copy]

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

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. A011848, A014493, A074378, A118277, A165998.

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

Adjacent sequences: A074374 A074375 A074376 * A074378 A074379 A074380

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